Introduction My final paper’s research objective is to locate the presence of information induced illiquidity in stock prices using CAPM. In order to do this I will attempt to locate and is possible correct for the presence autocorrelation. What is the CAPM theory? The CAPM theory, or Capital Asset Pricing Model theory, gives a precise prediction of an asset’s expected return based on its risk. (Bodie, 2007). The CAPM theory like any economic model in order to illustrate fundamental relationship amongst economic driver must simplify the real world with generalized assumptions. For CAPM, these assumptions are: first, investors are price takers. Second investors are myopic, that is, they plan their investments for only holding period. Third, investments are limited to a universe of publicly traded assets, which basically means non-tradable assets such as land and labor are not included. Fourth, investors suffer no transactions cost and taxes when investing. Fifth, all investors are mean-variance optimizers, that is, they would pick the highest ratio of returns they would get for an asset and the risk they undertake for the asset. Finally all investors analyze in the same way, and share the same view in investing. They are also privy to identical information about the market and available assets. Without going into the complicated mathematical derivations of CAPM, the implications and conclusions of CAPM are that: all investors will choose to hold the same portfolio, called the market portfolio, and will include all assets. This market portfolio is also a tangent point on the efficient frontier of investment that creates the highest capital allocation line. The systematic risk is the market portfolio variance, or in layman’s terms the risk of the market portfolio does not come from the risk of individual assets but from the market of individual assets. Lastly, the risk premium or the extra return to compensate for risk, of individual assets will be proportional to the risk premium on the market portfolio, and the beta coefficient of the asset relative to the market portfolio. Beta measures the extent to which returns on the stock and the market move together. The basic CAPM equation is:
where is the asset (a) expected return, is the risk free rate, is the market rate of return, and is the beta coefficient of asset (a) As mentioned, like all theories, in order to explain the basic rationale behind asset returns CAPM had to simplify the world. In particular, I would like to point out the sixth assumption of similar views and information availability. It is intuitive the not all investors share the same investment
view and/or is privy to the same information, in the real world, to make the same decisions. In short there is information asymmetry. Background One form of information asymmetry is that of an information lag. Information lag then is a form of illiquidity because relevant information about the asset is not yet reflected in the asset price. Basically information lag causes mispricing of assets. The motivation of finding the correct price is obvious. Mispricing would create undervalued or overvalued assets. This makes market’s and investor’s portfolio inefficient. Inefficiency in investing leads to losses for investors. In order to test for the existence and if possible correct the information asymmetry, I will estimate the beta of small capital stocks and see if there is autocorrelation. Autocorrelation of the beta coefficient according to the literature (Ibbotson, 1996) and intuitively is a sign of information lag. It is basically stating the beta of previous periods still affect the present period. Basically because there is a delay of previous time period’s information reaching the investor (because the investor is not privy to it or merely because there is a too high cost for the information), the investors’ reaction (as reflected on the asset’s price) is delayed as well. The use of small capital stocks is clear; because information about small capital stocks is not readily or quickly available for all the markets. As such the likelihood information lag, as represented by the presence of autocorrelation, is greater than picking well known (large) capital companies. It is important to note that in picking small companies there is an extra-benefit (and motivation) if the information lag is found or corrected. Small companies might carry greater risks than their beta suggests. This means that investors are over-valuing the small capital stocks, that is, investors are not being compensated for the extra risk they are taking when buying small capital stock. The implication of such as study is to stress the importance of information and timely information delivery. If estimation of important financial variable does cause mis-valuations, which cause financial losses, then policy makers have incentive to remove barriers which cause information asymmetry. Model Basing the model on the literature from Ibbotson et al. I start with the basic CAPM model:
I then find the risk premium of the asset by subtracting the risk free rate from the asset rate of return. It is important to note that the market risk premium is the market rate of returned subtracted the risk free rate of return.
