The Ahmetrics Solutions Library, (www.ahmetrics.com) contains exam practice questions for general statistics and econometrics which are available for instant download. The library is growing all the time, and you are welcome to contact us for customized solutions. (Customized solutions are hand written.) Solutions are produced in-house by staff who have experience of marking exams, and in our solutions we point out where applicable areas where students tend to trip up. Practice exam questions and solutions can be an aid to revising for exams, but if you do not understand a topic, you might consider benefitting from personal help from one of our stats/econometrics tutors. Sample Question. Consider the simple linear regression model
y i = α + βx i + u i , for
i = 1, . . . , n
where the errors are independent and distributed normally with zero mean and variance σ 2 . The following sum of squares are calculated from the raw data: ∑ xi = 70, ∑ yi = 136, ∑ xi2 = 712, ∑ yi2 = 2472, ∑ xi yi = 1319
(a) Find the least squares estimates for α and β and write down the fitted line. (b) Find the standard error for the least squares estimator for β . (c) Test the significance of β at the 5% significance level.
Sample Solution for Part (b) From part (a), the estimates are The standard error for β is
β = 2. 028 .
α = 1. 653,
seβ =
σ
2
∑ x 2i − nx 2
To find the standard error, we need the estimate of the variance of the error term 2
σ . For the simple linear regression model, the formula for 2
σ =
2
is:
RSS , n−2
where RSS denotes the residual sum of squares, namely 2
σ
RSS = ∑ u 2i .
Remark: Note in the formula for σ the term n − 2 . The 2 comes in because we have one explanatory variable and a constant - total of 2. If we have a model with 3 explanatory variables and an intercept term then we would replace n-2 by n-4 and so on, so it's not always n-2. This is something students may not realize.
Calculating this RSS is the key to solving the problem, and it's the place where students get stuck. We'll present the method that is commonly adopted by students, which is long and tedious. RSS = ∑ u 2i = ∑y i − α − βx i 2 Now if we had the raw data i.e. the list of data for x and y then we could plug it into the equation above to get the RSS. However, we can't use this method here, because we do not have the raw data. Instead we could expand the brackets to get the summation terms (the values we have) as follows:
∑y i − α − βx i 2
2
=
∑y2i − 2αy i − 2βx i y i + α 2 + 2αβx i + β
=
∑ y 2i − ∑ 2αy i − ∑ 2βx i y i + ∑ α 2 + ∑ 2αβx i + ∑ β
=
∑ y 2i − 2α ∑ y i − 2β ∑ x i yi + nα 2 + 2αβ ∑ x i + β ∑ x 2i ,
x 2i
2
x 2i ,
2
Summation rules used - if a is a constant then ∑ ax i = a ∑ x i (ie constant comes out of the summation sign); Also we have used the fact that the sum of a constant, which is a number that doesn't change as the counter i n changes is ∑ i=1 a = na. Now we are in a position to plug in the numbers given in the question: 2 RSS = ∑ y 2i − 2α ∑ y i − 2β ∑ x i y i + nα 2 + 2αβ ∑ x i + β ∑ x 2i ,
= 2472 − 2 ∗ 2. 028 ∗ 136 − 2 ∗ 1. 653 ∗ 1319 + 10 ∗ 2. 028 2 + 2 ∗ 2. 028 ∗ 1. 653 ∗ 70 + 1. 653 2 ∗ 712, = 15. 693. Phew! Because so many steps are needed using this method that along with the pressure in a statistics or econometrics exam, students all too frequently trip up along the way. Having RSS we can find: 2
σ =
RSS = 15. 693 = 1. 962 n−2 10 − 2
and so
seβ =
∑
σ x 2i
2
− nx
2
=
1. 962 = 0. 094 ■ 222
and we are done.Yippee! With this standard error we can go on to part (c) and
conduct the t-test. There is a quick and painless method to solve for the standard error which can be done without the need for the raw data, or manipulating summations. If you'd like to see the short cut, or if you want the solution for the entire question then go along to the Ahmetrics Solutions Library (www.ahmetrics.com) where you can download this and other econometrics/statistics exam solutions (answers). If you’d like to be kept in touch with solutions added to the library, then subscribe for updates via Ahmetrics Twitter (www.twitter.com/ahmetrics) Visit our website for other services: • tuition in stats, econometrics, CFA tuition, maths for school and university • view our short “how to…” videos for stats computing packages • support for projects • bookstore for recommended books.
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