Practice Exercises Problem 1: Consider multiple linear regression with E(Y | X = x) = Bo + Bix, + Box, and sample size n = 20. The Coefficient Table on the left shows the OLS estimates, while the ANOVA table on the right tests the hypotheses: H.: E(Y | X = x) = Bo VS H,: E( Y | X =x) = Bo + Bix,+ B2x2 Coefficient Table ANOVA Table Estimate Std. Error T-statistic p-value df SS MS F-statistic p-value Intercept 12.94346 7.183 1.8020 0.0893 Regression 23.583 11.791 5.172 0.0176 x1 0.534521 0.255 -2.0976 0.0512 Residuals 17 38.755 2.27973 x2 1.740149 1. 196 1.4551 0. 1638 Tota 19 62.338 (a) Based on the information from the two tables above, construct the ANOVA table for the hypotheses H.: E(Y | X =x) = Bo+ B2x2 VS H: E(Y | X = x) = Bo+ Bix,+ B2x2 (b) Based on the given ANOVA table and the results from part (a), construct the ANOVA table for the hypotheses H.: E(Y | X =x) = Bo VS H: E( Y | X =x) = Bo + B2x2 (c) What are the (i) Decision and (ii) Conclusion you would draw from the ANOVA table in part (b)? The F-table on the last page may be useful for this part. Problem2: Suppose X1, X2, ...Xn are known constants. Let y1, y2, ...n be independent random variables with mean 0 and variance 1. Let x =_ _x, and y = - Ey, n 1 1 (a) Simplify _ _( v. - y) and Z(v.-7) - Ex? + ny2. ( SXX ] ".). Problem3: Suppose M = 1 2 .). BI a = 2, B= B2 B3. (a) Let f(B) = B'MP -a'B. Express of (P) in terms of B1, B2 and B3. op (b) Find the values of (B1, B2, B3) which minimizes f(B) =B'MP -a's in part (a). (c) Let g(B) =(1 1 1)M'B. Express in terms of B,, B2 and B3. op Problem4: Consider a nx(p+1) matrix X. Let H = X(X' X) 'X' . (a) Simplify (I - H)X , where I is an identity matrix. (b) Is H a symmetric matrix? c) Compute tr(H). (d) Show that H3008-H. Page 2/4