Aenorm 61 by VSAE

Edition 61 Volume 16 Oct - 2008

The Livestock Collection Problem a real-world vehicle routing and inventory problem

Arnoud Boot about What is Happening to the Economy Switching between prediction rules by evolutionary selection: Can the stylized facts be explained? Operational risk according to Solvency II

Ageing, Schooling and Economic Growth: Standing on the Shoulders of Giants Testing for a unit root in time series with changing volatility An empirical model for the time allocation of parents

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Preface

Capatalism can be cruel 1

00% Pure and Natural” it said at the back of my orange juice. Apparently, being pure and natural is beneficial and can be used as a recommendation. However, in my opinion, nature isn’t good. Mushrooms exist that can kill a six feet man in a matter of hours. If you have just been bitten by a snake and are suffering excruciating pains you won’t think nature is good; it is cruel. It is for example a common misconception that predators only hunt the ill and the old. A fox will go after the slowest rabbit, not necessarily the ill one. As a rabbit, you do not have to be able to outrun the fox, you just have to run faster than your brothers and sisters. As I said before; nature is cruel. As long as resources are widely available nature will appear to be as peaceful as a mass at eastern. But when drought strikes and food is scarce, organisms have to compete to survive and the cruelty of nature becomes clear. In this respect a comparison can be made with the economy. When markets are booming, firms flourish. Investors battle to achieve the highest returns and risk is not a subject of discussion. However, you cannot have missed that the past period will not go down in history as the most profitable investor area. In this time of economic chaos, survival of the fittest applies to firms: only the companies that are best adapted to today’s circumstances will survive. Others go bankrupt or are taken over by their competitors. Capitalism is a system that has worked perfectly fine for the last 250 years (at least in the Western world). The US propaganda of the Cold War and the failure of the Russian communisms have convinced us that capitalism as a system is good. However, now that firms go down, people lose their jobs and cannot pay their mortgages any more. Still CEO´s and CFO´s receive bonuses equal to the GDP of small countries. The unequal distribution of wealth is a direct consequence of capitalism. When the greed of top-managers can cause families to be kicked out of their homes, the system does not feel right. Perhaps capitalism is not good. In a way it is cruel as well. As comes to Aenorm, we are trying hard to become better known internationally. In order to achieve this, we have visited international conferences in both Marseille and Maastricht. Here we have met a number of professors, PhD’s and other academics, some of which have or will contribute to Aenorm. Most members of the editorial staff are still in their bachelor phase. Therefore presentations like “Cross-Sectional Dependence Robust Block Bootstrap Panel Unit Root Tests”, although very interesting, have way too much adjectives to be comprehensible for us. However, I enjoyed visiting the congresses very much. The atmosphere combined with the kindness of both organization and participants is enough to make you look forward to the next one! Lennart Dek

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Aenorm 61

Contest List

Interview Arnoud Boot

Arnoud Boot is professor of corporate finance and financial markets at the University of Amsterdam. He is considered to be an authority in his field and has recently published many articles in Dutch newspapers about the Credit Crunch. Aenorm interviewed him about the current events. Lennart Dek and Raymon Badloe

Operational Risk According to Solvency II: SCR, Pillar 2 and Copula’s 9 The focus of this article is the quantification of operational risk according to the Solvency II requirements. These requirements are based on the QIS4 Technical Specifications by CEIOPS and the Operational Risk Questionnaire QIS4. As we read through these documents, we see that one of the major issues is the adequate treatment of the interdependencies between operational risk events. Marco Folmpers

Cover design: Michael Groen Aenorm has a circulation of 1950 copies for all students Actuarial Sciences and Econometrics & Operations Research at the University of Amsterdam and for all students in Econometrics at the Free University of Amsterdam. Aenorm is also distributed among all alumni of the VSAE. Aenorm is a joint publication of VSAE and Kraket. A free subsciption can be obtained at www.aenorm.eu. Insertion of an article does not mean that the opinion of the board of the VSAE, the board of Kraket or the editorial staff is verbalized. Nothing from this magazine can be duplicated without permission of VSAE or Kraket. No rights can be taken from the content of this magazine. © 2008 VSAE/Kraket

Testing for a Unit Root in Time Series with Changing Volatility 14 Since the 1970s it has become standard econometric practice to consider the possibility that economic and financial time series are the outcomes of nonstationary processes. The type of non-stationarity that has been investigated most extensively is that of integrated processes. This article reviews some recent results in the field of testing for a unit root in a changing volatility environment. Peter Boswijk

On the distribution of Traveling Salesman Problem Solutions among the Ranked Linear Assignment Alternatives 20 The Linear Assignment Problem is considered one of the fundamental combinatorial optimization problems in operations research. A problem closely related is called the Traveling Salesman Problem. In this article the following problem will be addressed: ‘How do the TSP solutions distribute among the LAP solutions when these are monotonously ordered in cost?’ Jannetje Veerman

On Density Forecast Evaluation

Traditionally, probability integral transforms (PITs) have been popular means for evaluating density forecasts. For an ideal density forecast, the PITs should be uniformly distributed on the unit interval and independent. However, this is only a necessary condition, and not a sufficient one, as shown by some simple examples. In this article, an alternative approach to density forecast evaluation is discussed, via the Kullback-Leibler information criterion (KLIC), and illustrate it with a small simulation study. Cees Diks

The Livestock Collection Problem - a Real-world Vehicle Routing and Inventory Problem 31 This article presents a description and two approaches to solve the Livestock Collection Problem (LCP), which is a real and complex routing and inventory problem. The LCP is taken from the Norwegian meat industry and consists of constructing a route plan for transporting animals from farms to a slaughterhouse. The transportation plan must provide the slaughterhouse with animals according to the production plan, and the capacity for keeping live animals at the slaughterhouse must not be exceeded. The combination of rules and regulations concerning animal welfare, together with constraints regarding production and inventory, makes the problem difficult to solve Johan Oppen

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An Empirical Model for the Time Allocation of Parents

Conventional models of labour supply describe how many hours individuals choose to work. In empirical research, the effects of individual wage rates and the household’s non-labour income are estimated. Typically, use is made of a microdata set, containing information about working hours, wage rates and incomes of a sample of individuals or households. The Nobel Prize winner Becker (1965) introduced a theory of time allocation that introduced the concept of household production. We estimate an empirical model for time allocation. Hans Bloemen

Volume 16 Edition 61 November 2008 ISSN 1568-2188 Chief editor: Lennart Dek Editorial Board: Lennart Dek

A Brief Reflection on Automatic Econometric Model Selection in a Fuel Price Elasticity Context. 38

Design: Carmen Cebrián

Automatrics is a program that automatically selects a model from data obtained through research. The idea that it is possible to insert data into a program and, with the simple click of a button,this program reports the best model obtainable, is intruiging. A problem with this approach is that the danger of data mining is always present. In this article the model and estimates based on extensive literature are compared to the estimates and model presented by Autometrics. Robert Schipperhein

Lay-out: Taek Bijman

Switching Between Prediction Rules by Evolutionary Selection: Can the Stylised Facts Be Explained? 44 In economics expectation formation and learning are of key importance to explaining the behaviour of consumers, households and producers. The methodology often employed for modelling and explaining this behaviour is based on the assumption of rational expectations. By means of a model with evolutionary switching between price prediction rules and heterogeneously behaving economic agents that were boundedly rational this article seeks to match the three stylized facts. Raymon Badloe

Estimation and Inference in Unstable Nonlinear Least Squares Models 50 Human skills, along with capital and technology, have rapidly progressed over the last two centuries. While this progress may be viewed as a continuous innovation, it is necessarily the case that from time to time, some major innovation occurs, and this innovation triggers a major change in the behavior of individuals and firms. In this article, three different statistical methods are proposed to test whether (multiple) breaks occur in the nonlinear behavior of the average individual. Otilia Boldea

A Generic Scheduling Method

Scheduling problems are known to be NP-hard; non-polynomial solvable. Though automatic scheduling could save much time, there still is no polynomial algorithm to solve a scheduling problem. Intensively studying a scheduling problem is very costly with regards to both the required time and specialists. This is the reason that a desire exists to develop a generic scheduling method. Petra Veerman

Ageing, Schooling and Economic Growth: Standing on the Shoulders of Giants 58 In this article the effects on the macroeconomic performance of open economy of demographic shocks of the type and magnitude the Netherlands is investigated. A simple analytical growth model which the schooling decision depends on the knowledge of their higher educated population results in higher economic growth.

a typical small we observed in is developed in teachers and a

Editorial staff: Raymon Badloe Erik Beckers Daniëlla Brals Lennart Dek Bas Meijers Jacco Meure Advertisers: Accenture Achmea All Options AON Delta Lloyd De Nederlandsche Bank IMC KPMG Mercer Michael Page PricewaterhousCoopers SNS Reaal TNO Towers Perrin Watson Wyatt Worldwide Information about advertising can be obtained from Tom Ruijter info@vsae.nl Editorial staff adresses: VSAE, Roetersstraat 11, 1018 WB Amsterdam, tel: 020-5254134 Kraket, de Boelelaan 1105, 1081 HV Amsterdam, tel: 020-5986015 www.aenorm.eu

Ward Romp

Puzzle 63 Facultative 64

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Interview

Interview with Arnoud Boot about What is Happening to the Economy

Arnoud Boot (1960) is professor of corporate finance and financial markets at the University of Amsterdam and director of the Amsterdam Center for Law & Economics (ACLE). He obtained his master in both business administration and marco-economics at the Tilburg University in 1983 and his PhD in Finance at the Indiana University in 1987. In 1998 he was appointed Vice Dean of the Faculty of Economics and Econometrics at University of Amsterdam, a position he held until 2000. Boot was during 2000 and 2001 partner in the consultancy firm McKinsey & Co. His daughter will start her study in econometrics next year, combining both the economics of her father and her personal favorite mathematics.

The subprime mortgage crisis already had its beginning in 2005 with the burst of the housing bubble in the United States. The housing market in the Netherlands has not severely suffered until now. Do you think the IMF was right in claiming the prices on the Dutch housing market are also blown out of proportion (April 2008) and the market could collapse just like it did in the United States and the United Kingdom? The IMF hit a raw nerve, given the emotional and the very strong and negative reaction of Dutch policy makers. The IMF study of the Dutch housing market sketched an image worse than the actual situation, since it forgot a number of aspects like the “Hypotheekrente aftrek”, i.e. the deductibility of the interest on mortgages, which make houses more affordable. But obviously the IMF is correct. A collapse of the Dutch housing market is potentially the biggest threat to the Dutch economy. The scarcity of housing is often used as an argument against this claim, since this would keep prices at the current high level. But the housing was already scarce ten years ago and the prices have increased enormously since then. If you believe in efficient markets, that scarcity was already priced in in the middle of the nineties. There are all kinds of economic indications that problems could occur in the housing market, as for example the increasing duration of houses standing on offer before being sold. In my opi-

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nion the Dutch housing market will cool down. We need to make sure this follows an orderly process. The cooling down of the housing market will have two negative effects. One is that it directly influences the wealth of house owners. The second is that it negatively affects financial institutions. Even in the Netherlands where we have granted mortgages in a rather prudent way, a downfall of house prices will have its effects on financial institutions. But this goes both ways: the problems of the financial institutions could spell disaster for the housing market as well. These two developments will strengthen each other. But one thing is certain; when the real estate market cools down, every financial institution will suffer. Obviously one of the most important causes of the crisis has been the lowered lending standards in the US making it for most people relatively easy to obtain credit to finance their mortgage. Have the supervisors of the financial markets in the United States failed by neglecting the consequences of this ‘very risky’ manner of lending? The supervisors have made mistakes. In particular the United States has a very complicated supervisory system, making it unclear who exactly is in charge. A nice analogy can be made with the 1920s. You think about the 1920s as the time of the depression, but it was a period of euphoria. The rise of technology and incre-

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Interview

ased use of electricity resulted in enormous economic growth in the aftermath of the First World War. This is similar to the 1990s, where developments in information technology also gave rise to overinvestments. This initially resulted in the .com bubble of around 2000.The banking system actually handled this collapse quite well and we did not face problems like those in the 1930s. This has put supervisors asleep. They thought the system had gained strength because it had become so much interrelated, which would spread the risks. Now we know the opposite is true. The interrelations in the financial world increase the risks, since the problems at one end of the financial system become the problems of all banks. Another problem is the macro-economic situation. Enormous amounts of money flow in and out the United States. This, primerily caused by the enormous US trade balance deficit, makes the financial sector too important. Rather than being facilitating, what you would expect from the financial sector, it now has a live on his own. Do you think the credit crunch will cause a culture shift in the United States as it comes to the roll of the financial supervisors? What you see is that every time we are faced with a financial crisis, the supervisors respond with very strong legislation. That will happen now too. Whether this legislation is any good, you never know. You only know that the legislature will get into action. In the 1930s commercial banks were separated from investment banks with the Glass-Steagell act of 1933. The strange thing is that the opposite is done now; the investment banks have been looking for cover to get access to deposits. The last thing is probably not the right thing to do now and the paradox is that in the 1930s the separation was not addressing the problems at that time. A lot of research has been conducted to the crisis of the 1930s and it has been shown that the separation was not the solution to the problem faced at the time. You will always have strange things going on when politicians get into action. Whether this changes things for the better, we do not know. The government of the US responded to the 2001 crisis, when the stock markets collapsed and all kinds of frauds like Ahold and Enron were revealed, with the Sarbanes-Oxley act (SOx) of 2002. This act resolved the conflicts of interests of accountants and held management responsible for all that went on in the company. Both CFOs and CEOs can now be criminally charged if something goes wrong with the internal contacts and cannot longer claim they did not know what was going on.

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However, one party was left out of the SOx; the credit rating agencies. The SOx had a chapter that said all kinds of problems existed with credit rating agencies, but governmental investigation was necessary before legislation could be introduced. These problems were the same then as they are today. For example ratings are conflicted. Objectivity may not exist because the agencies are paid by the parties they are rating. And in today’s crisis they were paid to give AAA ratings to all this subprime stuff. The US government investigation agency GAO carried out the investigation of the credit rating agencies in 2002. They came with a final report in 2003. This report was absolutely devastating for the agencies, which leads you to expect the US legislatures would come into action. But this was one year after the crisis and the frauds. Lobbyists were back at their seats and after a year of lobbying the US government failed to make any steps. The urgency of the crisis was gone, the Congress was bought and nothing happened. Now steps will be taken against the credit rating agencies, but this is after they caused or at least made worse both the crisis of 2001 and today’s crisis. The problem with the legislation in the US is that is too political. If you want changes you need a crisis to begin with. After the crisis has occurred, you have to move fast. If you take your time to think carefully about new laws, which is a valid point of view, the urgency for change is gone, lobbyists have secured the position of big companies and nothing will happen. It seems that the credit crunch strikes hardest in the United States of America. However, most European indices suffered more than both the Dow Jones and Nasdaq in the past year. Is there an explanation for this paradox? It is true that the American markets appear to have held better than the European ones. However, I am not sure whether this is a paradox. When comparing indices you have to account for exchange rate effects. Over the last year the dollar has weakened. So translated in euros the American indices have lost more than they appear to have lost in dollar terms. When accounting for the exchange rate effect, the loss of American market is almost identical to those of the European markets. Another aspect which has to be accounted for while comparing indices is their composition. For example the index of the Netherlands, the AEX, is heavily dominated by financials and Shell. This means the AEX will behave different than the complete broadband of stocks. While trying to solve the credit crunch, governments have to deal with the ‘moral hazard’; when helping banks with their li-

Interview

quidity problems; the banks are rewarded for bad behaviour, like speculation. The Fed allowing the bankruptcy of Lehman to happen seemed an attempt to deal with this problem. However, Lehman Brother was relatively only a minor player. The Fed did not dare taking the risk of a possible System Crisis by not helping one of the top four. Do these actions of the Fed signal that the top four banks may speculate and take irresponsible risks, but smaller banks cannot? I think that this is indeed a very big problem. You do not even have to look at the US, in the Netherlands the problem occurs as well. Minister Bos of Finance has said that systemic relevant banks will always be saved. If you are a small bank, you are probably not systemic relevant, so you better do not involve in risky affairs. Apparently, being big gives you an artificial advantage and actually may have perverse effects. A big bank can take risks and when those turn out good, they may keep the profit.

taking control can be dangerous because banks are lead by managers that may no longer be trusted by the financial markets. Markets have lost confidence in them. The combination of providing capital and taking control can restore the confidence in financial institutions and eventually resolve the crisis. In June you proposed three structural changes to the banking world to prevent a repetition of the credit crunch (NRC Handelsblad, 10 June). Since June the credit crunch has been on its worst yet. Did the events of the past months alter your opinion? Do you now believe that the credit crunch reveals the shortcomings of the capitalism and the entire system should be reviewed like president Sarkozy of France proposed earlier this year? Or does a derailment of the system causes the current crisis, but can we expect the system to work perfectly fine again in the future if we make some adjustments?

"Todays crisis is more than just a temporary backlash against capitalism" But when investments turn out to be bad, the losses are socialized. The American Congress accepted a proposal to give an injection of 700 billion dollars into the American economy. The European governments are also discussing plans to â&#x20AC;&#x2DC;saveâ&#x20AC;&#x2122; the European economy. Is it enough to just inject enormous amounts of money into the financial markets to solve the problems now that the confidence of investors and consumers has shrunk? The US was already very active before this plan. Liquidity injections of central banks have been very aggressively carried out for over a year. This did not help, because they do not solve the real problem. The real problem is that bad assets exist. This lowers confidence among banks to lend money to each other. The lack of confidence prohibits the liquidity injections to lead to more lending between banks. Weak institutions were forced to be taken over by stronger players. The new plan was actually not very good. It does not look too different now from what European governments are doing now with individual financial institutions. They nationalize them and take control. Just injecting enormous amounts of money without

The proposal of June is obviously what we need. Increasing the capital requirements and liquidity, transferring risk of financial transactions between banks to an organized exchange and unravelling interrelations between aspects of financial markets lowers the risk we will ever face a crisis like the credit crunch again. However, when you have pain in your back, it does not make sense to start doing exercises. You should have done them before the pain started. Now we have to wait. The three structural changes can only be implemented when the whole crisis is over. The crisis of 1929 made capitalism shook on its foundations. The same is said about the credit crisis of today. It may well be that in the future we will look at this moment as the end of a system that worked perfectly fine for 250 years, but in the end could not last. The model could be overturned at some point and a new model has to be introduced. Personally, I think that todayâ&#x20AC;&#x2122;s crisis is more than just a temporary backlash against capitalism.

