DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT3600B Linear Statistical Analysis
Class Test
April 12, 2022
1. You are given the following matrices computed for a regression analysis Y = β0 + β1X1 + β2X2 + ε with normal errors having common variance σ2.
The elements of the matrices are properly ordered according to the regression function given above.
(a) Obtain the sample size n and the sample mean of Y.
(b) Find the least squares estimate for β = (β0, β1, β2)
T
. Interpret the estimate for β2.
(c) Obtain the unbiased estimate (σ2) of σ2.
(d) Construct an ANOVA table for the regression analysis. Test whether there is a regression between the dependent and independent variables at the 5% level of significance.
If you cannot obtain σ2 in (c), you may take σ2 = 1.04 as an estimate of σ
2
for the following questions.
(e) Test, each at the 5% level of significance
(i) β2 = 0 or not.
(ii) β1 + β2 = 1 or not.
(f) Test at the 5% level of significance.
H0 : β2 = 0 and β1 + β2 = 1 versus H1: H0 is not true.
[Total: 65 marks]
2. Consider a simple linear regression model
yi = β0 + β1xi + εi, εi ~ N(0, σ2), i = 1, …, n
and ε1, …, εn are independent.
Consider n = 4 pairs of observations: (-1, y1), (-1, y2), (0, y3), (2, y4).
(a) Express the least squares estimators of β0 and β1 in terms of y1, y2, y3 and y4.
(Note: You may use the formula β = (X
TX)
-1 X
TY)
(b) Obtain the expression of the regression sum of squares SSR in terms of y1, y2, y3 and y4. (Note: You may use the formula, SSR = ∑n i=1(yi − y)2)
(c) Based on the expression obtained in (b), find E(SSR).
[Total: 35 marks]