where is the risk premium of the asset (small capital stock), is the risk premium of the market and is the asset (small capital stock) beta I will basically run a regression with the risk premium of the small capital stock as the dependent variable, the risk premium of the market as the independent variable and the beta my parameter. I will then test for asymmetry which related literature (Ibboston, 1996) has predicted and shown to exist. Since I am basing my model on CAPM, I will not be using any other variable (other than the presence of autocorrelation) to test for liquidity. However if I had been using APT (asset pricing theory) I would have tested for the presence of liquidity (or illiquidity) using bid-ask spread or volume. (Amihud, 1991) I do not feel I should be using any other variable because one of the implicit conclusions of CAPM is that asset prices already reflect relevant information on the asset. Data The data I will be using will be from Blomberg historical market data from 1998 to 2006, stratified into quarter (3 month) ranges. The small capital returns will be from actual returns of the S&P 600 Small Cap while the market return will be from actual returns of S&P 500. The S&P 600 Small Cap is a portfolio collection of small capital stocks, while the S&P 500 is a common proxy for market returns. The risk free rate is from the Federal Reserve, using historical Treasury bill rate with 3 month maturity. Both S&P 500 and 600 Small Cap are in dollar prices, I therefore had to change them into returns by subtracting previous prices from present prices and divide it by the previous prices. Mathematically:
where is the return, and is the price at time n. Of course during the time span of the series many extraneous (that is, unexpected or not within the regular economic/business cycle) events have occurred. Examples of such event would be the Internet Bubble Burst of 2001, the events of September 11, and the Iraq War which started in 2003. However as stated before these events should be reflected in the prices of assets not only because
of CAPM but because extraneous events are often large and very, very well known (that is, there is no delay involved). I am hypothesizing that the limited available data will make results diverge from previous studies (which I would like to point is an example of information asymmetry). My ideal data set would have included a longer time span of asset returns as well as a more formal grouping of capital size. Estimation, Results and Analysis1 The original estimation results are as follows: F-statistic
<.0001
R2
.7268
Beta Parameter Estimate
1.02882
Beta t Value
9.37
Beta p(t Value)
<.0001
First I tested the significance of the coefficients together. : β β 0; : ! " ! #$ % Second I tested for the significance of the coefficient. : 0 : & 0 I set my significance level to .05%. Since the F-statistic is <.0001 is less than the significance level then I can conclude that I can reject the null and say at least one coefficient is statistically different from zero. The probability of the Beta t-statistic as computed by SAS is <.0001, since it is less than the set significance level, I can reject the null hypothesis and say that the Beta coefficient is statistically significant from zero. As can be seen the R2 is also quite high (at .7268). This means that about 72.68% of the change is small capital risk premium is caused by changes in the market risk premium. Knowing the beta coefficient is significant and its estimated value is 1.02882, I can say that a percent change is the market risk premium leads to 1.02882 percent change in the small capital portfolio risk premium. I then tested for autocorrelation using SAS, specifically proc arima, proc autoreg with an nlag =1, durbin-watson and plotting.
1
Please refer to appendix for complete result and SAS code.
The results are as follows: proc arima chi-squared
.4429
proc autoreg t-Value nlag=1
-1.52
Setting the significance level at .05%, I looked at the proc arima results using decision rule, : $ ' : $ ' I concluded that since the p-value is greater than the significance level I cannot reject ( , and must say that there is no autocorrelation. However it is important to note that using proc autoreg the lowest lag tested is a value of nlag=6. Therefore a single period autocorrelationâ&#x20AC;&#x2122;s effect might be captured and thus undervalued by the other five periods using proc arima. I tested for autocorrelation next using proc autoreg with nlag=1. My results give a t-statistic Value of -1.52. If I use the significance level of .05% then I would have to reject once again the null hypothesis that there is no autocorrelation of order one, however if I looked at critical t-statistic Value at .10% (1.310) then I cannot reject the null hypothesis and must that say that there is autocorrelation of the first order. I give my t-statistic Value for autocorrelation order one, the benefit of the doubt because (1) the graph2 I plotted (the residual against time) showed some evidence of autocorrelation because of the concave pattern it showed. (2) Based on the literature there should be autocorrelation in beta estimate of small capital stocks. The model corrected for first order results are: F-statistic
.0016
R2
.7957
Beta Parameter Estimate
1.1189
Beta t Value
11.86
Beta p(t Value)
<.0001
First I tested the significance of the coefficients together. : β β 0; : ! " ! #$ % Next I tested for the significance of the coefficient.