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Actuarial Sciences

Operational Risk According to Solvency II: SCR, Pillar 2 and Copula’s The focus of this article is the quantification of operational risk according to the Solvency II requirements. These requirements are based on the QIS4 Technical Specifications by CEIOPS and the Operational Risk Questionnaire QIS4. As we read through these documents, we see that one of the major issues is the adequate treatment of the interdependencies between operational risk events. In sections 3 and 4 we will present a calculation methodology for the operational Value-at-Risk that takes these interdependencies into account: the copula. This methodology has been developed for banks on behalf of their Basel II models for operational risk (the so-called Advanced Measurement Approach), but it is also applicable to insurers for their Solvency II compliance.

Guidance for operational risk from Solvency II The solvability component for operational risk is one of the main constituents of the Solvency Capital Requirement (SCR) and is defined as the capital charge for “the risk of loss arising from inadequate or failed internal processes, people, systems or external events.”1 This definition is exactly the same as the definition that is used for operational risk within the Basel II rules for banks. A second analogy is that the risk measure is based on a one-year Value-at-Risk.2 A third resemblance is the simple approach that works with the help of a standard formula that is not risk-sensitive at all. In the case of Basel II, this simple formula (Basic Indicator Approach) calculates an operational risk charge based on net interest income, whereas the Solvency II simple formula is based on the addition of two components: • The first component is the basic operational risk charge for all business other than unitlinked business. This component is based on earned life and non-life premiums, or on the life and non-life technical provisions if this leads to a higher result. • The second component is the operational risk charge for the unit-linked business. This component is based on the amount of annual expenses incurred in respect of the unit-linked business.

Dr. Marco Folpmers FRM works for Capgemini Consulting. He is responsible for the financial risk management domain. Marco and his team have delivered Basel II projects within Europe at both large and small banks, especially Pillar II / Economic Capital implementations.

It is obvious that these formulas aim at an estimate for the operational risk capital charge based on earnings, expenses and technical provisions. Although the formula is very straightforward, it is far from risk-sensitive: the result is not dependent on the level of exposure to operational risk. The result won’t decrease as a result of an increased level of protection against this exposure, either. This has been brought to CEIOPS’ attention given the following remark:3 “The QIS standard formula for operational risk has been kept unchanged in QIS4, in spite of industry’s feed-back request that it be more risk-sensitive. CEIOPS considers that risk-sensitivity for operational risk should be in Pillar 2 only, consistent with the banking approach.” The reference that is made to the Basel II operational risk charge for banks is not completely correct since the Pillar 1 Advanced Measurement Approach for operational risk is risk-sensitive. However, it is essential that CEIOPS acknowledges that their proposed formula for the operational risk component within the SCR lacks risk-sensitivity.

1 CEIOPS, Techn. Specs QIS4, TS.VIII, B.1. 2 TS.VIII, A.5. The CEIOPS Solvency II VaR is calibrated at the 99.5% confidence level, whereas the Basel II AMA uses a 99.9% confidence level (see Basel II, International Convergence, art. 667). 3 TS.VIII.B.5.

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Actuarial Sciences

Risk

Frequency per year

Severity Minimum

Modus

Maximum

Theft of proprietary info

0.42

500

302,000

1,700,000

Sabotage of data and networks

0.42

125

63,000

98,000

System penetation by outsider

0.71

1,000

49,000

Insider abuse of Net access

1.58

1,000

295,000

Financial fraud

0.3

250

49,000

197,000

Denial of service

0.83

125

1,000

2,900,000

Virus

1.62

1,000

295,000

Unauthorized insider access

0.89

1,053

5,000

Telecom fraud

0.2

1,000

12,000

Active wiretapping

0.02

1250

30,000

34,000

Laptop theft

1.17

600

1000

98,000

Table 1: Frequency and loss amount statistics for each operational risk event within the IT domain

Apparently, CEIOPS links risk-sensitive methods to Pillar 2. The approach it adopts becomes clear from the Operational risk questionnaire QIS4, in which it asks about the record keeping of operational risk data (ORQ 8 a, c d) and the interrelations between risk events(ORQ 8 b): “Does your operational risk management system capture the interrelations between the various risks identified?“ CEIOPS does not mention specific interrelations but common examples are the dependency between system interruptions and fraud due to the temporary ineffectiveness of IT controls, and the dependency between a physical threat (power outage, hurricane) and theft of physical assets. Up to now we can conclude that CEIOPS uses a straightforward formula for the operational risk component of the SCR. The disadvantage is that the formula is not risk-sensitive. At the same time, CEIOPS anticipates on a risk-sensitive measure in Pillar 2, that uses recorded operational loss data and that takes interdependencies into account. Such an approach is far removed from the Pillar 1 formula. In the next sections, we will illustrate how such an approach works based on the operational risk modeling experience of the Basel II AMA approach. The VaR calculation for operational risk Since the copula approach is bottom-up, we need operational loss data that is categorized across operational risk events. The statistics that we need for the parameterization can be based on distributional statistics, expert judgment or a combination of both. Often the data

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that is available in the corporate loss database will not be entirely sufficient to calculate all parameters and expert judgment will always supplement these statistics. For each operational risk event we need statistics with regard to frequency of occurrence (per year) and the loss that arises once an operational risk event happens. In the example that we describe below, we use a Poisson distribution for the frequency and a triangle distribution for the loss amount (with parameters a, b and c for, respectively, minimum, modus and maximum). An example of the parameters needed is given in Table 1. With parameters λi and ai,bi,ci for, respectively, the Poisson frequency and triangle loss amount distributions (minimum, modus, maximum) for each operational risk event i,i=1,...,11, the operational expected loss can be derived analytically as: Op _= EL

∑ λ ⋅( i =1

ai + bi + ci ) 3

(1)

In our example the operational expected loss is € 1.5 mln. In order to compute the VaR a Monte Carlo simulation is set up. The activities that are carried out are illustrated in Figure 1. Within each simulation run instances are generated for each operational risk event according to the frequency parameter λi. For each instance of an operational risk event the loss amount is determined with the help of a sample from a triangle distribution that has been specified for the given operational risk event. For each simulation run, the loss amounts are added to compute a yearly loss amount. This procedure

Actuarial Sciences

Figure 2: Statistical loss curve (solid line: 99.9th percentile, dotted line: expected loss)

Figure 1: Activities for Monte Carlo simulation

is repeated for many simulation runs in order to derive the empirical distribution of the yearly loss amount. This empirical distribution is illustrated in Figure 2. With the help of the 99.9th percentile and the expected loss equal to € 1.5 million the VaR equals €5.6 million. We consider this VaR a

The VaR that we calculated above did not take event dependency into account. The copula approach that we illustrate here models the dependency by using correlated Poisson distributions for the frequencies. Since it is very difficult to generate samples from a multivariate Poisson distribution for dimensions larger that two,5 we start with a normal copula. Subsequently,

“Risk interdependencies have a considerable impact on the VaR.” lower limit since it does not take interdependencies into account; it simply assumes that all operational loss events are independent. The copula approach for correlated events The copula is a multivariate distribution that is characterized by a dependency structure. Examples of copulas that are often used within the financial risk domain are normal copulas and t copulas. The copula approach is currently the dominant approach to reflect the impact of interdependent risk event on the VaR. Marcelo Cruz states in this regard:4 “The study of risk interdependence is still at an early stage in operational risk due to the lack of data. However, there should be significant progress in the next few years. The use of copulas will be very important in defining the interrelationship between these extreme risks.”

we transform the marginal distributions to the Poisson distributions needed with the help of quantile transformation.6 This transformation does not preserve linear correlations but does preserve rank correlations (as measured by Spearman’s rho or Kendall’s tau). How the quantile transformation from a marginal normal distribution to a marginal Poisson distribution works is illustrated in Figure 3. For the sample from the multivariate normal distribution assumptions are needed with regard to the correlations between the operational risk events. Initially, these will be estimated with the help of expert judgment. However, this expert judgment can be guided by quantitative analysis of the key risk indicators, i.e. measures that are captured on a monthly or quarterly basis that reflect the exposure to operational risk (such as the number of IT breakdowns, the intensity of sickness leave, near-misses, et cetera).7 In a later development stage we can

4 Cruz, 2002, p. 217. 5 See Bouyé, e.a., 2001: ‘multivariate Poisson distributions are relatively complicated for dimensions higher than two.’ 6 For quantile transformation see McNeil, 2005, 186. 7 For Basel II Advanced Measurement Approach key risk indicators are mandatory, see BCBS, 2006, art. 675, and CEBS, CP10, art. 459-460. See also BCBS, 2006-2, p. 10 and CEBS, CP10, art. 457b-e.

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Actuarial Sciences

Figure 3: Transformation from normal to Poisson distribution

image that the correlations are computed with the help of data recorded in the corporate loss database. Suppose that we set the correlation coefficient between all 11 operational risk events equal to 0.3, then the statistical loss curve looks like the one depicted in Figure 4. With the help of the 99.9th percentile and the expected loss equal to € 1.5 million the VaR is now equal to € 6.5 million. The VaR has increased considerably by taking event correlations into account. This is a general phenomenon and the increase of the VaR can be explained with the help of the fact that correlated events cause a higher tail risk and, as a consequence, a larger VaR. For this reason, it is defensible that the Basel II AMA approach and Solvency II QIS4 requirements prescribe that the VaR calculation take correlated risk events into account. Without the correlations, the VaR is not conservative and, hence, a certain underestimate of the real risk. Conclusion In this article, we first described the QIS4 requirements for the operational risk component of the Solvency Capital Requirement (SCR). We also pointed out the differences between the SCR calculation and pillar 2 risk measures. The main differences are that pillar 2 risk measures have to be risk-sensitive and that they take correlated risk events into account. Subsequently, we demonstrated a pillar 2 methodology with the help of the copula approach. With the help of the normal copula and quantile transformation we generated correlated Poisson risk events for a Monte Carlo simulation. After the loss determination for the generated events (we used triangularly distributed losses) the VaR can be computed. An important result is that the risk interdependencies have a considerable impact on the VaR.

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Figure 4: Statistical loss curve for ρ = 0.3 (solid line: 99.9th percentile, dotted line: expected loss)

References (2006). International convergence of capital measurement and capital standards, BCBS. (2006). Observed range of practice in key elements of Advanced Measurement Approach (AMA), BCBS, 2 (October). Bouyé, E., Durrleman, V., Nikeghbali, A., Riboulet, G. and Roncalli, T. (2001). Copulas: an open field for risk management. (2006) Guidelines on the implementation, validation and assessment of Advanced Measurement (AMA) and Internal Ratings Based (IRB) Approaches, CEBS, CP10 (January) (2007). QIS4 Technical Specification, CEIOPS , (December). Cruz, M.G. (2002). Modeling, measuring and hedging operational risk, Wiley Finance. Di Clemente, A. and Romano C. (2003). A copula-extreme value theory approach for modeling operational risk, Working Paper, (September). Folpmers, M. (2008). A practical guide to measuring operational risk using subjective data through copulas and scenario analysis, The Journal of Operational Risk, 3 (Fall). Hoffman, D.G. (2002). Managing Operational Risk, New York: Wiley. Jorion, Ph. (2001). Value-at-Risk, 2nd edition, New York, McGraw-Hill. McNeil, A.J., Frey, R. and Embrechts, P. (2005). Quantitative Risk Management: Concepts, Techniques, Tools, Princeton.

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Econometrics

Testing for a Unit Root in Time Series with Changing Volatility Since the 1970s it has become standard econometric practice to consider the possibility that economic and financial time series are the outcomes of non-stationary processes. The type of non-stationarity that has been investigated most extensively is that of integrated processes, i.e., processes that can be made stationary by taking first differences. Fuller (1976) and Dickey and Fuller (1979) developed formal tests for the hypothesis that a time series needs differencing to become stationary. In autoregressive models, the null hypothesis of such tests is that the characteristic equation has a unit root (implying that the time series is integrated of order one), to be tested against the alternative that all roots are outside the unit circle (implying stationarity); hence the term unit root tests. The results of such tests are relevant not only for model selection, but also for investigating the validity of economic theories such as purchasing power parity or the expectations hypothesis of the term structure of interest rates.

Peter Boswijk is professor of Financial Econometrics at the Department of Quantitative Economics, University of Amsterdam. His research is on the econometric analysis of time series displaying nonstationarity and volatility clustering.

The Dickey-Fuller tests have been derived originally as likelihood ratio tests in autoregressive models with independent and identically distributed (i.i.d.) Gaussian innovations. However, for daily or weekly financial data such as exchange rates or interest rates, such assumptions are clearly violated: residuals from autoregressive models fitted to these data display volatility clustering as well as heavy tails, implying dependence in second moments, changing variances, and non-normal distributions. Similar issues arise in macro time series, many of which display a declining volatility since the early 1980s (a phenomenon known as the great moderation), so that models fitted to a longer sample period will display structural change in their residual variance. One might hope that tests are robust to such departures from standard assumptions, in the sense that the Dickey-Fuller test statistics might have the same asymptotic null distribution in the presence of non-normality and changing volatilities. Recent research, however, has shown that asymptotic null distributions are affected by persistent volatility changes such as level shifts or stochastic trends in the volatility. Moreover, even in cases where the asymptotic null distribution does not change, so that the asymptotic size of the Dickey-Fuller test is correct if

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standard tables are used, it may still be possible to obtain alternative more powerful tests by taking the changing volatility into account. This article reviews some recent results in the field of testing for a unit root in a changing volatility environment. I shall give a brief sketch of asymptotic properties of the Dickey-Fuller tests under standard assumptions, and indicate how these properties are affected by changing volatilities, either stationary or non-stationary. Some recent proposals to correct possible size problems and to improve the power of the tests are also discussed. Unit root tests Consider the first-order autoregressive model X t =φX t −1 + εt ,

t =1, … , n,

where {Xt, t=0,1,…,n} is an observed time series (treating the starting value X0 as fixed), and {εt} is an unobserved disturbance with mean zero. In practice the model will be extended by a constant and possibly a trend, and typically more lags will be needed, but I discuss only the simplest case for notational convenience. The null hypothesis is H0: Ф=1, to be tested against stationary alternatives H1: |Ф|<1 The DickeyFuller test statistic rejects for negative values of the t-statistic for γ=Ф-1=0 in the least-squares regression equation ΔXt=Xt-1+εt this t-statistic can be expressed as

(

)

−1/2

ˆ2 τ= X t2−1 ∑ t 1 Xt −1ΔXt , LS = σ ∑ t 1= n

Econometrics

σˆ 2 is the estimated variance of εt.

where

Under the null hypothesis, the model implies Xt=X0+∑ti=1 εi. It has been shown that under very general conditions on {εt}, an invariance principle applies to the stochastic part of Xt meaning that as n → ∞, n−1/2 X ⎢= n−1/2 ∑ i⎣=1⎦ εi + op (1) sn ⎥ ⎢ sn⎥

⎣

⎦

→

σW(s),

where t has been expressed as the integer part of a fraction s Є [0,1] of the sample size n, W(•) is a standard Brownian motion process on the unit interval, and → L denotes convergence 2 in distribution. Using consistency of σˆ this in turn can be shown to imply that under the null hypothesis, the limiting distribution of тLS can be characterized as τ LS

→

( ∫ W(s) ds ) 1

−1/2

∫

W(s)dW(s).

(1)

The right-hand side expression is a functional of the Brownian motion process, involving the Itô stochastic integral ∫WdW divided by the square root of its quadratic variation. Its distribution can be obtained via simulation; critical values have been tabulated by Fuller (1976). The asymptotic power function of the test is obtained from the distribution under local alternatives Hn: Фn=1+c/n where c≤0 is a non-centrality parameter. Under Hn we have

τ LS

→

( ∫ U(s) ds ) 1

−1/2

(2)

•∫ U(s)[dW (s) + cU(s)ds], 0

where U(s) is an Ornstein-Uhlenbeck process, satisfying dU(s)=cU(s)ds+dW(s) (which is mean-reverting if c<0). Simulating the rejection frequencies of the test for various values of c gives the asymptotic local power function. Stationary volatility Suppose now that the innovation εt has a timevarying (conditional) variance σt2, and hence volatility σt. An important example is the GARCH (generalized autoregressive-conditional heteroskedasticity) model

σ t2 =+ ω αεt2−1 + βσ t2−1,

which provides a convenient description for many daily financial time series. It is well known that such processes are weakly stationary if 0≤α+β<1 ; in that case the unconditional or mean variance is given by σ2=ω/(1-α-β)

Under slightly stronger conditions (in particular, if εt has a finite unconditional fourth moment), it has been shown that the variation in the volatility “averages out”, in the sense that P

n−1 ∑ t⎣ =1⎦ σ t2 ⎢ sn⎥

→

σ 2 s,

s Є[0,1],

(3)

where → P denotes convergence in probability. This in turn implies that the invariance principle holds, and hence the asymptotic null distribution and power function as described in (1) and (2) continue to hold. In other words, the Dickey-Fuller test is robust to such departures of the Gaussian i.i.d. assumption. The same result would hold if σt2 would display seasonal or more general cyclical variations; as long as the number of cycles increases proportionally with the sample size, the averaging condition (3) will hold, and hence the asymptotic distributions will not be affected. However, these results do not necessarily imply that the Dickey-Fuller test is the best possible approach. As is well known from any econometrics textbook, the presence of heteroskedasticity implies that we may improve estimator efficiency and hence test power by weighted least-squares, or more generally by applying maximum likelihood (ML). Gaussian ML estimation of unit root regressions with stationary GARCH has been investigated by Ling and Li (1998, 2003). Their results imply that for a tstatistic (or signed square root of the likelihood ratio statistic) тML, we have under H0, τ ML

→

( ∫ W(s) ds ) 1

−1/2

∫

W(s)dB(s).

(4)

Here B(s) is a second standard Brownian motion process, with cov(W(1), B(1))=ρ, which represents the correlation between εt and the “score” (a suitably defined derivative of the log-likelihood for observation t). This implies that the asymptotic null distribution depends on the nuisance parameter ρ it can be shown that the critical values lie between the critical values of the Dickey-Fuller test (corresponding to ρ=0) and the standard normal distribution (corresponding to ρ=1). In Boswijk (2001), I obtained an approximation of ρ as a function of the GARCH parameters (α,β), which allows us to estimate ρ and use this estimate to simulate the asymptotic null distribution (and hence p-values). The approximation implies that p↑0 as α+β ↓ 1 i.e., as we approach the integrated GARCH or IGARCH case, although the limiting case itself is not covered by the results given above (because they require finite variance and kurtosis for εt). Boswijk (2001) also shows that under local alternatives Hn,

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Econometrics

→

τ ML

( ∫ U(s) ds ) 1

−1/2

⎡ ⎤ c •∫ U(s) ⎢dB(s) + U(s)ds ⎥. 0 ρ ⎣ ⎦ 1

(5)

An interesting difference between (2) and (5) is that the non-centrality parameter c, which gives the test power by shifting the distribution under local alternatives to the left, is replaced by c/ρ This implies that we may expect the test based on тML to have larger power than тLS especially for smaller values of the correlation parameter ρ. Because the correlation gets smaller close to the IGARCH boundary, and the sum of empirical GARCH estimates αˆ + βˆ is often close to 1, one may expect substantial power gains from using the ML-based test statistic тML instead of the least-squares based Dickey-Fuller statistic тLS. Non-stationary volatility The results in the previous section imply that the IGARCH case is of particular interest regarding the power of unit root tests, but they also imply that the asymptotic theory sketched above no longer applies, because if α+β=1 the unconditional variance σ2 is infinite, and hence the averaging condition (3) is violated. Other cases of possible interest where this condition is violated correspond to persistent changes in the volatility, e.g., when σt changes from one value σa to another value σb at some time t*. To analyze the (near-) IGARCH case, Boswijk (2001) considered the GARCH parameters as sequences αn=n-1/2a and αn +βn=1+n-1b, where a and b are fixed constants, so that αn ↓ 0 and αn+βn↑1 as n → ∞ (note that the IGARCH model corresponds to b=0). The motivation for considering such sequences is that they occur naturally when we consider continuous-record asymptotics (where the number of observations within a fixed time span, and hence the observation frequency, increases as n → ∞). Based on the work by Nelson (1990), it has been shown that under these assumptions, L

σ ⎢sn⎥ → σ (s), ⎣

⎦

(6)

where σ(s) is a continuous-time stochastic volatility process. As analyzed by Cavaliere (2004), (6) also holds under a wide range of deterministically changing volatilities, although in those cases the limiting function σ(s) is a deterministic function as well. The result (6) implies that the averaging condition (3), needed for the invariance principle to hold, is violated. Instead, we now find, both under the null hypothesis and under local alternatives,

n−1/2 X ⎢sn⎥ ⎣

X (s),

→

⎦

where X(s) is a process satisfying the stochastic differential equation dX(s) = cX(s)ds + σ(s)dW(s), with W(s) again a standard Brownian motion. For the Dickey-Fuller test statistic, this implies under H0, p3,e= ω pt −1 + (1 − ω )p3,e t −1, t

(7)

with σ 2 = ∫0 σ (s)2 ds . The distribution of the right hand side expression now depends on the entire volatility process σ(s), which shows in particular that the critical values obtained by Fuller (1976) are no longer valid; the Dickey-Fuller test is not robust to this type of non-stationary volatility. Simulations in Boswijk (2001) and Cavaliere (2004) show that if one ignores this problem (and uses the tabulated critical values), then substantial size distortions (i.e., over-rejection under the null hypothesis) may result. Wild bootstrap If we have a consistent estimator of σ(s), then we may plug this estimate into (7) and obtain the distribution of the resulting functional by simulation; this approach is used by Boswijk (2001) and Cavaliere and Taylor (2007). It turns out, however, that a much simpler approach is possible using the so-called wild bootstrap, see Cavaliere and Taylor (2008). The idea is to compare the observed statistic тLS with a large number B of bootstrap replicates {τj*, j=1,...,B}, where τj* is based on a “resampled” time series Xjt*=Xj,t–1*+εjt*. In the usual bootstrap procedures, applicable to i.i.d. innovations, we would draw the bootstrap errors εjt* either from the empirical distribution of the least-squares residuals et (known as the non-parametric bootstrap) or from the standard normal distribution (parametric bootstrap). In the wild bootstrap however, the bootstrap errors are generated as εjt*= et•wjt, where {wjt} are i.i.d. draws from a distribution with zero mean and unit variance. This implies that εjt* is a serially uncorrelated sequence with E(εjt*) = 0 and var(εjt*) = E(et2) ≈ σt2, and this is sufficient to guarantee that the limiting distribution of τj* is the same as the asymptotic null distribution of тLS. Therefore, if we reject H0 when the bootstrap p-value (the percentage of {τj*, j=1,...,B} less than тLS) is lower than the significance level α, then the resulting test has asymptotic size α. Monte Carlo simulations in Cavaliere and Taylor (2008) show that this procedure is very effective in finite samples. Adaptive testing Although this bootstrap procedure effectively solves the size problem of Dickey-Fuller tests

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Econometrics

caused by non-stationary volatility, it is again not necessarily the most powerful unit root test. In Boswijk (2005), I addressed this issue by deriving the asymptotic power envelope, i.e., the maximum possible power, when the volatility σt is observed, of any test of H0 against local alternatives Hn plotted as function of the non-centrality parameter c. I showed that for realistic volatility processes, the power of the Dickey-Fuller test is substantially lower than the power envelope, implying some unexploited power potential. In the realistic case that the volatility process is unobserved, and we do not wish to impose a parametric volatility model such as GARCH, the volatility may be estimated consistently by non-parametric local averaging of squared residuals. The resulting test, which may be seen as a weighted-least-squares based Dickey-Fuller statistic with non-parametrically estimated weights, compared with simulated asymptotic critical values, obtains an asymptotic power function very close to the envelope. In Monte Carlo simulations, however, it appears that rather large sample sizes are needed to fully realize this power potential, and that for smaller samples, the proposed procedure may still suffer from size distortions. In order to address the latter problem, we are currently investigating the application of the wild bootstrap to this adaptive testing procedure.

Figure 1: US TBill rate (R3) and first difference (DR3), Sept. 1988 – Aug. 2008

short-run serial correlation) is given by тLS=1.195 with a conventional p-value of 0.68, so that we are not able to reject the null hypothesis. If we re-estimate the same ADF regression by maximum likelihood, allowing for GARCH(1,1) innovations, we find a signed likelihood ratio statistic of тML=-0.886 with a pvalue of 0.36. This p-value is obtained by si-

"Deviatons from the the long-run interest rate may persist for more than a decade" Empirical application and simulation To illustrate the size and power issues discussed in this article, we consider the behaviour of the US 3-month Treasury bill rate, over the last 20 years (5217 daily observations from September 1, 1988 through August 29, 2008). The time series and its first difference are depicted in Figure 1. From the top panel we observe that the interest rate displays only very weak mean reversion; if we believe in a long-run average interest rate of about 7%, then deviations from this level may persist for more than a decade. The first differenced series displays the usual signs of volatility clustering; in particular the increase in volatility over the last year (associated with the credit crisis) is notable. The augmented Dickey-Fuller (ADF) statistic (in a model with a constant term and 10 lagged differences, to allow for a non-zero mean and

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mulating the asymptotic null distribution in (4) using ρ=0.35 which is the sample correlation between the residuals and the scores in the GARCH model. Although the p-value has become a bit smaller by allowing for stationary GARCH, there clearly is still not much evidence of mean reversion. ˆ The estimated GARCH parameters equal â ˆ =0.848, so that their sum is =0.137 and β close to the non-stationarity bound (the estimates have been restricted to lie in the stationarity region). The estimated volatility based on the GARCH model is depicted in Figure 2, and shows a gradual decline for most of the period, but a clear increase in volatility in the last year. Therefore, we consider the possibility of nonstationary volatility. The wild bootstrap p-value of the ADF statistic is given by 0.67, hence very close to the p-value based on homoskedasticity. A similar figure is obtained if we simulate the

Econometrics

Figure 2: Estimated GARCH volatility of US TBill rate, Sept. 1988 – Aug. 2008

asymptotic null distribution in (7), taking the estimated volatility in Figure 2 as (a discretization of) the true volatility. Therefore, we may be tempted to conclude that in this case the volatility, although possibly non-stationary, has very little effect on unit root inference. However, if we follow the adaptive testing approach, which in this case boils down to applying weighted least-squares to the ADF regression, using the same GARCH volatility for the weights, then a statistic of –2.142 and a simulated p-value of 0.17 results. Although this p-value is still larger then a reasonable significance level, it does give an indication of somewhat stronger evidence of mean-reversion in the interest rate. It is always hard to illustrate size and power properties by an empirical application, which after all is only one realization from a distribution. Therefore, we conclude with some simulated asymptotic power curves of the tests considered here, using parameter values from the empirical application (i.e., ρ=0.35 and σ(s) is a continuous-time version of the estimated GARCH volatility in Figure 2). The results are depicted in Figure 3, where the power curves are plotted against minus the non-centrality parameter –c, ranging from 0 to 20. The solid curve gives the power function of the DF test under homoskedasticity; in the presence of non-stationary volatility as characterized by Figure 2, the power function drops to the shortdashed curve, which would be the asymptotic power of the wild bootstrap DF test in this case. On the other hand, if we apply the adaptive procedure we observe a serious increase in power, as indicated by the long-dashed curve. The dotted curve, finally, depicts the power function of the GARCH-based likelihood ratio test assuming ρ=0.35; although this seems to be by far the best result, finite-sample simulations indicate that asymptotics based on non-stationary volatility may give a better approximation of the true properties of the tests, so that the longdashed power curve may give a more realistic indication of the true power potential.

Figure 3: Asymptotic local power functions of various tests, as function of –c

References Boswijk, H.P. (2001). Testing for a unit root with near-integrated volatility, Tinbergen Institute Discussion Paper # 01-077/4. Boswijk, H.P. (2005). Adaptive testing for a unit root with non-stationary volatility, UvAEconometrics Discussion Paper 2005/07. Cavaliere, G. (2004). Unit root tests under timevarying variances, Econometric Reviews, 23, 259–292. Cavaliere, G. and Taylor, A.M.R. (2007). Testing for unit roots in time series models with nonstationarity volatility, Journal of Econometrics, 140, 919–947. Cavaliere, G. and Taylor, A.M.R. (2008). Bootstrap unit root tests for time series with non-stationary volatility, Econometric Theory, 24, 43–71. Dickey, D.A. and Fuller, W.A. (1979). Distribution of the estimators for autoregressive time series with a unit root, Journal of the American Statistical Association, 74, 427–431. Fuller, W.A. (1976). Introduction to Statistical Time Series, New York: John Wiley. Ling, S. and W.K. Li (1998). Limiting distributions of maximum likelihood estimators for unstable autoregressive moving-average time series with general autoregressive heteroskedastic errors, Annals of Statistics, 26, 84–125. Ling, S. and Li, W.K. (2003). Asymptotic inference for unit root with GARCH(1,1) errors, Econometric Theory, 19, 541–564. Nelson, D.B. (1990). ARCH models as diffusion approximations, Journal of Econometrics, 45, 7–38.

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On the distribution of Traveling Salesman Problem Solutions among the Ranked Linear Assignment Alternatives The Linear Assignment Problem is considered one of the fundamental combinatorial optimization problems in operations research. Operations researchers are concerned with finding optimal solutions for complex problems. These include decision variables, constraints and objective function. When the constraints and objective function only involve linear relationships among the decision variables, the problem is called Linear Programming. The integrality of the decision variables is often not guaranteed by the Linear Programming formulation but, if required, needs to be imposed. These so-called Integer Linear Programming problems are in general more complex than their Linear Programming counterparts. The Linear Assignment Problem is one special Linear Programming problem: its decision variables naturally assume only the values 0 and 1. Roughly speaking the Linear Assignment Problem deals with choosing n entries in a n x n cost matrix, one in each row and column, in such a way total costs are minimized. A problem closely related to the Linear Assignment Problem is called the Traveling Salesman Problem: a salesman wants to visit n cities in an optimal way. In this thesis the distribution of the solutions of the Traveling Salesman Problem among the Linear Assignment Problem alternatives is investigated. Therefore the Linear Assignment solutions are ordered in increasing costs.

Jannetje Veerman received her Bachelor degree in Econometrics and Operations Research, cum laude, from the Free University of Amsterdam in August 2008. This article is a summary of her bachelor thesis, which was written under supervision of dr. J.A. dos Santos Gromicho. In September she started the Master of Operations Research. She was one of the organisers of a symposium on E-commerce for Operations Research students in May 2008

The Linear Assignment Problem and Traveling Salesman Problem The linear assignment problem (LAP) is besides of mathematical interest of practical interest as well. In manpower management, the assignment problem is to assign N employees optimally to N different jobs. It is supposed that a numerical performance rating of all N2 employee-job combinations is known. The optimal solution are the N employee-job combinations which maximizes the sum of performances or minimizes the sum of costs. The matrix C = (cij)i;j=1,...,n is the cost matrix. In terms of the manpower assignment cij are the costs when employee i takes care of job j. Goal

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is to find a matrix X = (xij)i;j=1,...,n satisfying the constraints. This is a matrix in which only one element on each row and column equals 1 and all the other elements are 0. In terms of the personnel-assignment this is if xij == 1 then employee i gets job j. An X like this is therefore called an assignment. In general, there are a lot of feasible assignments for this problem. It can easily be seen that there are n! feasible solutions. Despite being large this number is finite and hence there is an assignment that minimizes the total cost. In 1946 Eastereld presented the first algorithm to solve the linear assignment problem. In 1955 H. W. Kuhn developed an improved method, called the Hungarian Method [2]. This algorithm is specic to the assignment problem and more efficient than solving the LAP as a general LP problem. For more details, see the complete thesis [5]. In some cases the assignment problem needs to have an extra constraint. For example, it is not possible to have employee i assigned job j together with employee k assigned job l (iâ&#x2030; k; jâ&#x2030; l). In this case it would be helpful to have a

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list with all assignments of the original problem and their total costs. The assignment satisfying all the constraints with the lowest cost level is chosen. This approach yields the optimal solution for the constrained problem. Katta G. Murty developed an algorithm for ranking all the assignments in non decreasing order in [4]. Murty used the Hungarian Method to identify the optimal solution. Following a few simple steps the second best solution is found. By repetition of the steps all assignments are identified. This algorithm is described in [5]. For the Traveling Salesman Problem the matrix with distances between cities is used as cost matrix, being the objective the minimization of the total distances. The Traveling Salesman Problem (TSP) has a (natural) constraint which prohibits the salesman to visit a city more than once. The optimal solution is a single loop between n cities. Again, there is more than one feasible solution for the TSP. Starting with the first city (i.e. first row) the next city can be chosen from n-1 other cities. When the second city is chosen, the third city can be chosen from the remaining n-2 cities. Continuing like this yields there are (n-1)! feasible solutions for the TSP because the choice of the first city is irrelevant. Clearly, a solution of the Traveling Salesman Problem is a feasible solution of the Linear Assignment Problem. These are the so-called circular permutations. On the other hand, a solution of the LAP is not necessarily a solution of the TSP. As shown above, if the problem is of dimension n there are n! assignments and (n-1)! solutions of the TSP. This yields (n-1)! in n! assignments are solutions of the TSP or 1 in n assignments is a solution of the TSP. In this thesis the following problem will be addressed: ‘How do the TSP solutions distribute among the LAP solutions when these are monotonously ordered in cost?’ Computational results For the purpose of this thesis, the ranking algorithm of Murty is implemented in MatLab, a high-level programming language to perform numerical computing. This section presents the results obtained. Some remarks can be done at this stage. At first, there are more efficient algorithms developed which can be used to solve the discussed problem. However, these improvements are beyond the scope of this thesis. Hence, Murty’s algorithm is used to rank all the assignments and to find a least cost assignment in a given matrix a LAP code of Jonker & Volgenant [1] is used as a black box. Secondly, the example in [5] showed that by repeating the steps of the algorithm of Murty some linear assignment problems arise more

Figure 1: Typical results for a matrix, a symmetric matrix and a matrix with distances of dimension 6

than once. After the u-th best solution is identified, the algorithm creates a new list in which the u+1-th best solution has to be found. This list is slightly different from the previous list. Therefore, it is more efficient to store the obtained results than to solve the same problems again. However, this isn’t included in our implementation in MatLab. For the magnitude of the problems used in the thesis, it suffices.

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Initial efforts To get insight in the distribution of Traveling Salesman solutions among linear assignment solutions the results of different cost matrices are described. Because the number of assignments becomes very large when the dimension of the matrix increases, all matrices have dimension 6 or smaller. For the aim of this thesis, this is not a big restriction. Furthermore, the matrices contain no fractions because the code used to create the LAP MatLab extension can only deal with integers. At first, matrices of dimension 4, 5 and 6 with random integers 0,..., 10 are used to run the program. Secondly, matrices with random integers 0,...,10 are multiplied by their transpose. This results in symmetric matrices. Finally, n coordinates in the Euclidean space are randomly generated. The distances between these n points, which can be thought of as cities, are calculated and rounded off. Typically for a distance matrix is the symmetry and the zero-diagonal: the distance between city i and city i is 0. The program returns all assignments ordered by their costs and a zero-one vector. In case an element in the vector equals 1, this assignment is a solution of the TSP. In the figures the zeroone vector has been plotted in a bar graph. The x-axes run from 0 to n! and represent the number of the assignments. Hence, the distribution of the TSP solutions in the assignments is visualized. In the first picture the TSP solutions seem randomly distributed among all assignments. This is not surprising, the elements of the cost matrix were chosen randomly. The second picture shows a more surprising result; the density of TSP solutions is in the bigger in the first assignments, i.e. in the best assignments. There is no obvious cause for this particular distribution. Further investigation may possibly give more insight. The TSP solutions in the final picture are more concentrated between the assignments with highest costs. The reason for this are the zeros on the diagonal. The best assignments will include such a diagonal element, because this results in low costs. Since choosing element (i, i) means the salesman has to travel from city i to city i, this prohibits the solution to be a solution of the TSP. So after a couple of assignments with a diagonal element, other assignments will show up in the list. These assignments are more likely to be a solution of the TSP. Adapted diagonal From now on, the diagonal elements in all matrices are replaced by enormous costs. The expectation is that the TSP solutions are among

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Figure 2: Typical result for random asymmetric, symmetric and distance matrix with huge costs on the diagonal, dimension 6

the cheapest assignments. In all graphics in figure 2 the TSP solutions are situated among the assignments with lowest costs. It is interesting to visualize the costs of the assignments. The program is asked to plot the costs of the first 50 assignments and mark the TSP solutions with a circle. The graphics donâ&#x20AC;&#x2122;t show surprising results. The costs grow slowly in all matrices. There are more assignments with same costs in the random and distances matrix than in the symme-

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In table 1 the mean number of the first TSP solution of different matrices is given. Because of the huge computation time 5 matrices have been created. Of these matrices the number of the first TSP solution has been identified and of these numbers the average has been taken. This has been done for dimension n = 10 and n = 20. Matrix

Dimension n

Mean number of first TSP solution

Asymmetric random

5.5

Asymmetric random

10.4

Symmetric random

120.6

Symmetric random

Took too long

Distances

204.6

Distances

Took too long

Table 1: Number of first TSP solution

The matrices which are used to get the results as shown in the table, all had huge diagonal elements. By doing so, the TSP solutions are more likely to arise in the first (i.e. best) ranked assignments, see figure 2. Nevertheless, the first TSP solution in a 20 x 20 matrix cannot be found in case the matrix is symmetric, like the symmetric random matrix and the distances matrix. After 30 minutes just the 450 best solution of one matrix were found. In this respect, it should be mentioned that the inefficiency of our program plays an important role. This result causes some trouble for the Traveling Salesman Problem; in this problem the corresponding matrix will often be symmetric. For dimension n = 10, the results are comparable. It can be (carefully) concluded that in the ranked assignments of a asymmetric matrix a TSP solution can be found in an early stage. Figure 3: Costs of the rst 50 assignments and an indication when the assignment is a TSP solution of a random asymmetric, symmetric and distance matrix with huge diagonal elements

tric matrix. There is no obvious reason for this. In these examples the first TSP solution is the fourth, ninth and ninth assignment in the ranked assignments. These solutions have costs which are about the same as the better LAP alternatives. As stated above the total number of assignments explodes when the dimension of the cost matrix increases. However, ranking all assignments until the first TSP solution has been found should be possible in most matrices because this solution can arise in an early stage.