2
Please refer to appendix for graphs as well
: 0 : & 0 I set my significance level to .05%. Since the F-statistic is .0016 is less than the significance level then I can conclude that I can reject the null and say at least one coefficient is statistically different from zero. The probability of the Beta t-statistic as computed by SAS is <.0001, since it is less than the set significance level, I can reject the null hypothesis and say that the Beta coefficient is statistically significant from zero. As can be seen the R2 is also quite high (at .7957), though still very much similar to original. Knowing the beta coefficient is significant and its estimated value is 1.1189, I can say that a percent change is the market risk premium leads to 1.1189 percent change in the small capital portfolio risk premium. I would also like to note that in neither the original nor the corrected modelâ&#x20AC;&#x2122;s intercept coefficient were significant at any reasonable levels. [β t-statistic Value original =0.1422, β t statistic Value original .2103: I believe the differing results show that autocorrelation and thus information lag is present in the model. First the R2 of the corrected is larger suggesting greater explanatory ability of the corrected. Not only that the beta coefficient estimate also increased from 1.02282 to 1.1189. From the literature (Ibbotson, 1996) it can be seen the small capital stock beta estimates are too low because of unrepresented risk (unrepresented, because of information illiquidity). While decimal changes in the beta coefficient do not seem much, it is important remember that this beta affects the pricing of assets. Even minute changes of prices when used in transaction of very large amounts can result into large losses or very profitable arbitrage opportunities. Finally I attempted to correct my model by mimicking my main literature, which was adding a lagged beta coefficient in order to capture the information illiquidity. Mathematically: ; < , ,
where the lagged beta coefficient and lagged risk premium is noted with a t 1. The results of my attempt at following correction, as well as the tests I did to check for the existence of problems in the new model are as follows: F-statistic
<.0001
R2
.7490
Beta t=0 Parameter Estimate
1.01967
Beta t=0 t Value
9.44
Beta t=0 t p(Value)
<.0001
Beta t=-1 Parameter Estimate
-0.09376
Beta t=-1 t Value
-.87
Beta t=-1 t p(Value)
0.3932
proc autoreg t-Value nlag=1
-1.13
proc corr rho
.5639
White-Pagan chi-squared
.2420
Breush-Godfrey chi-squared
.7435
First I tested the significance of the coefficients together. : β β 0; : ! " ! #$ % Next I tested for the significance of the coefficient. : 0 : & 0 I set my significance level to .05%. Since the F-statistic is <.0001 is less than the significance level then I can conclude that I can reject the null and say at least one coefficient is statistically different from zero. The probability of the Beta t=0 t-statistic as computed by SAS is <.0001, since it is less than the set significance level, I can reject the null hypothesis and say that the Beta t=1 coefficient is statistically significant from zero. The probability of the Beta t=-1 t-statistic as computed by SAS is .3932, since it is more than the set significance level (by a large amount, I have to add), I cannot reject the null hypothesis and say that the Beta t=-1 coefficient is not statistically significant from zero. As can be seen the R2 is also quite high (at .7490). This means that about 74.90% of the change is small capital risk premium is caused by changes in the market risk premium. Knowing the beta t=0 coefficient is significant and its estimated value is 1.01967, I can say that a percent change is the market risk premium at t=0 leads to 1.01967 percent change in the small capital portfolio risk premium. Beta t=-1, on the other hand, when estimated was found to be not statistically significant from zero so we can say that the previous period market risk premium does not influence small capital portfolio risk premiums significantly. I then tested for the presence of autocorrelation of the first order, multi-correlation and heteroscedasticity. For the first order autocorrelation test since the calculated t-statistic Value is less than, even the 90% confidence interval critical value of 1.390, I would say that there is no first
autocorrelation. For multicollinearity, since rho is .5639 is less than our standard .70, I would have to say that there is no significant multi-correlation between the t=0 and t=1 market risk premiums. Finally using the heteroscedasticity tests of White-Pagan and Breush-Godfrey I found that neither have a value that is less than any reasonable significance level, therefore I must conclude that there is no heteroscedasticity in the model. [Note that as before I did not test for misspecification, the rationale behind this (1) I am just basically following a model created by more knowledgeable financial economist. (2) The basic model CAPM, is a fundamental theory of finance.] My results seem to deviate from the literature (Ibbotson, 1996) and contradict each other. First their proposed correction did not work for me. I would however point to data difference (not to mention expertise difference). For example the literature (Ibbotson, 1996) had a longer time span as well as capital weighted not only between small and large, but into ten groups from smallest to larger capital stocks. Second while the Beta t=-1 t-statistic is insignificant there is no first order autocorrelation which seems to suggest that while insignificant the Beta t=-1 Coefficient does capture some of the effect of previous period information lags. Conclusion As can be seen autocorrelation does exist for small capital stocks, making their CAPM beta too small. This means small capital stocks are often overvalued; and investors are not being compensated for risks which they are undertaking when buying such stocks. My results follow previous studies as well as follow intuitive financial logic. If I could do research project again, I would want to try to get better data, specifically data from a longer time span as well as to be able to separate different size capital stock into different sets, so that may be able to see if there is (and if there is, the extent of) information illiquidity. I would hypothesize that larger and more known company stocks have less or no autocorrelation having less information lag, while the smallest capital company stock has the most autocorrelation (e.g. order beyond the first) because intuitively they would have the most information lag. In the real world, if I had better data and more time, such information as that mis-estimation beta would be tremendously useful for finance companies like hedge funds or mutual fund. If I were working in mutual funds that are heavy in small company stocks, for instance, a better beta could save these companies large amounts of money. Another example of possible use this re-estimated beta is in trading; traders who see the overvaluation of stock prices before other traders are presented an incredibly profitable arbitrage opportunities.