Conclusion The problem addressed in this article was: â&#x20AC;&#x2DC;How do the TSP solutions distribute among the LAP solutions when these are monotonously ordered in cost?â&#x20AC;&#x2122; In the first place, it can readily be concluded that 1 out of n assignments is a solution of the Traveling Salesman Problem. By implementing K.G. Murtyâ&#x20AC;&#x2122;s algorithm for ranking linear assignments on the computer, this question can now be answered. For matrices with low dimensions, all assignments can be found quickly. Because the total number of assignments explodes when the size of the matrix increases, matrices with higher dimensions are difficult to handle.

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Among the assignments of random, asymmetric cost matrices the TSP solutions are randomly distributed. In these matrices the first TSP solution is found between the assignments with lowest costs. This is why the computer can deal with greater matrices if they are asymmetric. The TSP solutions of matrices which are symmetric, like a matrix with distances between cities, are in the assignments with highest costs. To find the first TSP solution in a matrix of dimension greater than 10, this causes trouble. A solution for this is to replace the diagonal elements of the cost matrices by huge costs. Now, the TSP solutions are between the linear assignment solutions with low costs. Hence, the program is more able to return the first TSP solution in a reasonable time. However, in a symmetric matrix of dimension 20, the first TSP solution cannot be found even after running the program for more than 30 minutes. If interested, further research can be done by improving the MatLab code. As stated above, storing the obtained results in each stage increases the efficiency and the speed of the program. Probably the first TSP solution (or even all) can then be identified for larger matrices. In addition, one can improve the code in case the cost matrix is symmetric. In that case the two assignments with rows and columns inverted succeed each other.

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References Jonker, R. and Volgenant A. (1987). A shortest path algorithm for dense and sparse linear assignment problems. Computing, 38, 325-340. Kuhn, H. W. (1955). The Hungarian Method for the Assignment Problem. Nav. Res. Log. Quart, 2, 83-97. Munkres, J. (1957). Algorithms for the Assignment and Transportation Problems. Journal of the Society for Industrial and Applied Mathematics, 5(1), p 32-38. Murty, K. G. (1967). An Algorithm for Ranking all the Assignments in Order of Increasing Cost. Operations Research, 16(3), 682-687. Veerman, J. M. A. (2008). On the distribution of Traveling Salesman Problem solutions among the ranked Linear Assignment alternatives. Bachelor thesis Econometrics and Operations Research, VU Amsterdam.

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On Density Forecast Evaluation Traditionally, probability integral transforms (PITs) have been popular means for evaluating density forecasts. For an ideal density forecast, the PITs should be uniformly distributed on the unit interval and independent. However, this is only a necessary condition, and not a sufficient one, as shown by some simple examples. I discuss an alternative approach to density forecast evaluation, via the Kullback-Leibler information criterion (KLIC), and illustrate it with a small simulation study.

Cees Diks studied theoretical physics at Utrecht University and obtained a PhD (1996) in mathematics from Leiden University for his research on nonlinear time series analysis. In 1998, after a two-year postdoctoral research fellowship at the University of Kent at Canterbury (UK), he joined the Center for Nonlinear Dynamics in Economics and Finance (CeNDEF) group of the Faculty of Economics and Econometrics, where he is now an associate professor.

Comparing density forecasts If we would like to predict a currently unknown random variable, such as tomorrow’s closing value of the AEX index, based on currently available information, there are many different ways to do this. One way would be to predict the mean of the future random variable, given the information available to us (a point predictor). Another way would be to give an interval in which we think it is likely (say, at a confidence level of 95%) that the unknown future variable will fall (an interval predictor). Here we will be concerned with the most detailed type of forecast, namely estimates of the entire conditional distribution of the future random variable, given the available information. Typically this distributional forecast is given in terms of a probability density function, hence the name density forecast. Naturally, we would prefer an accurate forecast over an inaccurate one, so it is natural to develop methods that compare the accuracy of competing density forecasts.

To fix the notation, consider a univariate time series process {Yt}, t ‫ א‬Ժ . For simplicity I consider one-step-ahead forecasts only. The density forecast for Yt+1, made at time t, is based on the information Ft available at time t. Besides the past observations Ys, s ≤ t, the information set may also contain exogenous variables known at time t. A one-step-ahead density forecast is a mapping from the information set to a probability density function (pdf) for Yt+1, i.e. given Ft, it gives a density forecast for Yt+1 in the form of a pdf, denoted here by ft,t+1(y). Ideally the pdf ft,t+1(y) is a good approximation to the true conditional pdf, gt,t+1(y) of Yt+1 given Ft. Clearly it is of interest to asses the quality of density forecasts. If we have a density forecast we are often interested in its quality relative to the true conditional density. This leads to the area of goodness-of-fit tests. The issue of how to select one of several alternative density forecasts that at our disposal leads to forecast selection procedures. There is a branch of econometric literature that builds on density forecast evaluation along the lines proposed by Diebold et al. (1998). The idea is to transform the predictive densities into a sequence of PITs, defined as Ut+1 = Ft,t+1(Yt+1), y

where Ft,t+1(y) = ∫ ft ,t +1(s)ds , the cumulative dis−∞ tribution function (CDF) of the density forecast made at time t. In the ideal case that the den-

Forecast

density forecast for Yt+1 when μt ~ N(0, 1), determined exogenously, is part of Ft

N(μt, 1)

N(0, 2)

III

(N(μt, 1) + N(μt + тt, 1))/2, where тi = ±1, each with probability 1/2.

N(μt + δt,

σ t2 ) , where (δ , σ t2 ) = (1/2,1), (-1/2,1) or (0, 1.69), each with probability 1/3. t

Table 1: Competing density forecasts for Yt+1, which is N(μt, 1) distributed, where μt ~ N(0, 1) is part of the information set Ft.

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ॱ

Figure 1: Uniform histograms PIT for density forecast I up to IV.

sity forecast is correct, a standard result in probability theory states that the sequence of PITs consists of independent uniform random variables on the interval [0, 1]. This observation has led to a methodology where one assesses the accuracy of density forecasts by investigating the sequence of PITs. Formal tests for uniformity and serial independence applied to observed PIT sequences are then used to judge the accuracy of density forecasts. As illustrated by an example by Hamill (2001), however, the uniformity and independence of the PIT sequence is only a necessary, and not

led the historical forecast. It is like predicting the weather for++ tomorrow on the basis of the weather on the same calendar day in the past 100 years. Forecast III involves an irrelevant variable тt, and hence is called an unfocused density forecast. Forecast IV uses three incorrect density forecasts, which each are chosen with equal probability. This example was constructed to illustrate that PITs can be independent (they are so by construction here) and have a textbook uniform distribution, even if they are far from ideal. Figure 1 shows the estimated density of the respec-

“It is like predicting the weather for tomorrow on the basis of the weather on the same calendar day in the past 100 years” a sufficient, condition. In other words, there are density forecasts that are not ideal, but still have a sequence of PITs that are uniformly distributed on [0, 1] and serially independent. Gneiting et al. (2007) give a number of additional examples. Following these authors I here would like to consider the following example. The data Yt+1 are independently drawn from N(μt, 1), where μt is an exogenous N(0, 1) variable (generated by nature), which is part of the information Ft, available at at time t. Table 1 shows four competing density forecasts. Density forecast I corresponds exactly to the conditional density of Yt+1 given μt, hence it is the ideal forecast. Forecast II simply only uses the marginal density of {Yt}, and could be cal-

tive PITs for each of these forecasts, based on 10,000 simulated values of Yt. The histograms are practically flat. In fact, for forecasts I – III, it can be shown analytically that the PITs are distributed uniformly on the unit interval. It can be shown that the distribution of the PITs for forecast IV, which corresponds with the example given by Hamill (2001), has small deviations from uniformity, but this is not visible in Figure 1 due to the accuracy of the histogram. My aim is to show that an approach using scoring rules automatically resolve the issues associated with PITs, at least when we want to compare competing pairs of density forecasts. This is illustrated numerically using the fore-

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Econometrics

casts from the above example. An additional theoretical advantage to a score-based approach is that it is easily extended to a multivariate setting.

introduce a scoring rule for the competing forecasts, and then test the hypothesis that they are performing equally well. The reason for using the logarithmic scoring rule here is that, as argued above, the ideal forecast should, on average, receive a larger logarithmic score than any deviating forecast.

dt = ln(f1/f2) = ln f1t,t+1(Xt+1) − ln f2t,t+1(Xt+1) Consider the null hypothesis is that the models receive the same average score: H0 : E[dt] = 0. The null hypothesis can be tested using the standardized sample mean of the scores where is σ�� a heteroskedasticity and covariance ro_ bust estimator of the asymptotic variance of �

-1.4189

-1.7665

-15.304

-1.5298

The log-likelihood score associated with a forecast density ft,t+1(x) is

For two competing density forecasts, say f1 and f2, we define the score difference as

E[St]

For later reference I calculated numerical values of the true average scores for the four competing models. The results are shown in Table 2. From this table we can observe against which alternative the test should reject the null hypothesis, ideally. For instance, if we compare forecast I with forecast II, the test should reject the null of equal predictive ability in favour of forecast I, since forecast I actually has a higher average score than forecast II. In general, the test should ideally reject the null against the

0.8

0.6 power

St = ln ft,t+1(Xt+1)

In this section I investigate the behaviour of the test for equal predictive ability by applying it repeatedly to the example forecasts given in Table 1. The actual data are generated in correspondence with the ideal forecast: Yt+1~ μt, where μt ~ N(0, 1) is drawn by nature, and known at time t.

Suppose we would like to measure, in the KLIC sense, which of two competing density forecasts is closer to the ideal density forecast, then we can use the testing methodology proposed by Giacomini and White (2006). The idea is to σ�

III

A small simulation study

Tests for equal predictive ability

_ �

(a HAC estimator). Under the null hypothesis of equal predictive ability Tn is asymptotically standard normally distributed. If one of the competing models is outperforming the other, then a two-tailed test ideally rejects, and the sign of the test statistic indicates which of the two models is receiving statistically significantly higher scores, on average. This is how such a test would be used in practice. In the simulations below I report size-power plots for the onesided test, in order to keep track of the rejection rates corresponding with each of the tails of the distribution.

� �� KLIC��

�� √�

Table 2: Average log-likelihood score for the four models.

A popular scoring rule for density forecasts is based on the Kullback-Leibler information criterion (KLIC). Let Y have density g(y) The KLIC for a density f(y) relative to the true density g(y) is defined as

The KLIC is a divergence measure between the densities f and g, which means that it is nonnegative, and zero only if f and g coincide. Of course, in practice the true density g is not known, but if we subtract two KLICs, the unknown density g drops out: KLIC(f1, g) − KLIC(f2, g) = ॱ (ln f1(Y) − ln f2(Y)). This motivates the use of the logarithmic scoring rule for comparing competing density forecasts.

Forecast

0.4 I -- II I -- III I -- IV III -- II IV -- II IV -- III

0.2

0.1

0.2

0.3

0.4

0.5 size

0.6

0.7

0.8

0.9

Figure 2: Size-power plots for the test of equal expected log-scores for each of the six pairs of processes. The lines are labeled by the two density forecasts that were being compared. The power plotted is that of rejecting the null of equal predictive ability against the model being mentioned first receiving better scores, on average. Sample size n = 50.

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Econometrics

alternative that the model with higher average score in Table 2 performs better. To make the interpretation of the size-power graphs easier, for each pair of density forecasts tested against each other, I report the power in terms of rejections of the null hypothesis against this ‘correct’ alternative. Figure 2 shows the size-power plots of all pairs of density forecasts, based on 1000 replications,for a sample size n = 50. The horizontal axis represents the nominal size, while the curves plotted correspond with the power (fraction of rejections) observed among the 1000 replications, for each particular nominal size. For all pairs of competing density forecasts involving the ideal density forecast (I) the test clearly has power against the null hypothesis, in favour of the alternative that forecast I performs better. Likewise, the test has power against the ‘best’ density forecasts involving the other pairs, except for the pair III–IV, for which the power almost coincides with the nominal size (i.e. a size-power plot along the diagonal). The reason is that the average logarithmic scores are so close for these two forecasts (see Table 2) that much larger sample sizes are required to detect the difference in performance. Additional simulations (not shown) indicate that the sample size should be of the order of 50,000 to obtain considerable power for this pair of density forecasts. Summary There are various ways to evaluate density forecasts. Among the most popular methods are those based on the sequence of probability integral (PIT) transforms, and score-based methods. In this paper I have illustrated some of the limitations of PIT-based methods by means of examples showing that for density forecasts that are far from ideal, the sequences of PITs can have the same properties as those for the ideal density forecast. Subsequently I have shown that score-based methods can distinguish between the predictive ability of these competing density forecasts.

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References Diebold, F.X., Gunther, T.A. and Tay, A.S. (1998). Evaluating density forecasts with applicationsto financial risk management. International Economic Review, 39, 863–883. Giacomini, R. and White, H. (2006). Tests of conditional predictive ability. Econometrica, 74, 1545–1578. Gneiting, T., Balabdaoui, F. and Raftery, A. E. (2007). Probabilistic forecasts, calibration and sharpness. Journal of the Royal Statistical Society, Series B, 69, 243–268. Hamill, T.M. (2001). Interpretation of rank histograms for verifying ensemble forecasts. Monthly Weather Review, 129, 550–560.

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The Livestock Collection Problem - a Real-world Vehicle Routing and Inventory Problem This article presents a description and two approaches to solve the Livestock Collection Problem (LCP), which is a real and complex routing and inventory problem. The LCP is taken from the Norwegian meat industry and consists of constructing a route plan for transporting animals from farms to a slaughterhouse. The transportation plan must provide the slaughterhouse with animals according to the production plan, and the capacity for keeping live animals at the slaughterhouse must not be exceeded. The combination of rules and regulations concerning animal welfare, together with constraints regarding production and inventory, makes the problem difficult to solve. Only relatively minor problems can be solved to optimality. In the case of the larger problems that affect abattoirs one has to use heuristic solution methods.

Problem description The LCP can be viewed as an extension of the well-known and well-studied Vehicle Routing Problem (VRP) [10]. This problem deals with the allocation of transportation tasks to a fleet of vehicles and the simultaneous routing for each vehicle. The VRP, first described by Dantzig and Ramser [2], is a difficult optimization problem with high industrial relevance. In the classical VRP the goal is to design a least cost set of routes where customers are visited exactly once, all starting and ending at a central depot. The total demand of all customers on a route must be within the vehicle capacity, which is assumed to be the same for all vehicles. Extensions to the basic VRP In order to describe the Livestock Collection Problem we need to make some extensions to the basic VRP. Some of these extensions are quite standard and can be found in many VRP applications, while others are specific to our version of the LCP. Tour length and duration. Animals are not allowed to stay on the vehicle for more than eight hours. This is a constraint imposed rigorously by Norwegian law and it cannot be violated in a feasible solution. There are no theoretical restrictions on the tour length, but the eight-hour rule in practice also limits the length of tours. Vehicle fleet and loading. Transported live animals are divided into types and categories. A type represents the species of the animal. All types are divided into categories predomi-

Johan Oppen is Assiciate Professor in logistics at Molde University College, Norway. He has a bachelor degree in computer science and mathematics from the same school from 2002, and a master degree in logistics from 2004. He got a PhD in logistics, also at Molde University College, in 2008. The title of his PhD thesis is â&#x20AC;&#x153;Models and Solutions for Rich Logistic Problems".

nately by age, size and gender. For reasons of animal welfare the mixing of different animal types, and to a certain degree categories, in the same compartment of a vehicle is not allowed. The categories require different amounts of floor space and height in the vehicle during transport. In this study we examine four animal types consisting of 16 different animal categories in total. Norwegian animal trucks are normally divided horizontally into three sections. Most vehicles are outfitted to stack pigs and sheep in two tiers, while cattle need the full height of the vehicle. When a truck transports only one type of animal, normally desirable for animal welfare reasons, it is relatively easy to calculate the legal load of the vehicle. On occasion two or three types of animals are transported by the same vehicle, which makes the loading problem more cumbersome. This situation typically occurs in remote rural areas, where it is often difficult to compose full loads of only one animal type. The vehicle capacity then depends on the sequence in which the farms are visited on each tour. Vehicles are loaded starting at the front and working backwards. Each individual section is loaded from the top downwards. Only one non-empty compartment is accessible in the

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available in the lairage at all times. Therefore, the routes have to be planned in such a way that unwanted stops of the production process are avoided, since such stops are very costly. One must also avoid situations in which animals are held in lairage too long before slaughter lest the lairage capacity be exceeded. Solution methods Figure 1: A vehicle is divided into sections and compartments

vehicle at any time, as one cannot load animals into a compartment by passing through another compartment already in use. Figure 1 shows an example of how a vehicle may be divided into sections and compartments. Problems where different types of load have to be kept in separate compartments during transport are often referred to as loading problems (refer to [1] and [9]). The LCP differs from most loading problems in that not all compartments of the vehicle are accessible all the time. Multiple tours per vehicle per day. In contrast to a standard VRP, each vehicle may be used for multiple tours each day. Time is added between consecutive tours to allow for the cleaning of the vehicle.