As can be seen even the smallest difference in significant financial variables can creates large losses for the market. Beyond monetary losses I hope this paper illustrated the value of information, specifically the value of right information at the right time. As such I hope that agencies such as the SEC would promote policy to make information not only cheaper but available for more people as well.
Sources: Amihud, Yakov and Haim Mendelson. 1991.Liquidity, Asset Prices and Finacial Policy. Financial Anlyst Journal. Ibbotson, Roger et al. 1996. Estimates of Small Stock Betas are Much Too Low. Pastor, Lubos and Robert Stambaugh. 2001. Liquidity Risk and Expected Stock Returns. The Center for Research in Security Prices. Bodie, Zvi et al. 2007. 7th Ed Investments. McGraw-Hill Irwin. Liquidity Risk. Wikipedia. http://www.wikipedia.com Bloomberg Financial Terminal. Bloomberg 2006-2007. Federal Reserve Statiscal Release. http://www.federalreserve.com/releases/h15/data.htm
Appendix 1: Raw SAS Code data liq; infile "c:\E\liq.prn"; input q sp sc i id rsp rsc ksp ksc; /*playing with possible variables*/ lksp = lag(ksp); sumksp = lksp + ksp; proc print; proc reg data=liq; ORIGINAL: model ksc = ksp; /*autocorrelation test*/ output out=liq r=ehat; data liq2; set liq; ehat=ehat; proc plot data=liq2; plot ehat * q = "X"; proc arima data=liq; identify var=ehat nlag=6; /*heteroskedasticity test*/ proc plot data=liq2; plot ehat * ksp; /*lag of one autocorrelation*/ proc autoreg data=liq; TESTORIGINAL: model ksc = ksp /nlag=1; proc autoreg data=liq; TESTORIGINAL2: model ksc = ksp /nlag=2; proc autoreg data=liq; ALSODW: model ksc = ksp/dw; /*mimic ibbotson*/ proc reg data=liq2; IBBOTSON: model ksc = ksp lksp; output out=liq2 r=ehatc; data liqc; set liq2; ehatc=ehatc; /*test*/
proc autoreg data=liqc; AUTOCORRTEST: model ksc = ksp lksp/nlag=1; proc plot plot plot
plot data=liqc; ehatc * q = "Y"; ehatc * ksp = "H"; ehatc * lksp = "K";
proc model data=liqc; parms b0 b1 b2; ksc = b0 + b1*ksp + b2*lksp; fit ksc/white pagan = (1 ksp lksp); proc corr ; var ksp lksp; run;
Appendix 2: Raw SAS Output The SAS System Obs
q
sp
sc
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
1998.25 1998.50 1998.75 1999.00 1999.25 1999.50 1999.75 2000.00 2000.25 2000.50 2000.75 2001.00 2001.25 2001.50 2001.75 2002.00 2002.25 2002.50 2002.75 2003.00 2003.25 2003.50 2003.75 2004.00 2004.25 2004.50 2004.75 2005.00 2005.25 2005.50 2005.75 2006.00 2006.25 2006.50 2006.75
1133.84 1017.01 1229.23 1286.37 1372.72 1282.71 1469.25 1498.58 1452.60 1436.51 1320.28 1160.33 1224.42 1040.94 1148.80 1147.39 989.82 815.28 879.82 848.18 974.50 995.97 1111.92 1126.21 1140.84 1114.58 1211.92 1180.59 1191.33 1228.81 1248.29 1294.83 1270.20 1335.85 1418.30