In many cases, one may choose between two main types of methods to solve an optimization problem. Exact methods find optimal solutions, whilst heuristic methods provide no guarantees on solution quality. In general exact methods are limited to solving minor problems and relatively simple models. On the other hand, many heuristics have been quite successful in finding appropriate solutions for many instances of complex problems. To find solutions to the LCP, both a Tabu Search [4] heuristic [7] and an exact method have been developed. Both methods are based on column generation [3]. The heuristic represents a possible way of handling real-world instances of the problem. The exact algorithm, though yielding valuable insight, is capable of solving only small instances.

"Tabu Search heuristic generates solutions that are between 12% and 21% better than manual ones" Precedence constraints. The different farms visited may have different health statuses, which impose restrictions on the order of visits. Farms with males used for breeding can only be visited by a vehicle that is empty and clean. This means that such farms must be visited first in a tour. Farms with animals that are infected with a disease may also deliver animals for slaughter. However, in this case the vehicle must be cleaned and disinfected before it may visit other farms. Farms with infected herds thus have to be the last stop on the tour. In [8], a similar problem involving transportation of live pigs between farms subject to veterinary restrictions is described and solved. Inventory. In addition to the constraints normally associated with VRPs, constraints are needed to control the inventory level. The production process at the slaughterhouse should run smoothly throughout the day. This requires a sufficient number of every type of animal be

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Tabu Search. The Tabu Search heuristic developed for the LCP can be summarized as follows. First, an initial solution is constructed. This is done by approximating truckloads based on clustering together orders of the same animal type that are close geographically. These loads are then distributed on vehicles and days such that the production and inventory constraints are not violated. However, the initial solution may violate precedence constraints, vehicle capacity constraints and time related constraints. The search then proceeds by repeatedly moving one customer from one tour to another. Infeasible solutions are allowed during the search, and a penalty regime is applied to guide the search back into the feasible part of the solution space. Non-improving moves are allowed while attributes of recently performed moves are prohibited for a time to prevent the search from going back to recently visited solutions. The best feasible solution found during the search is saved and the search is stopped

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once a predetermined number of iterations or time limit is reached. Column generation. Column generation is a solution method based mainly on Linear Programming [5]. A feasible solution to a routing problem is a collection of feasible routes. An optimal solution is then an optimal subset of the set of all feasible routes. For most practical problem instances, the set of all feasible routes is extremely large. The basic idea in column generation is to work with only a small subset of routes and to generate and add only those new routes that improve the solution. Finding new promising routes correspond to solving shortest path problems in the network of the problem. For more details about this algorithm for the LCP, refer to [6]. Computational results Computational tests have been conducted to test the performance of both the Tabu Search heuristic and the column generation algorithm. Solutions found by the heuristic have been compared with real transportation plans generated by manual systems that are used by the industry today. The results from [7] show that solutions generated by the Tabu Search heuristic are between 12% and 21% better the manual ones when measured by total distance driven by all vehicles. The computational testing of the exact solution procedure shows that the column generation algorithm is currently capable of optimally solving instances with up to approximately 25 orders. Conclusions and further work We have presented the Livestock Collection Problem, which is a complex routing and inventory problem that affects the meat industry today. Though heuristics are needed to find solutions to real-world instances of the LCP, exact methods are also of interest as they allow for the study of the characteristics of such a problem. There are concrete plans for further work on the LCP. These will focus on shared vehicle fleet planning that allows simultaneous collection and delivery of animals to more than one slaughterhouse, improved treatment of travel costs and travel times in the road network and the possibility of using ferries. In addition, some of the results from the project are to be included in existing commercial software and thus made available to the industry.

References Brown, G.C., Ellis, C.J., Graves, G.W. and Ronen, D. (1987). Real-time, wide area dispatch of mobil tank trucks. Interfaces, 17, 107-120. Dantzig, G.B. and Ramser, J.H. (1959). The truck dispatching problem. Management Science, 6, 80-91. Desaulniers, G., Desrosiers, J. and Solomon, M.M., editors (2005). Column Generation. Springer. Glover, F. and Laguna, M. (1997) Tabu Search. Kluwer academic publishers. Matousek J. and GĂ¤rtner, B. (2007). Understanding and Using Linear Programming. Springer. Oppen, J. (2008). Models and Solutions for Rich Logistic Problems, PhD thesis. Molde University College. Oppen, J. and LĂ¸kketangen, A. (2008). A tabu search approach for the livestock collection problem. Computers & Operations Research, 35(10), 3213-3229. Sigurd, M., Pisinger, D. and Sig, M. (2004). Sceduling transportation of live animals to avoid the spread of diseases. Transportation Science, 38, 197-209. Cole Smith, J. (2003). A genetic algorithm approach to solving a multiple inventory loading problem. International Journal of Industrial Engineering, 10, 45-54. Toth, P. and Vigo, D., editors (2002). The Vehicle Routing Problem. Society for Industrial and Applied Mathematics, Philadelphia.

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Econometrics

An Empirical Model for the Time Allocation of Parents Conventional models of labour supply describe how many hours individuals choose to work. In empirical research, the effects of individual wage rates and the household’s non-labour income are estimated. Typically, use is made of a micro-data set, containing information about working hours, wage rates and incomes of a sample of individuals or households. All nonworking time is treated as ‘leisure’, from which individuals derive utility. That non-working time may consist of a variety of different activities such as doing housework, taking care of the children or ‘pure’ leisure activities, is ignored. The Nobel Prize winner Becker (1965) introduced a theory of time allocation that introduced the concept of household production. We estimate an empirical model for time allocation. We distinguish three different uses of non-labour time: housework activities, childcare, and leisure.

Hans Bloemen is associate professor in Economics at the Free University, Amsterdam. He studied econometrics at Erasmus University and wrote his PhD thesis at Tilburg University. He is doing empirical micro-econometric research in the field of labour supply, structural job search models, wealth and consumption, retirement, and the labour market behaviour of couples.

Theory of time allocation In the spirit of Becker, we begin by constructing a theoretical model of time allocation. In this model, we look at the preferences of a couple such as a husband and wife. Each spouse within a couple derives utility from his or her own private consumption, but also from the consumption of housework services and childcare services. Moreover, spouses derive utility from leisure time and time spent on their children. Housework and childcare services can be produced by the household member themselves, using their time inputs. Alternatively, they can be bought in the market. However, spending more time on household production diminishes the amount of time available for market labour or leisure. Market labour is the most important source of income for most households. The household is restricted in its choice by the available budget. Private consumption and services bought in the market are expenditures, while income comes from doing paid work and some amount of non-labour income. The total time that can be spent on the various activities is limited due to the natural restriction of having only 24 hours in a day. The different time uses have different impacts: time at labour brings in income, but is not consumed (i.e. it is not included in the utility function); time at hou-

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sework is productive, as it is used to produce housework services, but it is not consumed; time spent on childcare time is both productive and it is consumed; leisure time is consumed only. In the empirical analysis, the emphasis is on measuring the impacts of the wage rates and the non-labour income on the allocation of time. The theoretical model, under certain assumptions, can be used to predict the impacts of these variables on the time usage. For nonlabour income the model predicts a negative effect on paid-work time for each spouse, and a positive effect on childcare time, while for housework no effect is expected. We are unable to always predict whether the effects of wage rates of the husband and wife will have a positive or negative impact on time usage. This is partly due to opposing income and substitution effects. For instance, if someone’s wage rises, all non-market activities become more expensive, so we may expect an increase in paid-work time (the substitution effect). But the income will increase, and therefore someone may like to increase the leisure time or the time spent on childcare, so we may expect a decrease in paid work time (the income effect). A similar story holds for an increase in the wage rate on the time that someone spends on their children: there are opposite income and substitution effects and the total effect may be positive or negative. The model unequivocally predicts that an increase in the wage rate leads to a decrease in time spent on housework. Whether an increase in the husband’s (wife’s) wage rate leads to an increase or decrease in the wife’s (husband’s) time spent on childcare depends on whether the husband’s and wife’s childcare are substitutes or complements in the production of childcare services.

Econometrics

Empirical analysis: data and econometric methods The data used in this study is the French Time Use Survey from 1998. Data on time usage is collected by a diary method. Respondents participating in the survey fill out a diary for a specific day. They report their time spent on various activities in intervals of ten minutes. For instance, it is recorded whether one is at (paid) work, cleaning, shopping, taking care of the children, etc. We have aggregated all time uses into the three categories of paid work, housework, and childcare. In the analysis, pure leisure is treated as the residual category, much like in the conventional labour supply model. The survey also collects information on earnings, income, characteristics of the individuals such as education level, age, number and ages of children, etc. The data are used in the estimation of the econometric model. This econometric model consists of several equations and must capture several features of the data construction. The dependent variables of interest are the time uses of the spouses: the paid work time, housework time, and childcare time of both the husband and the wife. This provides six equations from which to begin. As explanatory variables we would like to include the wage rates of each spouse, the household’s non-labour income, and various background characteristics. However, there are two problems with including wage rates. A priori it is unlikely that wage rates are uncorrelated with the error terms of the time use equations. Moreover, the wage rates are unobserved for non-employed individuals. Simply excluding non-employed individuals from the sample would lead to selectivity bias, as was shown by Heckman (1979). To overcome this problem we add two wage equations to the model, one for the husband and one for the wife. We allow the errors of the wage equations to be correlated with the errors of the time use equations. We also add two (husband and wife) employment equations to the model: equations for the binary indicator that shows whether someone is employed or not. By allowing the errors of the employment equation to be correlated with the wage equation, we aim to correct for selectivity bias: wage rates of employed workers may be different from the (latent) wage rates of non-employed workers, for reasons that cannot be observed. Finally, we also allow the error terms of the time use equations to be correlated with each other and between spouses. There are various interpretations for the possible correlations in time usage between spouses. For instance, there may be positive assortative mating: two people with similar, possibly unobserved, characteristics or preferences tend to form couples, and this may be reflected in correlated time allocation outcomes. Another econometric problem is the

fact that we use diary information obtained at a specific day. Therefore, it is very well possible that some individuals have not devoted any time to a specific activity such as housework. Moreover, non-employed individuals by definition spend no time on paid work. Thus, in the econometric specification we have to allow for corner solutions at zero. This can be done by defining equations for time usage in a similar manner to the ‘Tobit’ model. In summary, we obtain a system of ten equations consisting of six Tobit-like time use equations, two wage equations, and two (Probit-like) employment equations. We allow the error terms of all equations to be correlated and will estimate the correlation coefficients. We assume that the error terms follow a multivariate normal distribution. Under this assumption, we are able to construct the likelihood function and estimate the model parameters by maximum likelihood. In the likelihood function, the unobserved wage rates for the non-employed need to be removed through mathematical integration. Moreover, for observations with corner solutions, we need to compute probabilities, which is achieved by integrating density functions. Since the total density function of our model is ten dimensional, this may imply that we need to perform ten-dimensional integration. This problem can be overcome by using Simulated Maximum Likelihood: by drawing random numbers from the underlying density function, we can simulate the multidimensional probabilities, which greatly reduces the computational burden (see Börsch-Supan and Hajivassiliou, 1993). The model’s parameters are estimated using a numerical optimization method that maximizes the simulated likelihood function. Main results Estimates of the model lead to parameter estimates for the parameters of each equation’s explanatory variables and for the variance-covariance matrix of the error terms. The detailed presentation and discussion of the results can be read in Bloemen and Stancanelli (2008). Here we present the effects of the wage rates and non-labour income on time usage. We have summarized the results in terms of elasticities. In brackets are the standard errors caused by the use of estimated parameters in computing the elasticities. From the effects of non-labour income we can identity the sign of the income effects. Nonlabour income has a negative impact on paid work for both husband and wife. This is in accordance with the theoretical predictions. Nonlabour income has a positive impact on the amount of time spent on the children, although the impact is estimated somewhat less precisely for men than for women. This is in accordance with the model, which assumed that childcare

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Econometrics

Elasticity of:

Wage husband

Wage wife

Non-labour income

0.41**

-0.01

-0.10**

Paid work: Paid work husband

(0.14)

(0.09)

(0.02)

Paid work wife

-0.07

1.40**

-0.08**

(0.12)

(0.3)

(0.02)

Total paid work time, for both spouses

0.25**

0.47**

-0.10**

(0.1)

(0.14)

(0.01)

-0.56**

0.49**

0.08**

(0.27)

(0.16)

(0.04)

Housework time: Housework husband Housework wife Total housework time, for both spouses

-0.09

-0.74**

0.01

(0.06)

(0.16)

(0.01)

-0.18**

-0.51**

0.03*

(0.07)

(0.13)

(0.01)

-0.52

0.57**

0.08*

(0.34)

(0.21)

(0.05)

-0.07

-0.31

0.04**

(0.1)

(0.21)

(0.02)

0.17

-0.11

0.05**

(0.12)

(0.18)

(0.02)

Childcare time: Childcare time husband Childcare time wife Total childcare time, for both spouses

standard errors in parentheses **: significant at 5% level; *: % significant at 10% level Table 1: Elasticities

time is ‘consumed’ by the parents. Non-labour income has a significant positive impact on the housework time of the husband. Thus, the assumption that the husband does not derive utility from time spent on housework is not supported by this result. For the husband, there may be a ‘hobby’ element in performing certain housework tasks. We do not see this for the wife. The husband’s own wage has a positive impact on paid work, and a negative impact on housework. The former implies that the substitution effect is stronger than the income effect. The latter is in accordance with the prediction of the theory. The wage rate of the husband does not significantly influence the time he spends on childcare, although the sign of the estimated elasticity is negative. This may suggest that substitution and income effects cancel each other. For the wife, we see that a higher wage rate increases her paid work time and decreases her housework time. The magnitudes of the elasticities are larger for the wife than for the husband. Further, we do not find that the wage rate of the wife has a significant effect on her childcare time. There are no significant cross-wage effects on paid work: the paid work time of the husband (wife) barely responds to a change in the wage rate of the wife (husband). The husband will spend more time on housework and childcare if the wage of the wife incre-

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ases. In contrast, the housework and childcare time of the wife do not respond to a change in the wage rate of the husband. References Becker, G. S. (1965). A Theory of the Allocation of Time, The Economic Journal, 75(299), 493-517. Bloemen, H.G. and E.G.F. Stancanelli (2008). How do Parents Allocate Time? The Effects of Wages and Income, Discussion Paper 2008079/3, Tinbergen Institute, Amsterdam. Börsch-Supan, A. and V. Hajivassiliou (1993). Smooth unbiased multivariate probability simulators for maximum likelihood estimation of limited dependent variable models, Journal of Econometrics, 58(3), 347-368.

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Econometristen

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Actuarissen

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Beleidsmedewerker vermogensbeheer

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Econometrics

A Brief Reflection on Automatic Econometric Model Selection in a Fuel Price Elasticity Context. In December 2005, the VSAE (econometrics study association of the UvA) organised a trip to London and one of the study related activities was a lecture at the University of Oxford on automatic model selection using PcGets. Having just completed my first master’s courses I did not at that time fully understand all that was stated. Despite this lack of understanding, or my physical condition at the time, I was captured by the idea that it was possible to insert data obtained through research into a program and, with the simple click of a button, this program would report the best model obtainable with the present data. I embraced the idea that econometric modelling could be automated. This meant I would not have to do much of it in the future and could spend my time solely on interpreting the results. On the other hand, the word “data-mining” engaged my mind and would not abate.

Robert Schipperhein obtained his master degree in Econometrics at the University of Amsterdam in September 2008. He has been a member of the 2006 VSAE board, during this board membership he was the chief editor of Aenorm. He co-organised the 2006 and 2007 editions of the Econometric Game and contended for the University of Amsterdam in the 2008 edition.

Then two years later I needed to come up with a topic for my thesis. At an internship at the Ministry of Finance I was asked to investigate the price elasticities of motor fuels. Due to the abundant research available on this topic, it was interesting to compare my estimates (based on the extensive literature) to the estimates and model presented by Autometrics. This article is a summary of the second half of my thesis in which I detail the aforementioned comparison. What is Autometrics? Autometrics is the most recent automatic model selection program. It was created by Jurgen Doornik, a student of econometrics at the UvA. He introduced it in a 2007 online paper. Autometrics is a successor of PcGets, which was previously explained by Jennifer Castle in the 52nd edition of Aenorm. Because of limited space I will only briefly explain the Autometrics algorithm. Further information is available by consulting the two articles mentioned above. The General Unrestricted Model (GUM) is the starting point for the model reduction procedure and covers the entire dataset. This GUM

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must provide sufficient information on the process being modelled and (has to be statistically well behaved.) The latter property is checked by series of diagnostic tests. For each insignificant variable the GUM defines a path: delete that variable and continue with backward deletion, one variable at a time. Each path is followed until termination, leaving only significant variables. The process then tests the resultant path with respect to the GUM to prevent too great a loss of information. This terminal is also subjected to diagnostic tests and if rejected, the path is followed in the reverse direction until a suitable model is found. Usually several terminal models are selected by the procedure. In this case Autometrics combines the elements of the selected terminals and creates a new GUM, after which the procedure starts all over again. When the next GUM equals the former one, and still several terminals remain, an information criterion of choice is applied to choose between them. The last important elements of the systematic approach of Autometrics are the ‘pre-search’ possibilities. Lag-length pre-search checks the significance of lags up to a manually decided length. The program deletes the lags of a certain variable up to the first significant one. This ensures that the number of variables in the first GUM is reduced and improves efficiency. A process called ‘bunching’ also reduces the number of paths. The program is designed to begin by attempting to remove blocks of variables that are ‘highly insignificant’. If this is successful the number of branches is reduced. Because of the

Econometrics

Variable

formula form

software acronym

entity

gasoline consumption

qbenz

1000 kg

diesel consumption

qdies

1000 kg

price of gasoline

pbenz

€/100 liter

price of diesel

pdies

€/100 liter

vehicle stock

wagenpark

income

bni

€ 1.000.000

population

bevolking

inflatie weging

CPI(1965 =1)

inflation

cpi

Table 1: data summary

Lag-length pre-search and the block deletion, Autometrics is able to handle GUMs with more variables than observations. Data mining and its risks Early work on data mining by Lovell (1983) introduces data mining by six quotes. The two most relevant to this discussion are: “The precise variables included in the regression were determined on the basis of extensive experimentation (on the same body of data) . . . .” “The method of step-wise regression provides an economical way of choosing from a large set of variables . . . those which are most statistically significant . . . .” The last quote describes precisely the core purpose of automatic econometrics model selection software like Autometrics. Lovell warns of two possible problems when data mining is used. In an experimental setting he shows that the significance of tests suffers when data mining is performed. This is caused by the large number of tests performed on the same data. Moreover, the researcher who uses a data mining approach is uncertain to select the correct (true) model. Lovell tries three basic data mining strategies in an experimental environment. The best two strategies have a success rate of almost 70%.

Classical econometric research begins with an hypothesis based on theory, earlier research or common sense. This hypothesis is then tested using data and statistical tests. The problem with data mining is that it is difficult to properly test the validity of a model obtained from data based on common sense. Once you believe in data mining, and why would you use data mining software if you were not a believer, you will always find a way to explain its results. With a success rate of 70%, many ´wrong´ results will be published. There is further uncertainty regarding the claimed success rate as other sample properties such as size surely affect it. A brief summary of the ´manually obtained´ results With the data presented in table 1 the following two partial adjustment models emerged from the literature and withstood extensive testing:

q g = b1 + b2 pg + b3i + b4c + b5q−g1 + b6 pd + e (1) qd = b1 + b2 pd + b3i + b4c + b5q−d1 + b6 pg (2) + disturbances The variables in these equations are corrected for inflation and population growth, which are not included explicitly. Logs have been taken of all variables. All data is for the Netherlands and aggregated yearly. Consistent data is available from 1975 to 2007. The results of my research on price elasticity are summarised in table 2 below. For further information regarding these results and how they were obtained, please refer to my thesis. Autometrics in action As with any process, output is influenced by input. Autometrics is not designed to distinguish between log-linear specification and linear specification. Moreover, the researcher must choose to use level or differenced input. By making these choices the researcher directs the program towards a certain class of models. To be able to compare the Autometrics results with the earlier results, Autometrics should have the possibility to construct the partial adjustment models (1) and (2). The PcGive–12 guide,

price elasticity of gasoline

dependent variable

short term

long term

short term

long term

gasoline

-0.401

-0.954

0.168

0.490

(0.091)

(0.193)

(0.057)

(0.105)

diesel

0.247

1.343

-0.263

-1.410

(0.192)

(1.653)

(0.085)

(1.132)

Table 2: price elasticities

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Econometrics

the initial dataset are reported in table 3. Run 1 includes only the level variables. Run 2 includes the lagged dependent. Run 3 includes all variables and their first lags. Every successive run adds the lags of the previous level. An ‘a’ means the level variable is in the final equation, the ‘b’ means the first lag is included, and so on. Furthermore, the number of included observations, number of included (lags of) variables and the Residual Sum of Squares (RSS) of the final equation are presented.

which includes Autometrics, recommends the use of first differences when the data is integrated of order 1. Such is the case in this instance. However, if the programme is directed to use the data in such a way that a Partial Adjustment Model does or does not result dependant on the researcher’s preference, the automatic model selection program has less selection to do itself. Autometrics should find the right model of its own accord as that is the purpose for which it was designed.

"I was captured by the idea that with the simple click of a button, the program would report the best model" To keep the possibility of the equations (1) and (2), but not manually skew Autometrics towards this result, level log-specification of the data is chosen. Moreover, the data inserted is not corrected for inflation or population growth. Both correction variables are inserted in the equation individually to give Autometrics all possible freedom.

Although the number of observations declines by one every successive run, the RSS is indicative of the statistical quality of the model. The fourth run has the best RSS, although run 5 and 6 have more freedom and should therefore be able to statistically outperform run 4, instead of indicating an RSS that is twice as large. Moreover, as opposed to earlier runs, the last two runs eliminate the number of vehicles as a relevant variable. This is not only counterintuitive from the point of view of the modeller, who would expect it to be a very important variable. It is also counterintuitive from a statistical point of view because a decreasing number of observations had a negative influence on the power of the significance tests. An important reason for this ‘strange’ behaviour of Autometrics could well be the fact that run 5 and 6 contain more variables than observations while the earlier runs have more observations than variables. Autometrics is able to handle

The next decision is the number of lags to include in the GUM. With the small number of observations, more lags will increase the challenge for Autometrics and may make it impossible to automatically select a model. Including only the lagged dependent will direct Autometrics towards my outcome. To explore this dilemma, consider the gasoline equation. When we include all relevant variables up to a certain lag length, we can see what final equations Autometrics provides. The results of running Autometrics with an increasing number of lags included in Variable

Run 1

qb p

Run 2

Run 3

Run 4

Run 5

Run 6

d a,d,e

a,b,c

a,b

b a,c

a,b,c

a b

b,e

a,b

a,d

constant

pcpi

trend

# observations

# of var (inc. Lags)

RSS

0.00672

0.00384

0.000629

0.001402

0.00129

Table 3: models estimated by Autometrics starting from different GUMs

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Econometrics

Short term

Long term

gasoline

diesel

gasoline

diesel

Section 2

0.007

0.022

0.042

0.146

Autometrics

0.001

0.003

0.012

Table 4: RSS of the models

more variables than observations by creating blocks of variables, which it then treats as one variable. The program searches for blocks that can be deleted as a whole, decreasing the number of variables until the ‘normal’ situation of more observations than variables is reached once again. Considering that this setting offers more challenges than a low number of observations, it would seem wise to work with more observations than variables. Running Autometrics twice, with both gasoline and diesel consumption as dependent variables, yields the two models presented in the Appendix as A.1 and A.2. The final equation for the diesel model does not include a lag of the dependent variable and is therefore not an error correction model. Moreover, the Lagrange Multiplier test rejects the presence of first order autocorrelation in the final gasoline equation. This indicates that the model chosen by Autometrics is improperly specified. Contrary to the final Gasoline model, in which some lag of all the inserted variables remains significant, the Diesel model contains only 5 of the 9 possible variables. Autometrics therefore shows that it ‘believes’ the processes that determine gasoline and diesel consumption are quite different. A feature exploited in the first part of my research is the correction for population growth and inflation. The inclusion of this correction term is based on economic theory and Autometrics will not include it unless imposed. Since correcting for inflation and population growth is considered valid, helping Autometrics by imposing these criteria could improve its results. The final results are beyond the scope of this article, but the selected models differ significantly from A1 and A2. Comparison of Autometrics results with the Partial Adjustment Models The performance of automatic model selection algorithms can only be determined by comparing their outcomes to manually obtained results. There has been extensive research on fuel consumption. The model and estimates obtained manually are in line with the mass of earlier research and therefore reliable enough to compare with the results generated by Autometrics.

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Equations (1) and (2) include only the first lag of the dependent variable, while the models generated by Autometrics also use second order lags. This causes a difference in the number of observations between the two approaches. The performance of the models can only be compared by RSS if the number of observations is equal. Therefore, we use 28 observations to estimate both models, with only Autometrics using the 1975 observation. For all four equations the resulting RSS is presented in table 4. Note that calculating the RSS for the four models is done with four different dependent variables. This means that the diesel and gasoline statistics cannot be compared to one another. The Section 2 gasoline statistics on the errors can be compared to the Autometrics gasoline residuals. This is because equation 1 can be rewritten as: Pg I log Qg = log N + b1 + b2 log cpi + b3 log P N * P cpi

Qg C Pd b4 log + b5 log −1 + b6 log cpi + e N N P

In this equation the Log N on the right hand side has a coefficient of one. Estimating this equation (with the same dependent variable as the Autometrics model) leads to the same coefficients and, more importantly, to the same residuals as estimating equation 2. The same reasoning is applied to modelling diesel. Obviously the RSS of Autometrics’s choices are superior to my own model. This is of no surprise as Autometrics is designed to find the model of best fit. However, note that the gasoline equation suffers from an AR-test rejection. The RSS in table 4 are obtained for all models with the same number of observations. Another interesting feature is the correlation between the residuals of the estimation of two equations modelling the same consumption by another strategy. Do they tend to over- or underestimate for the same observations? The correlations between the residual series are 0.11 for gasoline and 0.23 for diesel. Both correlations are very small and I have therefore concluded that the models are indeed very different. In short, on statistical grounds it may be concluded that the results of conventional modelling and Autometrics differ and that Autometrics finds a superior fit. However, the use of the model is more interesting. Most of the time we do not wish to know how the past developed. Understanding the past is just a means to understand the future.

Econometrics

gasoline

diesel

PAM’s

0.027

0.010

Autometrics

0.034

0.109

Table 5: RMSE of the forecasts

With this small number of observations I was able to save only one observation and add two others because I took so much time writing this thesis. Forecasts can be done for 2005-2007. The forecasting errors can be compared between the two methods for the two dependent variables. PcGive reports the Root Mean Square Error (RMSE) when forecasting. The definition of the RMSE is: RMSE =

⎡1 ⎤ h ⎢ ∑ k=1(y k - fk )2 ⎥ ⎢⎣ h ⎥⎦

1/ 2

In this case h=3 because there are three of sample observations. yk+fk is the error of the forecast. These are again comparable because of the reasoning given above. As becomes clear from table 5, Autometrics is less accurate at forecasting than my manually obtained models. Correcting the data for population and inflation does improve the forecasting results of Autometrics. The RMSE is better than Autometrics’ earlier results, but still worse than the manual results presented in table 5. If both my own model and Autometrics make a forecast of the 2006 Gasoline tax revenues based on the data until 2004, the forecast of Autometrics is 80 million further off than my own model. This is on top of the fact that the manual model estimate itself is off by roughly 220 million. The partial adjustment model performs better when estimating diesel consumption, but the results of Autometrics in this case are much worse.

is complementary to Autometrics, has not been tried. Last but not least, I have done little to guide Autometrics with any principles of common sense. There are possibilities to lock certain key-variables that should be in the final model or to insert less ‘raw’ data. Examples of the researcher helping the software are using first differences and inserting data that guides to an error correction model. Final remarks For more information on my research please ask for my thesis (robert.schipperhein@gmail. com). If you have become interested in automatic econometric model selection and are in need of a thesis topic, I have quite a number of ideas I was unable to include in my own thesis. Feel free to contact me since lots of research can be done to map the current use of these algorithms or even improve them. Serious development of these algorithms has been as recent as 1999 to 2007. Applied work that makes use of either PcGets or Autometrics is still scarce to my knowledge. References Castle, J., (2006). Automatic Econometric Model Selection using PcGets, AENORM, 52, pp.4346. (online available; http://www.aenorm.nl/ artikelen/52-castle.pdf) Doornik, J.A., (2007). Autometrics, Department of Econometrics, University of Oxford, http:// www.economics.ox.ac.uk/hendryconference/ Papers/Doornik_DFHVol.pdf, 2008-06-06. Lovell, M.C., (1983). Data Mining, The Review of Economics and Statistics, 65, pp. 1-12.

Conclusion The previous section compared the Autometrics results to my own manually obtained results. Although a single investigation based on a very small sample, the research revealed again the typical pit fall of data mining. Statistically Autometrics outperformed my own research. Contrary to this, the out of sample forecast showed better results for the ‘classical’ approach. This provides further evidence that sound research is not possible without some knowledge about the topic of interest and some common sense. Some remarks in favour of Autometrics have to be made though. The dataset used in my research is very small, with only 30 observations to start with and 28 observations if lags are accounted for. Moreover Co-integratedVAR-modelling, another feature of PcGive that

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Switching Between Prediction Rules by Evolutionary Selection: Can the Stylized Facts Be Explained? In economics expectation formation and learning are of key importance to explaining the behaviour of consumers, households and producers. Hommes et al. (2007) investigated these phenomena in a demand-supply cobweb setting by letting participants predict prices one period ahead in a laboratory experiment. The methodology often employed for modelling and explaining this behaviour is based on the assumption of rational expectations. Consequently, the rational expectations hypothesis was tested by Hommes et al. (2007) and resulted into three so-called “stylized facts”. By means of a model with evolutionary switching between price prediction rules and heterogeneously behaving economic agents that were boundedly rational this article seeks to match the three stylized facts.

Raymon Badloe currently studies Actuarial Science at the University of Amsterdam and works for Hewitt Associates in Amsterdam. During his study he has contributed to the organisation of the Econometric Game 2007, the Financial Econometric Project and several other VSAE committees. This article is a summary based on his bachelor thesis written under the supervision of prof.dr. Cars Hommes and dr. Mikhail Anufriev.

This article first discusses the laboratory experiment conducted by Hommes et al. (2007) and the economic model in the second section. The third section discusses the stylized facts and in the fourth section price prediction rules are discussed. Switching and evolutionary selection is treated in the fifth section. Section six discusses the simulations and in the final section conclusions are drawn. Laboratory experiment Rational expectations (RE) was first introduced by Muth (1961) becoming the dominating paradigm once Lucas (1972) made applications to macroeconomics. RE implies economic agents to have perfect knowledge of the economy and its underlying process. The assumptions made by RE often did not coincide with the behaviour of agents observed in laboratory experiments, which lead the way towards the concept of bounded rationality. More recently Hommes et al. (2007) conducted an experiment in a demand-supply cobweb set-

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ting in which participants were predicting prices one period ahead. The cobweb model describes price fluctuations for a consumption good that takes one period of time to produce and cannot be stored. The model also exhibits self-reversing expectations feedback i.e., a high (low) price expectation leads to high (low) production by market clearing resulting in a low (high) realized market price. The participants were not informed about the fundamentals of the economic model underlying the experiment; they did however know the following: • The price was determined by market clearing; • Supply and demand of the good were subject to market uncertainties, but the distribution of these exogeneous shocks was not specified; • The participants could observe both their past price prediction as well as the realized market prices as a time series. Having limitations on their knowledge the predicted price of the participants had to be between 0 and 10, consequently making the realized price also to be within this interval. These predicted prices also affected the acquired earnings of a participant. The following payoff function specifies the earnings for participant i in period t as a function of the quadratic prediction error:

{

}

= Π i ,t max 1300 − 260(pt − pie,t )2 , 0 .

(1)

Econometrics

The maximum earnings of a participant were e equal to 1300 euro’s and pi ,t denoted participant i’s price prediction in period t, with pt the realized price in period t. The aggregate realized price is determined by the expectations of all 6 participants combined. The cobweb model dynamics were adjusted to investigate stable, unstable and strongly unstable situations. Proceeding one could examine the behaviour of economic agents in different situations as in complex economic markets of today. The described setting depicts a situation as discussed by Sargent (1993) and establishes a useful testing frame for the RE hypothesis. The next section will elaborate upon the economic model more thoroughly.

and also needs to be decreasing in the realized price. We could represent the demand function to be a linear decreasing function with a demand shock added in period t. This results into the following demand function: D( pt ) = a − bpt + ηt ,

the demand shock ηt will be normally distributed and is important, because it deals with uncertainty in demand. The parameters a and b will be fixed at successively 13.8 and 1.5 like in the original experiment. Taking into account market clearing the realized market price generated in our experiment reads as follows: n

Economic model Forming expectations about price developments gives producers the opportunity to determine their optimal supply. In the cobweb model these expectations are formed one period ahead and the best production decision arises from profit maximization. Assuming the producers’ cost function to be strictly convex we could define the supply function to be nonlinear and incree asing in the expected price pi ,t . Making these assumptions we choose the following supply function of one producer: S = (pie,t ) tanh(λ( pie,t − 6)) + 1,

(2)

price prediction done at time t-1 by with pie,t the participant i. The parameter λ tunes the nonlinearity and therefore the stability within the model. In accordance with the values of λ used in Hommes et al. (2007) we will use the same values, namely 0.22, 0.5 and 2 for respectively the stable, unstable and strongly unstable case. Consumer demand is determined by the realized price pt, rather than the expected price

PRICE_GROUP2 EXP21 EXP22

EXP23 EXP24 EXP25

EXP26

+ εt.

(4)

The RE hypothesis could be tested in our controlled environment by examining whether or not it would be able to describe the price prediction behaviour of the participants and the realized market prices. Hommes et al. (2007) focused on three main characteristics, namely (1) are the realized market prices biased, (2) do price fluctuations show excess volatility i.e., the extent in which price movements are larger than movements in underlying economic fundamentals and (3) can realized prices be predicted. These were respectively measured by the mean, variance and autocorrelations of the prices. Making use of the RE benchmark a comparison was made with the mean, variance and autocorrelations observed in the experiment for the

Three stylized facts

i =1

The number of producers is n=6 and the noise εt = ηt /b is normally distributed, with variance σ ε2 = 0.25 .

a − ∑ S( pie,t )

= pt

(3)

PRICE_GROUP4 EXP41 EXP42

EXP43 EXP44 EXP45

EXP46

Figure 1: Price fluctuations and individual predictions done in the experiment.