191.54 151.17 177.36 161.07 185.52 176.20 197.79 208.95 210.70 217.29 219.59 204.78 232.41 192.77 232.18 247.92 231.33 187.86 196.62 184.78 220.98 236.10 270.42 286.66 296.35 291.60 328.80 321.26 333.10 350.20 350.67 394.83 375.97 371.78 400.20
i
id
5.12 4.74 4.50 4.57 4.72 4.82 5.36 5.86 5.86 6.18 5.94 4.54 3.57 2.69 1.72 1.83 1.73 1.66 1.21 1.15 0.94 0.96 0.91 0.95 1.29 1.68 2.22 2.80 3.04 3.49 3.97 4.63 4.92 4.93 4.97
0.051 0.047 0.045 0.046 0.047 0.048 0.054 0.059 0.059 0.062 0.059 0.045 0.036 0.027 0.017 0.018 0.017 0.017 0.012 0.012 0.009 0.010 0.009 0.010 0.013 0.017 0.022 0.028 0.030 0.035 0.040 0.046 0.049 0.049 0.050
11:06 Wednesday, December 19, 2007
rsp
rsc
ksp
ksc
lksp
sumksp
0.029 -0.103 0.209 0.046 0.067 -0.066 0.145 0.020 -0.031 -0.011 -0.081 -0.121 0.055 -0.150 0.104 -0.001 -0.137 -0.176 0.079 -0.036 0.149 0.022 0.116 0.013 0.013 -0.023 0.087 -0.026 0.009 0.031 0.016 0.037 -0.019 0.052 0.062
-0.046 -0.211 0.173 -0.092 0.152 -0.050 0.123 0.056 0.008 0.031 0.011 -0.067 0.135 -0.171 0.204 0.068 -0.067 -0.188 0.047 -0.060 0.196 0.068 0.145 0.060 0.034 -0.016 0.128 -0.023 0.037 0.051 0.001 0.126 -0.048 -0.011 0.076
-0.022 -0.150 0.164 0.001 0.020 -0.114 0.092 -0.039 -0.089 -0.073 -0.140 -0.167 0.020 -0.177 0.086 -0.020 -0.155 -0.193 0.067 -0.047 0.140 0.012 0.107 0.003 0.000 -0.040 0.065 -0.054 -0.021 -0.003 -0.024 -0.009 -0.068 0.002 0.012
-0.098 -0.258 0.128 -0.138 0.105 -0.098 0.069 -0.002 -0.050 -0.031 -0.049 -0.113 0.099 -0.197 0.187 0.049 -0.084 -0.205 0.035 -0.072 0.187 0.059 0.136 0.051 0.021 -0.033 0.105 -0.051 0.006 0.016 -0.038 0.080 -0.097 -0.060 0.027
. -0.022 -0.150 0.164 0.001 0.020 -0.114 0.092 -0.039 -0.089 -0.073 -0.140 -0.167 0.020 -0.177 0.086 -0.020 -0.155 -0.193 0.067 -0.047 0.140 0.012 0.107 0.003 0.000 -0.040 0.065 -0.054 -0.021 -0.003 -0.024 -0.009 -0.068 0.002
. -0.172 0.014 0.165 0.021 -0.094 -0.022 0.053 -0.128 -0.162 -0.213 -0.307 -0.147 -0.157 -0.091 0.066 -0.175 -0.348 -0.126 0.020 0.093 0.152 0.119 0.110 0.003 -0.040 0.025 0.011 -0.075 -0.024 -0.027 -0.033 -0.077 -0.066 0.014
The SAS System
11:06 Wednesday, December 19, 2007
The REG Procedure Model: ORIGINAL Dependent Variable: ksc Analysis of Variance
Source
DF
Sum of Squares
Mean Square
Model Error Corrected Total
1 33 34
0.28408 0.10677 0.39085
0.28408 0.00324
F Value
Pr > F
87.80
<.0001
41
42
Root MSE Dependent Mean Coeff Var
0.05688 -0.00897 -634.01913
R-Square Adj R-Sq
0.7268 0.7186
Parameter Estimates
Variable
DF
Parameter Estimate
Standard Error
t Value
Pr > |t|
Intercept ksp
1 1
0.01496 1.02882
0.00995 0.10979
1.50 9.37
0.1422 <.0001
The SAS System Plot of ehat*q.
R e s i d u a l
‚ 0.10 ˆ ‚ ‚ ‚ ‚ ‚ ‚ 0.05 ˆ ‚ ‚ ‚ ‚ ‚ ‚ 0.00 ˆ ‚ ‚ ‚ ‚ ‚ ‚ -0.05 ˆ ‚ ‚ ‚ ‚ ‚ ‚ -0.10 ˆ ‚ ‚ ‚ ‚ ‚ ‚ -0.15 ˆ ‚
Symbol used is 'X'.
X
X
X
11:06 Wednesday, December 19, 2007
X X
X X
X
X
X
X X
X X
X X
X
X
X X
X X X
X X X
X
X
X
X X
X X
X
X
43
‚ ‚ ‚ ‚ ‚ -0.20 ˆ ‚ Šƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒ 1997.50 1998.75 2000.00 2001.25 2002.50 2003.75 2005.00 2006.25 2007.50 q The SAS System
11:06 Wednesday, December 19, 2007
44
The ARIMA Procedure Name of Variable = ehat Mean of Working Series Standard Deviation Number of Observations
-227E-20 0.055232 35
Autocorrelations Lag
Covariance
Correlation
0 1 2 3 4 5 6
0.0030505 0.00078967 0.00064959 0.00044646 -0.0000946 0.00029317 0.00019843
1.00000 0.25886 0.21295 0.14635 -.03101 0.09611 0.06505
-1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 | | | | | | |
. . . . . .
|********************| |***** . | |**** . | |*** . | *| . | |** . | |* . |
"." marks two standard errors
Inverse Autocorrelations Lag
Correlation
1 2 3 4 5 6
-0.17753 -0.14788 -0.07748 0.15877 -0.07322 -0.04141
-1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 | | | | | |
. . . . . .
****| ***| **| |*** *| *|
. . . . . .
| | | | | |
Partial Autocorrelations Lag
Correlation
1 2 3
0.25886 0.15642 0.06522
-1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 | | |
. . .
|***** . |*** . |* .
| | |
Std Error 0 0.169031 0.180002 0.187061 0.190304 0.190449 0.191829
4 5 6
-0.11987 0.10224 0.04605
| | |
. . .
**| |** |*
The SAS System
. . .
| | |
11:06 Wednesday, December 19, 2007
45
The ARIMA Procedure Autocorrelation Check for White Noise To Lag
ChiSquare
DF
Pr > ChiSq
6
5.83
6
0.4429
--------------------Autocorrelations-------------------0.259
0.213
0.146
The SAS System Plot of ehat*ksp.
R e s i d u a l
‚ 0.10 ˆ ‚ ‚ ‚ ‚ ‚ ‚ 0.05 ˆ ‚ ‚ ‚ ‚ ‚ ‚ 0.00 ˆ ‚ ‚ ‚ ‚ ‚ ‚ -0.05 ˆ ‚ ‚ ‚ ‚ ‚ ‚ -0.10 ˆ ‚ ‚ ‚ ‚ ‚ ‚
-0.031
0.096
0.065
11:06 Wednesday, December 19, 2007
Legend: A = 1 obs, B = 2 obs, etc.
A A
A
A
A A
A
A A
A
A
A A A
A
AA A
A
A
A
A
A
A A A
A
A A A
A A
A
A
46
-0.15 ˆ ‚ A ‚ ‚ ‚ ‚ ‚ -0.20 ˆ ‚ Šƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒ -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 ksp The SAS System
11:06 Wednesday, December 19, 2007
The AUTOREG Procedure Model Dependent Variable
TESTORIGINAL ksc
Ordinary Least Squares Estimates SSE MSE SBC Regress R-Square Durbin-Watson
0.10676819 0.00324 -96.299123 0.7268 1.4059
DFE Root MSE AIC Total R-Square
33 0.05688 -99.40982 0.7268
Variable
DF
Estimate
Standard Error
t Value
Approx Pr > |t|
Intercept ksp
1 1
0.0150 1.0288
0.009948 0.1098
1.50 9.37
0.1422 <.0001
Estimates of Autocorrelations Lag
Covariance
Correlation
0 1
0.00305 0.000790
1.000000 0.258863
-1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 | |
Preliminary MSE
|********************| |***** |
0.00285
Estimates of Autoregressive Parameters
Lag
Coefficient
Standard Error
t Value
1
-0.258863
0.170751
-1.52
47
Yule-Walker Estimates SSE MSE SBC Regress R-Square Durbin-Watson
0.09661393 0.00302 -96.172211 0.7957 1.7977
DFE Root MSE AIC Total R-Square
The SAS System
32 0.05495 -100.83825 0.7528
11:06 Wednesday, December 19, 2007
48
The AUTOREG Procedure
Variable
DF
Estimate
Standard Error
t Value
Approx Pr > |t|
Intercept ksp
1 1