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stable, unstable and strongly unstable treatment. Statistical t-tests were employed to test whether the RE benchmark mean and variance were equal to the realized mean and variance of the price in the experiment. The autocorrelation coefficients of the prices were examined by looking at their patterns within a confidence interval. The results of these tests are more easily interpreted by examining figure 1. The left figure exhibits individual price predictions and the realized market price in a stable treatment. The figure on the right depicts individual price predictions and the realized market price in the strongly unstable treatment. From the previously described tests and these figures the following three “stylized facts” have resulted: 1 The sample mean of realized market prices is close to the RE benchmark. 2 The sample variance of realized market prices is close to the RE benchmark in the stable treatment, but is significantly larger than the RE benchmark in both the unstable and strongly unstable treatment. 3 The realized market prices exhibit almost no linear autocorrelation. These three stylized facts indicate that the RE benchmark could explain the outcomes of a stable cobweb experiment well. However, it fails to do so in the unstable and strongly unstable case. Different models with homogeneous expectations could not completely explain these results. It therefore seems reasonable to assume that agents are boundedly rational. These agents could also use several prediction rules, since there are many possible rules one could use an evolutionary selection measure to decide upon the use of a rule. Evaluating the prediction rules on relative performance and allowing agents to “switch” between these rules accordingly it need not necessarily be the case that agents use the same rule at once making heterogeneous expectations a reasonable assumption. We will make these assumptions and integrate them into our model of evolutionary switching that will be developed below. Price prediction rules Under uncertainty individuals often rely on simple decision heuristics to make price predictions. As we are trying to gain insight on the prediction behaviour in various situations and match the stylized facts we obviously should include simple prediction rules. One of the simplest rules is a constant prediction rule, which were also observed in the experiment. We then

choose two constant rules being above and below the RE steady state price: p1,e t = c1

(=5)

(5)

p2,e t = c2

(=7)

(6)

When producers use the last price observed and change their prediction in this direction we come across the adaptive expectations rule: p3,e= ω pt −1 + (1 − ω )p3,e t −1, t

(7)

with ω є [0,1] a parameter determining the extent of the prediction adjustment in the direction of the last price. An obvious prediction rule would be one based on the average of all past prices up to time t-1, but we will make use of the last two prices: e p4, = vpt −1 + (1 − v )pt −2 , t

(8)

with v є [0,1] to describe a rule that mostly decreases the variance of the price and lowers the autocorrelations. Our most sophisticated rule is the sample autocorrelation learning rule (SAC-rule) and has the following form: e p5,= βt −1(pt −1 − α t −1 ) + α t −1, t

the parameter α t −1 =

1 t −1 ∑ pi t i =0

(9)

is the sample average and t −2

βt −1 =

∑ (p i =0

− α t −1 )(pi +1 − α t −1 ) t −2

∑ (p i =0

− α t −1 )2

the first order autocorrelation coefficient and always belonging to the interval [-1,1]. The SAC-rule could clear a time series of any first order autocorrelation, however taking some time to have effect. This rule will therefore play a very important part in matching the stylized facts as there was no autocorrelation present in the experiment. Model of evolutionary switching1 Predicting prices often is done by relying on →

previously realized prices pt ∈ {pt , pt −1,..., p0 } . Subsequently, these prices are used in predictors Hj є {H1,H2,...,Hn} which lead to estimations of future prices. Selecting the price predictor that results into the best estimates is a selection process that could be defined as evo-

This section focuses on continuous fractions which is of essence in our model. For a treatment of fixed fractions one should consult the thesis of Daan in ‘t Veld entitled: “A model with Heterogeneous Beliefs for Expectation Behavior in Cobweb Markets”. 1

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Econometrics

a − 6∑ ni ,t S( pie,t )

i =1

+ εt ,

(10)

where K=5 and equals the number of price prediction rules considered. Assessing performance of prediction rules will be done by adjusting equation (1) to resemble an increase in performance once the forecasting error is lower:

π i ,t −1 = −( pt −1 − pie,t −1 )2

(11)

Producers could decide on whether to give more weight on a more recent payoff of function (11), resulting in the following fitness measure:

Ui= wUi ,t −2 + (1 − w)π i ,t −1. ,t −1

(12)

The parameter w є [0,1] is the memory parameter meaning for w=0 there is “no memory” and the performance is minus the last prediction error. On the other hand if w=1 the producers have got “complete memory” and the performance is measured at time t-2. Fractions of producers change asynchronously according to their relative performance. These fraction dynamics will be described by the multinomial logit model, resembling the probability that a producer chooses strategy j. The following expressing denotes the fraction updating: 2

n= δ ni ,t −1 + (1 − δ ) i ,t

e K

γ Ui , t −1

∑e

γ U j , t −1

Figure 2: Simulated realized prices in stable (left) and strongly unstable (right) situation

lutionary selection. Applying this concept to our economic model is done by letting fractions of producers ni,t select the prediction rule that performs best at time t. The fractions are continuously updated in accordance with the “switching” between prediction rules by producers. The updating changes the realized market price and equation (4). A fraction ni,t of producer i at time t uses prediction rule i that predicts a price, determining optimal supply:

(13)

j =1

The parameter δ є [0,1] determines the extent of asynchronous updating and for δ=0 we are dealing with synchronous updating. Choosing δ=0.8 we have that 80 percent of the producers do not change their current heuristic at time t. The parameter γ≥0 is the intensity of choice parameter and measures how sensitive producers are to the difference in performance of the prediction rules. Equation (10) depends on the fractions ni,t at time t. In turn these fractions depend on the fitness at time t-1 and these depend on the market price at time t-1. The realized market price pt determines the new fractions and these lead to a new realized market price pt+1. Having finalized our model we will now proceed and try to match the stylized facts. Simulations The aim of our simulations2 is to match the stylized facts for a time series of 50 periods. First a “benchmark case” will be considered after which we will examine some extreme parameter cases and try to change our parameter according to our findings. The benchmark case is established by concentrating on the parameters in our model that are of the upmost importance, namely γ,w,ω,δ and v. We will examine the stable and strongly unstable case and try to match the stylized facts for these cases before afterwards adjusting our parameter values such that the unstable case can also be matched. Having found reasonable values for the benchmark case we were able to match the mean in both the stable and strongly unstable situation, but the variance was in both cases too low. The first order autocorrelation was matched in both

The simulations were programmed using the language Lua in E&F Chaos.

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Econometrics

.35

.30

.25

.6 .5

.20

.15

.10

.05 .00

.1 .0

ADAPT_EXP AVER_EXP

C1_5 C2_7

SAC_LEARN

ADAPT_EXP AVER_EXP

C1_5 C2_7

SAC_LEARN

Figure 3: Simulated evolution of fractions in stable (left) and strongly unstable (right) situation

cases, but higher order autocorrelations were very large in the unstable case. Proceeding by looking at two extreme cases, where respectively w=0 or δ=0 we were also not able to match the stylized facts. However, we did recognize that it could be possible to match the stylized facts by adjusting our parameter δ. We eventually get figure 2 for the realized prices in the stable and strongly unstable situation. The switching behaviour of producers is depicted figure 3, from which it is apparent that the SAC-rule and adaptive expectations rule are used the most in the stable case (left). In the strongly unstable case the switching increases, as the value of δ=0.2. Conclusion Modelling the stylized facts resulting from the experiment of Hommes et al. (2007) was pursued by building a model of evolutionary switching between prediction rules. Assuming agents to be boundedly rational and having heterogeneous expectations was crucial to this end. Determining the appropriate prediction rules was done by examining the effect of the prediction rules on the mean, variance and autocorrelations of the realized prices. Having built the model of evolutionary switching we were able to match all the stylized facts in the stable situation, as the mean and variance were similar to those of the experiment. Just like in the experiment there was no significant autocorrelation present. The unstable situation could only be matched for the mean and autocorrelations, since the variance was much lower than in the experiment. The strongly unstable case was matched for the mean, variance and first order autocorrelation. However, autocorrelations of orders higher than one remained significant. To match the stylized facts entirely further research has to be done regarding the autocorrelations by developing price prediction rules that wash away all the autocorrelations. Also

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November 2008

determining the price prediction rules of which the model should be comprised requires further examining. References Hommes, C.H., Sonnemans, J., Tuinstra, J. and Velden, H. van de (2007). Learning in cobweb experiments, Macroeconomic Dynamics 11(S1), 8-33. Hommes, C.H. (2006). Heterogeneous Agents Models in Economics and Finance, In Handbook of Computational Economics II: Agent-Based Computational Economics, edited by Leigh Tesfatsion and Ken Judd, Elsevier, 1109-1186. Hommes, C.H. (2008). Bounded Rationality and Learning in Complex Markets, forthcoming, Handbook of Economic Complexity, edited by J. Barkley Rosser, Jr., Cheltenham: Edward Elgar. Lucas, R.E. (1972). Econometric testing of the natural rate hypothesis, The Econometrics of Price Determination, 50-59. Muth, J.F. (1961). Rational expectations and the theory of price movements, Econometrica 29, 315-335. Sargent, T.J. (1993). Bounded Rationality in Macroeconomics, Clarendon Press, Oxford.

Wat doe je? als je weet dat de beroepsbevolking steeds kleiner wordt

De Nederlandse bevolking vergrijst, mensen leven gemiddeld langer en er worden minder kinderen geboren. Dit betekent dat de pensioenlasten zullen stijgen. Dit heeft ook ingrijpende gevolgen voor een verzekeringsmaatschappij. Binnen Achmea is de actuaris de aangewezen persoon om te berekenen en uit te leggen wat de vergrijzing betekent: wat zijn de financiële effecten van weten regelgeving? En wat betekent dit voor de prijs en premiestelling van producten? En hoe gaan we de in- en externe rapportage hierover zo efficiënt mogelijk inrichten?

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O ntz or g e n i s e e n we r kwoord

Econometrics

Estimation and Inference in Unstable Nonlinear Least Squares Models One doesnâ&#x20AC;&#x2122;t need any complicated data analysis to assert that human skills, along with capital and technology, have rapidly progressed over the last two centuries. While this progress may be viewed as a continuous innovation, it is necessarily the case that from time to time, some major innovation occurs, and this innovation triggers a major change in the behavior of individuals and firms. In general, an innovation by itself cannot promote progress unless a critical mass of people and firms adapt their behavioral patterns to incorporate its benefits into their every day life. Moreover, while the government and central banks can design policies to speed up the adaptation process of individuals to important changes, it is helpful if they knew the approximate period the average individual needs to incorporate this change in his or her environment.

The present article builds around this central idea of unknown time of change. While we may know approximately when technology has changed drastically, when climate change started to become an important issue, when the Federal Reserve Bank declares an economic recession, it usually takes time for the average individual to react to these changes. If central banks or environmental authorities knew in advance the duration of this lag, they could design more efficient policies in the sense that these policies would already incorporate a certain anticipated change.

at a monthly, quarterly or sometimes only yearly frequency, parameter changes often look abrupt rather than smooth, and the time of change used to be identified by prior identification of kinks in the data. Since in the past, these kinks were viewed as features of a questionable long-term unpredictable process in which all (cyclical) shocks have permanent effects on the economy, Perronâ&#x20AC;&#x2DC;s (1989) argument that they are in fact sudden parameter changes (temporary in nature because they would last until the next change occurs) was reassuring. However, inter alia Zivot and Andrews (1992) showed that identifying the kinks by eye-balling procedures is not necessarily enough to discriminate between different processes all characterized by kinks, and there is a need to rigorously assess evidence for and identify unknown change-points, if any, endogenously, through an appropriate statistical procedure. Moreover, as more data becomes available, it is plausible to encounter more than one unknown changepoint in the data.

1995 Nobel prize winner in Economics, Robert E. Lucas, has long recognized this issue. However, economic literature has lacked for a long time the statistical methodology to investigate all the aspects of unknown change-point problems.

In summary, the econometric literature was in need of rigorous statistical procedures to test for multiple unknown change-points, and to locate them once enough evidence for a certain number of break-points has been established.

In econometric literature, change-point problems are often modeled as changes in the parameters of the average individualâ&#x20AC;&#x2122;s optimization problem of maximizing e.g. lifetime consumption subject to a lifetime budget constraint. Due to macroeconomic data availability

Borrowing from the statistical literature of identifying change-points in e.g. temperature series, for very simple behavioral patterns, Bai and Perron (1998) propose a statistical procedure to find out through a certain data set whether any (possibly) multiple changes occur, and to date

Otilia Boldea is currently Assistant Professor in the Dept. of Econometrics and Operations Research, Tilburg University. She obtained her ME and PhD in Economics from North Carolina State University in 2008, and her current research is build around time series questions such as distinguishing between nonlinearity, nonstationarity, breaks and unit roots by means of rigorous econometric methodologies.

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Econometrics

these changes when they are present. They mathematically prove that this procedure attains some desired degree of accuracy needed e.g. for policy makers to get a (historical) idea about the lags with which some important policy they introduce will affect the average person’s consumption and investment decisions, and hence the economy. However, their model is ‘linear’, e.g. it is assumed that a change in the interest rate changes the parameters of a consumption decision of the average individual (an increase in the interest rate would in general motivate people to save more and consume less today) in the same way whether he is young or old, or that the Fed would respond by changing their policy in the same way in economic recessions versus expansions. This thesis extends the statistical procedures of Bai and Perron (1998) to ‘nonlinear’ models, e.g. where a change in the interest rate changes the consumption decision of an individual more when he is older than when he is younger, or when the Fed responds asymmetrically to expansions versus recessions. The extension to nonlinearity is important, as most economic models are inherently nonlinear, and researchers could up to now at their best linearize them to approximate behavior in the presence of major changes (‘change-points’, ‘breaks’ or ‘break-points’). In the present work, we propose three different statistical methods to test whether (multiple) breaks occur in the nonlinear behavior of the average individual, and we show that they have some desired degree of accuracy both mathematically and through simulation studies. The first test is for no breaks against a known number of breaks. Since knowing the number of breaks a priori is not a very likely setting, we also propose a test for no breaks versus an unknown number of breaks, and a test for an additional break. The latter is more useful in practice, since it can be used recursively until no more evidence of breaks is found. All these tests have non-standard features, but the decision rules are similar to that of Bai and Perron (1998). The novelty of our approach lies in showing with different statistical tools that the decision rules are indeed similar. If the researcher doesn’t find any breaks, he or she can assume that the change in policy was not drastic enough to shift the behavioral pattern of the average individual in an important fashion, where important is defined e.g. in terms of the consumption response to interest rates, or in terms of the interest rates response to output. The absence of breaks should not be interpreted as a negative finding, but rather as a feature of the data: the average consumer will still adjust its consumption response to a

change in the interest rate, but in the same way as before, with no sudden change in his or her response, likely due to a critical mass of people believing that the change in policy is more or less in line with their expectations. If, on the contrary, there is evidence of breaks, the test we developed for an additional break allows us to establish the number of breaks in the data. Knowledge of the number of breaks allows us to propose a very simple method to date these breaks. More exactly, we propose searching over all possible break configurations until the right one is found; once the breaks are dated, they are imposed and all the conclusions of the model can be carried out as if the breaks were known a priori. This is not a new result in the literature but is different than eyeballing because it involves a rigorous statistical procedure. The novelty of our approach is showing both theoretically and through simulations that this is possible in a ‘nonlinear world’ as well because the break-points are detected with high accuracy even for small samples. The accuracy is statistically derived through bridging between two ‘worlds’: a linear one with breaks, and a nonlinear one with no breaks, and simulations show that only once in one hundred occurrences would we date the break incorrectly, and if so, it would be only one time period off the true one. Regarding computational aspects, it might seem that searching over all break configurations is time consuming when the dataset contains many observations (e.g. many years, quarters, months). However, the nature of our problem allows us to use a dynamic programming algorithm that greatly simplifies the computation so that such caveats are eliminated. Dating the breaks is helpful for policy makers not only from a historical perspective, but also from a prediction point of view, since for accuracy of prediction, dating the last break is crucial. Through simulations, we show that our method works well for significant changes (breaks) in behavioral patterns of the average individual, and is also not completely uninformative for small changes. As an application, we use data on US interest rates, inflation, aggregate income rates and other macroeconomic indicators to test whether breaks occurred in the behavioral patterns of the average consumer. More specifically, we use a so-called ‘interest rate reaction function’, model that is frequently used by the Fed to assess the response of interest rates to inflation and production (output, income) gap. Our nonlinear setting allows us to consider cases where this response is asymmetric in the sense

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Wat als zijn verzekeringsmaatschappij ook niet gezond is? Die duik naar de bal had hij toch iets te wild ingezet. En die tegenstander was steviger dan hij dacht. Het resultaat: zes weken gips en fysiotherapie. Vervelend, maar het is geen ramp. Hij is immers goed verzekerd. Maar wat als zijn verzekeringsmaatschappij geen geld uitkeert? Omdat het er simpelweg niet is? Daarom houdt de Nederlandsche Bank (DNB) toezicht op de soliditeit van financiële instellingen. We stellen eisen aan verzekeraars, banken en pensioenfondsen en houden de vinger aan de pols. Toezicht houden is niet de enige taak van DNB. Als onderdeel van het Europese Stelsel van Centrale Banken dragen we ook bij aan een solide monetair beleid en een zo soepel en veilig mogelijk betalingsverkeer. Zo maken we ons sterk voor de financiële stabiliteit van Nederland. Want vertrouwen in ons financiële stelsel is de voorwaarde voor welvaart en een gezonde economie. Wil jij daaraan meewerken? Kijk dan op www.werkenbijdnb.nl. | Juristen | Accountants | Actuarissen

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Econometrics

that the Fed’s response will be asymmetric in a recession versus an expansion (e.g. when production and national income are high compared to when they are low). More exactly, we expect the Fed would reduce interest rates more to get the economy out of a recession as compared to the increase in interest rates needed to stop an overheated expanding economy. In addition to the asymmetry mentioned above, the nonlinear design we consider allows us to nest sudden changes with smooth transitions from one regime to another (e.g. recession to expansion or vice-versa), and this transition can be generated by a certain variable (other than time) reaching a threshold. This feature is empirically attractive due to the fact that

line with other nonlinear studies. We also find that the transition is quite different before and after the change, implying that the beginning of 1980’s has indeed marked a significant change in the structure of the US economy. In conclusion, we provide a comprehensive battery of tests and estimation procedures to assess the evidence for breaks and to date them if they occur. While our method was illustrated through two very specific examples, it has a much further reach. At a regional level, it could also be used to assess evidence for the lags with which major changes in the tax code can affect tax payers, changes in voting laws can affect voters, a.s.o. Moreover, our method’s reach branches out of economics, since it can be used

“Only around 1985 a critical mass of individuals has decided to change its behavior in reaction to economic events” the usual confusion between nonlinearity and breaks no longer occurs, since our procedure can nest both. Using US data from 1984-2002 to make our results comparable to other studies, we find evidence of at least one break-point in 1985, presumably some years after the oil price shocks at the beginning of 1980’s, and the drastic interest rate policy changes the Fed employed around that time. We check the robustness of our findings by comparing our model to a model of smooth transition where the main variable causing the transition is time, hence where the average individual would adapt gradually to changes in the policy regime because overall each agent takes a long time to adjust. We find that a sudden change approach is more accurate in this setting, meaning that only around 1985 a critical mass of individuals has decided to change its behavior in reaction to economic events. This is not surprising since most economists agree about the occurrence of the 1980’s ‘Great Moderation’, a big change in the structure of the economy manifesting itself in less volatile interest rates, followed (later) by less volatile output and a stable economy.

e.g. to assess major changes and more accurately predict nonlinear models for pollution levels, temperature, humidity and other factors of interest for climate change monitoring both at a local and at a global level. References Bai, J. and Perron, P. (1998). Estimating and Testing Linear Models with Multiple Structural Changes, Econometrica, 66, 47-78. Perron, P. (1989). The Great Crash, the Oil Price Shock and the Unit Root Hypothesis, Econometrica, 57, 1361-1401. Zivot, E. and Andrews, D. W. K. (1992). Further Evidence on the Great Crash, the Oil Price Shock, and the Unit Root Hypothesis, Journal of Business and Economic Statistics, 10, 251270.