0.0161 1.1189
0.0126 0.1002
1.28 11.16
0.2103 <.0001
The SAS System
11:06 Wednesday, December 19, 2007
The AUTOREG Procedure Model Dependent Variable
TESTORIGINAL2 ksc
Ordinary Least Squares Estimates SSE MSE SBC Regress R-Square Durbin-Watson
0.10676819 0.00324 -96.299123 0.7268 1.4059
DFE Root MSE AIC Total R-Square
33 0.05688 -99.40982 0.7268
Variable
DF
Estimate
Standard Error
t Value
Approx Pr > |t|
Intercept ksp
1 1
0.0150 1.0288
0.009948 0.1098
1.50 9.37
0.1422 <.0001
Estimates of Autocorrelations Lag
Covariance
Correlation
0 1 2
0.00305 0.000790 0.000650
1.000000 0.258863 0.212945
-1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 | | |
Preliminary MSE
|********************| |***** | |**** |
0.00278
Estimates of Autoregressive Parameters Standard
49
Lag
Coefficient
Error
t Value
1 2
-0.218372 -0.156417
0.177395 0.177395
-1.23 -0.88
Yule-Walker Estimates SSE MSE SBC Regress R-Square Durbin-Watson
0.09444644 0.00305 -93.361474 0.7764 1.8450
DFE Root MSE AIC Total R-Square
The SAS System
31 0.05520 -99.582867 0.7584
11:06 Wednesday, December 19, 2007
50
The AUTOREG Procedure
Variable
DF
Estimate
Standard Error
t Value
Approx Pr > |t|
Intercept ksp
1 1
0.0134 1.0825
0.0148 0.1043
0.91 10.37
0.3717 <.0001
The SAS System
11:06 Wednesday, December 19, 2007
51
The REG Procedure Model: IBBOTSON Dependent Variable: ksc Analysis of Variance
Source
DF
Sum of Squares
Mean Square
Model Error Corrected Total
2 31 33
0.28664 0.09605 0.38269
0.14332 0.00310
Root MSE Dependent Mean Coeff Var
0.05566 -0.00635 -876.16460
R-Square Adj R-Sq
F Value
Pr > F
46.26
<.0001
0.7490 0.7328
Parameter Estimates
Variable
DF
Parameter Estimate
Standard Error
t Value
Pr > |t|
Intercept ksp lksp
1 1 1
0.01512 1.01967 -0.09376
0.01028 0.10801 0.10827
1.47 9.44 -0.87
0.1515 <.0001 0.3932
The SAS System
11:06 Wednesday, December 19, 2007
The AUTOREG Procedure
52
Model Dependent Variable
AUTOCORRTEST ksc
Ordinary Least Squares Estimates SSE MSE SBC Regress R-Square Durbin-Watson
0.09604677 0.00310 -92.488636 0.7490 1.4406
DFE Root MSE AIC Total R-Square
31 0.05566 -97.067718 0.7490
Variable
DF
Estimate
Standard Error
t Value
Approx Pr > |t|
Intercept ksp lksp
1 1 1
0.0151 1.0197 -0.0938
0.0103 0.1080 0.1083
1.47 9.44 -0.87
0.1515 <.0001 0.3932
Estimates of Autocorrelations Lag
Covariance
Correlation
0 1
0.00282 0.000570
1.000000 0.201939
-1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 | |
Preliminary MSE
|********************| |**** |
0.00271
Estimates of Autoregressive Parameters
Lag
Coefficient
Standard Error
t Value
1
-0.201939
0.178813
-1.13
Yule-Walker Estimates SSE MSE SBC Regress R-Square Durbin-Watson
0.090501 0.00302 -90.942769 0.7984 1.7456
DFE Root MSE AIC Total R-Square
The SAS System
30 0.05492 -97.048211 0.7635
11:06 Wednesday, December 19, 2007
The AUTOREG Procedure
Variable
DF
Estimate
Standard Error
t Value
Approx Pr > |t|
Intercept
1
0.0153
0.0125
1.23
0.2299
53
ksp lksp
1 1
1.0740 -0.1047
0.1079 0.1079
The SAS System Plot of ehatc*q.
0.10
0.05
R e s i d u a l
0.00
-0.05
-0.10
-0.15
9.96 -0.97
<.0001 0.3395
11:06 Wednesday, December 19, 2007
54
Symbol used is 'Y'.
‚ ‚ ˆ ‚ ‚ ‚ ‚ Y Y Y ‚ Y Y ‚ Y ˆ Y ‚ Y Y ‚ ‚ Y Y ‚ Y Y Y Y ‚ Y ‚ Y Y Y ˆ Y Y ‚ Y Y ‚ ‚ ‚ Y Y ‚ Y Y ‚ Y ˆ Y ‚ ‚ Y ‚ Y ‚ ‚ Y ‚ ˆ ‚ ‚ ‚ Y ‚ ‚ Y ‚ ˆ ‚ Šƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒ 1997.50 1998.75 2000.00 2001.25 2002.50 2003.75 2005.00 2006.25 2007.50 q
NOTE: 1 obs had missing values. The SAS System Plot of ehatc*ksp.
11:06 Wednesday, December 19, 2007
Symbol used is 'H'.
55
0.10
0.05
R e s i d u a l
0.00
-0.05
-0.10
-0.15
‚ ‚ ˆ ‚ ‚ ‚ ‚ H H H ‚ H H ‚ H ˆ H ‚ HH ‚ ‚ H H ‚ H H H H ‚ H ‚ H H H ˆ H H ‚ H H ‚ ‚ ‚ H H ‚ H H ‚ H ˆ H ‚ ‚ H ‚ H ‚ ‚ H ‚ ˆ ‚ ‚ ‚ H ‚ ‚ H ‚ ˆ ‚ Šƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒ -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 ksp
NOTE: 1 obs had missing values. The SAS System Plot of ehatc*lksp. ‚ ‚ 0.10 ˆ ‚ ‚ ‚ ‚
K
K
11:06 Wednesday, December 19, 2007
Symbol used is 'K'.
K
56
0.05
R e s i d u a l
0.00
-0.05
-0.10
-0.15
‚ K K ‚ K ˆ K ‚ K K ‚ ‚ K K ‚ K KK ‚ K ‚ K K K ˆ K K ‚ K K ‚ ‚ ‚ K K ‚ K K ‚ K ˆ K ‚ ‚ K ‚ K ‚ ‚ K ‚ ˆ ‚ ‚ ‚ K ‚ ‚ K ‚ ˆ ‚ Šƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒˆƒƒ -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 lksp
NOTE: 1 obs had missing values.
1 obs hidden. The SAS System
11:06 Wednesday, December 19, 2007
The MODEL Procedure Model Summary Model Variables Parameters Equations Number of Statements
Model Variables Parameters Equations
1 3 1 1
ksc b0 b1 b2 ksc
The Equation to Estimate is
57
ksc =
F(b0(1), b1(ksp), b2(lksp))
NOTE: At OLS Iteration 1 CONVERGE=0.001 Criteria Met. The SAS System
11:06 Wednesday, December 19, 2007
58
The MODEL Procedure OLS Estimation Summary Data Set Options DATA=
LIQC
Minimization Summary Parameters Estimated Method Iterations
3 Gauss 1
Final Convergence Criteria R PPC RPC(b1) Object Trace(S) Objective Value
0 0 10094.73 0.749851 0.003098 0.002825
Observations Processed Read Solved Used Missing
35 35 34 1
The SAS System
11:06 Wednesday, December 19, 2007
The MODEL Procedure Nonlinear OLS Summary of Residual Errors
Equation ksc
DF Model
DF Error
SSE
MSE
Root MSE
R-Square
Adj R-Sq
3
31
0.0960
0.00310
0.0557
0.7490
0.7328
Nonlinear OLS Parameter Estimates
Parameter
Estimate
Approx Std Err
t Value
Approx Pr > |t|
59
b0 b1 b2
0.015122 1.019668 -0.09376
0.0103 0.1080 0.1083
Number of Observations Used Missing
1.47 9.44 -0.87
Statistics for System
34 1
Objective Objective*N
0.002825 0.0960
Heteroscedasticity Test Statistic DF Pr > ChiSq
Equation
Test
ksc
White's Test Breusch-Pagan
0.1515 <.0001 0.3932
6.72 0.59
5 2
0.2420 0.7435
The SAS System
Variables Cross of all vars 1, ksp, lksp
11:06 Wednesday, December 19, 2007
The CORR Procedure 2
Variables:
ksp
lksp
Simple Statistics Variable ksp lksp
N
Mean
Std Dev
Sum
Minimum
Maximum
35 34
-0.02326 -0.02429
0.08885 0.08997
-0.81400 -0.82600
-0.19300 -0.19300
0.16400 0.16400
Pearson Correlation Coefficients Prob > |r| under H0: Rho=0 Number of Observations
ksp
ksp
lksp
1.00000
-0.10253 0.5639 34
35 lksp
-0.10253 0.5639 34
1.00000 34
60