The sudden change we find does not impede us to consider a smooth transition regime before and after the change, but in each case this transition is caused by a certain variable such as the lagged change in interest rates reaching a certain threshold. This transition feature is in

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ORM

A Generic Scheduling Method Scheduling problems are known to be NP-hard; non-polynomial solvable. Though automatic scheduling could save much time, there still is no polynomial algorithm to solve a scheduling problem. This is the reason that a number of specific scheduling problems have been studied intensively over the last few decades. Different methods are developed to resolve specific problems, such as branch-and-price for the aircrew-scheduling problem. As scheduling methods have been developed solely to address specific problems, a wide variety of scheduling methods exists. Variants of linear programming, constraint programming, dynamic programming, vehicle routing heuristics and local search methods were used to solve scheduling problems. A scheduling problem has to be studied exhaustively before it is possible to select a scheduling method or before it is clear that a new algorithm must be developed. Intensively studying a scheduling problem is very costly with regards to both the required time and specialists. This is the reason that a desire exists to develop a generic scheduling method.

Petra Veerman has just finished her masterâ&#x20AC;&#x2122;s thesis for the Masterâ&#x20AC;&#x2122;s of Operations Research and Management. She wrote her master thesis during an internship at Logica under the supervision of dr. C.W. Duin of the University van Amsterdam (UvA.) This article is a global summary of her thesis.

Scheduling problem The first step in developing a generic scheduling method is to define the scheduling problem. The most general description of the scheduling problem is assigning the resources to tasks over time and place. Figure 1 provides a schematic overview of the scheduling problem. Resources can be people, machines or raw materials. The first two are renewable; after one task they can be utilised to perform another. Raw materials are in most cases non-renewable, which means that the lifetime of these materials do not extend further than one task. This study concentrates on renewable resources.

Figure 1: A schematic problem overview

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Perhaps this can be extended to non-renewable resources in further research. Each resource has one or more qualifications and availability specifications in time and place. It is possible that the execution of a task starts on a specific time, but it can also be the case that the time-interval in which the task must be performed is larger than its duration. In this research results are evaluated for tasks that have a specific starting time, but the scheduling method can be extended to tasks that have a time-interval in which to be performed. Each task needs resources with specific characteristics. These characteristics are represented in the specifications of the task. The place at which a task is performed can vary from an entire city to solely within a section of a company. In this research all tasks are executed at a single place where all resources are available. The scheduling method presented below can easily be extended to a situation of multiple places of execution. So this research aims to solve a scheduling pro-

ORM

blem in which the tasks have a specific execution time and are all performed at the same place. The required resources are all renewable. Scheduling method It is difficult to work with a general problem description and even the problem considered in this research has many exceptions. The scheduling method developed in this study was tested on the problem described above, but has been developed in such a way that it can also be used to solve other scheduling problems. The scheduling method is a combination of the methods used for specific scheduling problems. This approach combines all the good qualities of the specific methods. The scheduling method that then arises is called the variable and value ordering heuristic. This method first orders the variables and then selects only the ‘best’ one. In the terms of scheduling problems the variable is a task. Selecting a task is done based on objectives, which can differ across different scheduling problems. Once a variable has been selected, the method selects a value for it. This value represents one of the possible resources for the task. Before this resource is definitive, a number of constraints are checked. All possible constraints can be taken into account. In the case of scheduling problems an example would be availability constraints. If the resource for the task passes the constraint checks, the domains of other variables must be updated; a resource is no longer available for another task on the time the selected task is executed. Once completed, the next variable can be selected and this process continues until values are selected for all variables. Figure 2 is a schematic overview of the variable and value ordering heuristic.

This scheduling method allows flexibility of objectives when selecting a variable, in setting the objectives that determine a value for that variable and in the constraints that are to be checked. That is why this scheduling method can be used for a general scheduling problem. Tests and results Two test cases were developed in order to test the variable and value ordering heuristic. These test cases represent scheduling problems similar to the problem described above. The first test case is small, containing only a few tasks and resources. This makes it possible to intensively analyze the results. The second test case involves many tasks and resources and so enables us to analyze the computation time. An early test shows that the objectives on which the selection of a variable and a value are based have an influence on the schedule. Further research leads to the following objectives: • Objectives to select a variable: o Task that utilises the least amount of resources to perform o Task with the longest duration • Objectives to select a value: o The resource with the lowest workload o Number of other possible tasks able to be performed with this resource A second test with these new objectives gave better results; in very little time almost all tasks were scheduled. Finally the results can be further improved by adding a local search method to the variable and value ordering heuristic. The local search method that is most suitable for scheduling problems is the simulated annealing heuristic.

Figure 2: A schematic overview of the variable and value ordering heuristic

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In this method a neighborhood is defined by one of the following four possible changes: • Delete a task from the current solution • Mutual replacement of tasks • Delete all tasks of a resource • Replace all tasks of a resource with that of another resource A neighborhood is randomly selected. If the new solution passes the constraint check then the deleted tasks are again planned with the variable and value ordering heuristic. Conclusion The variable and value ordering heuristic is a flexible scheduling method based on a combination of positive qualities of methods provided by the solutions to other specific scheduling problems. Though at first sight the variable and value ordering heuristic demonstrates promising results, further testing of the scheduling method is required. The results must also be compared to traditional methods before a definitive conclusion can be drawn with regards to the performance of the scheduling method.

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Wie schrijft blijft? © 2007 KPMG Staffing & Facility Services B.V., een Nederlandse besloten vennootschap, is lid van het KPMG-netwerk van zelfstandige ondernemingen die verbonden zijn aan KPMG International, een Zwitserse coöperatie. Alle rechten voorbehouden.

Schrijf je scriptie of afstudeeropdracht bij KPMG. Eerlijk is eerlijk, niet iedere tekst is even goed. Maar vaak zit er iets slims of moois tussen. En heel soms iets onvergetelijks. Zo is dat op de wc-deur van een kroeg en zo is dat bij KPMG, waar studenten als jij een scriptie of afstudeeropdracht kunnen schrijven. Zo’n scriptie of afstudeeropdracht is een ideale manier om kennis te maken met KPMG. Misschien zelfs het begin van een prachtige carrière: geef je al schrijvend blijk van passie voor het vak, dan moet je maar eens serieus overwegen om te blijven. Meer weten? Kijk op www.kpmg.nl/stages.

A U D I T  TA X  A DV I S O R Y

Actuarial Sciences

Ageing, Schooling and Economic Growth: Standing on the Shoulders of Giants In a recent paper Ben Heijdra (University of Groningen) and I investigate the effects of demographic shocks on the macroeconomic performance of a small open economy. The shocks are of the type and magnitude observed in The Netherlands in the 20th century. For the analysis we developed a simple analytical growth model in which the schooling decision depends on the knowledge of the teachers and a higher educated population results in higher economic growth

Ward Romp is a postdoctoral researcher at the Faculty of Economics and Business of the University of Amsterdam. His research focuses on the macroeconomic of ageing and pensions. His position is financed by MN Services and Netspar.

Ageing The western world is ageing rapidly. For example, in the Netherlands, life expectancy at birth rose from 71 years in 1950 to 80 years in 2007, whilst the average number of children per woman dropped from 3.1 to 1.7. Because infant mortality stayed relatively constant during that period, the increase in longevity must be attributed to reduced adult mortality. A similar demographic pattern can be observed for most OECD countries. Demography The first step in the analyses was to get the ageing process right. There are two workhorse models in the macroeconomic literature to analyse ageing: the discrete time DiamondSamuelson model and the continuous time Blanchard-Yaari model. Both models have their strengths and weaknesses. The DiamondSamuelson model is easy to generalise such that it fits observed mortality patterns, but the number of variables grows exponentially in the number of age groups. These models tend to become so complex that the term ‘black box’ is usually applicable. The second group of models builds on the Blanchard-Yaari continuous time framework. The strength of these models is their tractabi-

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lity. Unfortunately, these models are tractable because of the so-called perpetual youth assumption. Once born, an individual has a constant probability of dying, which implies that the expected remaining lifetime is the same for everybody. This is not an attractive feature if one wants to analyse ageing. Figure 1 shows the surviving fraction of Dutch males born in 1930. The dotted line indicates the best approximation of the perpetual youth model. It is clear that the fit is not great. In a previous paper (Heijdra & Romp 2008) we extended the standard Blanchard-Yaari framework with a non-constant mortality rate. The solid line in Figure 1 shows the surviving fraction according to the so-called GompertzMakeham law of mortality. Although not perfect, the fit is much better, especially for the ages 20–60. We showed that – as long as the discount rate is constant – solving the model boils down to being able to quickly calculate a demographic discount function. This function calculates the present value of an income stream discounted with an actuarially fair discount rate. The beauty is that this function reduces to widely used functions in statistics such as the gamma function for a number of wildly used mortality profiles among which the already mentioned Gompetz-Makeham. So solving the extended Blanchard-Yaari model with a Gompertz-Makeham mortality function boils down to being able to calculate the gamma function and there are very efficient algorithms to do this! With the Heijdra-Romp extension of the Blanchard-Yaari model we can analyse the effects of ageing in a simple, tractable model.

Actuarial Sciences

Marginal costs Marginal benefits Low Mortality

Figure 1: Surviving fraction (without child mortality)

Schooling The second building block of our analysis concerns the engine of growth during the demographic transition and possibly also in the long run. Following Lucas, we assume that the purposeful accumulation of human capital forms the core mechanism leading to economic growth. More specifically we assume that people accumulate human capital (they go to school) by engaging in full-time educational activities early in life. The start-up education period is chosen optimally by each individual and labour market entry is assumed to be irreversible. Depending on the parameter setting, the human capital production function (or training function) may include an intergenerational external effect of the “shoulders of giants” variety,

Marginal benefits High Mortality Optimal

Time spent

Schooling Figure 2: Mortality and optimal schooling at school

value of the higher wage increases. Figure 2 shows the effect of a lower mortality rate on the optimal schooling decision. The horizontal axis shows the time spent at school. The dotted line through the origin depicts the marginal costs. It is upward sloping because the wage increases as someone is ‘wiser’. The curved lines show the marginal benefits of going to school. As the mortality rate is lower (the straight line), the optimal time spent at school increases.

"if mortality rates drop, people will spend more time at school" as first proposed in an overlapping generations context by Azariadis and Drazen (1990). With an operative externality, an individual’s training function depends positively on the economywide stock of human capital per worker in that individual’s birth period. That is, the smarter the teachers are, the more you learn spending one more year at school. As always in economics, people go to school as long as the marginal benefits exceed the marginal costs. The marginal benefits of spending one more year at school consist of the present value of a higher wage. The more you know, the more productive you are, the more you earn (that’s the assumption). The marginal costs of going to school consist of the net wage you could have earned (minus a possible scholarship). Apparently the costs of going to school do not depend on mortality, but the benefits are. If one expects to live longer, the actuarially fair discount rate drops and the present

Schooling and growth In Figure 2 we showed that if mortality rates drop, people will spend more time at school. This in itself leads to a more productive labour force, but also increases the effectiveness of schooling of the young – the standing on the shoulders of giants effect. There is however an offsetting effect. If people live longer, population growth increases and total knowledge is spread over more people. Our interpretation is that only a fraction of the people may work in the education sector and more people increases the student/teacher ratio and hence lowers the quality of the schooling system. The total effect of longevity on economic performance is ambiguous. Whether or not the externality might lead to a permanent change in economic growth depends on the strength of the ‘standing on the shoulders’ effect. In the paper we show that only for a knife-edge case where this intergeneratio-

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Actuarial Sciences

nal externality parameter is exactly one, that smarter teachers may lead to higher growth forever. This is the case usually considered in the literature. However, using the recent empirical study by de la Fuente and Doménech (2006), we argue that a plausible value for the intergenerational externality parameter lies between 0.27 and 0.40, i.e. nowhere in the vicinity of the knife-edge case. The factual evidence points firmly in the direction of positive but strongly diminishing returns to the intergenerational external effect. Ageing Given that schooling does not have a permanent growth effect, we used our model to study the transition from one steady state to another after a baby bust and a longevity shock. A reduction in the birth rate and an increase in longevity (due to reduced adult mortality) both increase the steady-state per capita human capital stock but have opposite effects on the population growth rate. Using a plausible calibration of the model, we demonstrate that the effects of the baby bust on human capital and the labour force participation rate are quantitatively significant. In stark contrast, even a rather large longevity shock hardly affects these variables at all. The effect is much larger for the baby bust because the drop in the population growth rate reduces required human capital investment (i.e. human capital investment that is needed to endow each newborn worker with the same amount of this type of capital). In contrast, for the longevity boost, the schooling effect increases human capital per worker but the slight increase in the population growth rate decreases it somewhat, rendering the total effect small.

nerations of finitely lived agents facing realistic age-dependent mortality profiles. Among other things, the paper highlights the crucial role played by the strength of the intergenerational external effect in the training function faced by individual agents. Provided this external effect is non-zero, as the empirical evidence suggests, the vintage nature of the model gives rise to very slow and rather complicated dynamic adjustment. This feature of the model may help explain why robust empirical results linking education and growth have been so hard to come by. References Azariadis, C., Drazen, A. (1990). Threshold externalities in economic development. Quarterly Journal of Economics, 105, 501–526. Boucekkine, R., de la Croix, D. and Licandro, O. (2002). Vintage human capital, demographic trends, and endogenous growth. Journal of Economic Theory, 104, 340–375. de la Fuente, A. and Doménech, R. (2006). Human capital in growth regressions: How much difference does data quality make? Journal of the European Economic Association, 4, 1–36. Heijdra, B. J. and Romp, W. E. (2008). A lifecycle overlapping-generations model of the small open economy. Oxford Economic Papers 60, 89–122.

In our numerical analysis we extend the literature in that we are able to compute the transitional dynamics also for shocks affecting the optimal schooling period (such as the fiscal and longevity shocks). In contrast, Boucekkine et al. (2002, pp. 363-365), only show the adjustment path in the (endogenous) growth rate following a drop in the birth rate. Such a shock leaves the optimal schooling period unchanged, so that all transitional dynamics are entirely attributable to changes in the growth rate of the population. In our model we find that, for all shocks considered, the transitional adjustment is rather slow and often non-monotonic. Transition periods of more than 100 years are not uncommon. Conclusion We have studied how demographic shocks affect the macroeconomic performance of a small open economy populated by disconnected ge-

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Puzzle

Puzzle At the end of this Aenorm, itâ&#x20AC;&#x2122;s time for a bit of relaxing with a few puzzles. But let us first give the solutions to the puzzles of the previous edition.

A Chinese switch-word puzzle

Making a perfect square The solution is given out below. First clip off the little pieces 1 and 2 and pack them into the center. Then cut the zig-zag steps and move piece 4 down one step. The four pieces will fit together so as to make a perfect square.

Not only numbers inflict a great challenge to econometricians, but also words are quite challenging as this puzzle demonstrates. There is supposed to be a letter placed upon each of the twelve movable blocks in the picture above, which reading from top downward spell a correct word. Can you find a word, which by sliding the blocks into the horizontal groove makes the same word from left to right? Although this puzzle is originally Chinese, answers to it are preferred in English or Dutch. Solutions

Six bottles of beer on the wall The professor sold the 13 and 15 gals of wine at 5 Euro per gallon = 140 Euro. He also sold the 8, 17 and 31 gals of beer at 2,50 Euro = 140 Euro. So he had the 19 gals barrel left, which was worth 47,50 Euro or 95 Euro according to whether it contained beer or wine.

Solutions to the two puzzles above can be submitted up to December 1st. You can hand them in at the VSAE room; C6.06, mail them to info@ vsae.nl or send them to VSAE, for the attention of Aenorm puzzle 59, Roetersstraat 11, 1018 WB Amsterdam, Holland. Among the correct submissions, one book token will be won. Solutions can be both in English as in Dutch.

American election With the upcoming US presidential elections in mind here is a nice opportunity for us econometricians to show the American people an example of how a fair count of the votes is performed. This time there are not two but four candidates to become president of the United States. A total of 5,219 votes were cast and the winner exceeded his opponents by 22, 30 and 73 votes. How many votes did they all receive?

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Facultive

Free University Amsterdam

University van Amsterdam For the VSAE as well the new study year has started. In August we began this year with the introduction of our new freshman to the VSAE and the university. More than a hundred new econometricians-to-be have started their study at the University of Amsterdam this year. Perhaps not all of them will become Master of Science, but hopefully lots of them do. In October we organized the National Econometricians Football Tournament. Approximately 200 students in econometrics gathered in Amsterdam to find out who had the best team. After a long day of hard work, a team from Tilburg beat a team from our own university in the final. They celebrated this victory in the centre of Amsterdam until the last train home departed. In November a group of 50 VSAE-members will visit Prague for five days. After Milan and Copenhagen in the past two years, we hope that this trip will turn out equally successful! The “Actuariaatcongres” – a congress on actuarial sciences – is scheduled for the 17th of December. The central theme of the congress is ‘The Transparency of Insurance Products’ and it will be held at the Okura Hotel in Amsterdam. Agenda

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Agenda 13 November 2008 Inhouse day Saen Options 26 November 2008 Bowling with the VSAE

7-24 January 2009 Study Trip

20 – 25 November Short Journey Abroad

In the upcoming months we have planned a bowling tournament with our fellow students of the VSAE and Saint Nicholas will visit to Kraket. Last but not least we organize a study trip. In January a group of seventeen of our best students will go to New York and Atlanta to visit universities and companies.

December 2008 Saint Nicholas

11 November Monthly free drink

17 December Congress on Actuarial Sciences: Transparency of Insurance Products

With the start of a new academic year the study association Kraket has a new board. The board now consists of only four members instead of five in the past years. The academic year has started very well for Kraket, with a lot of new members and a successful introduction period. Our first activities were a success too. In September we organized both a kart and a pool tournament. A month later the National Econometricians Football Tournament took place, which was successfully hosted by our colleagues of the VSAE.

The

((4 + 3)² + 4) / 1/4 = - - 11 + 6³ = - - -

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√576 + 11 x 3² = - - √196 = - -

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