Problem 3: In the multiple regression model M1 given in Problem 1, it is suspected that the regressors may have an interaction to affect the response variable. Construct a statistical test to address the question of whether the potential interaction deserves attention. Use Problem 1 shown in the image.

Page 1 625.661 Statistical Models and Regression Test 1 for Modules 1, 2, 3, 4 Do all the problems (2 pages) below. Provide intermediate steps and state the assumptionls) for each intermediate step of your work. 1. In a multiple linear regression analysis, (31,-, x1i,x2i),i = 1, ...n , are statistically independent and satisfy the model (M1) given by: y=50+51x1+l82x2+5. where the response variable y is continuous, the regressor vector X = (x1,x2)' has mean vector (juxl, llxz)' and positive definite variance-covariance matrix 2x , the random errors 8,- conditional on X,- are statistically independent and normally distributed with mean zero and variance 0'2 which does not depend on X. a) b) Consider the case that the value of 02 is known; that is, it is given. 1) Construct a statistical test for testing H0: 31 = z = 0, using the known 02, and the a-level rejection region of the test. [5 points] 2) Construct a statistical test for testing H0: 31 = [32, using the known 02, and the cr- level rejection region of the test. [5 points] 3) Are there any differences in testing between al) and 32) above? why or why not? [5 points] Consider the case that the value of 02 is unknown; that is, it needs to be estimated. 1) Construct a statistical test for testing H0:B1 = z = 0 and the at-level rejection region of the test. [5 points] 2) Construct a statistical test for testing each individual regressor; that is, for the ith regressor (i = 1, 2), Hm: Bi = 0 and the a-Ievel rejection region of the test. [5 points] 3) In b2) above, derive the probability of falsely rejecting at least one H0,- , i = 1, 2. In b1) above, derive the probability of falsely rejecting at least one H0,- , i = 1,2. [7 points] If ,80 may or may not be zero in the model M1, construct analysis-of-variance (or sum-of- squares) table for this multiple linear regression analysis. [3 points] Page 2 2. After trying many multiple linear regression analyses based on the model M1 in Problem 1 above, a data analyst obtained the following statistics. The regression sum of squares 55(30) by excluding the two regressors x1,x2 . The regression sums of squares 55090), 55(0, Bl), by excluding the regressor x2 . The regression sums of squares 55080), 53(0, 52), by excluding the regressor x1 . The regression sums of squares 35(50),SS(BO, 51,32); that is, include all the regressors. a) Discuss with mathematical proof whether the hypothesis that no regressor has a nonzero effect on the response variable can be tested. [5 points] b) Discuss with mathematical proof whether 53(30. BI. [32) = $3090.31) + 55(30. 192) . [5 points] c) Discuss with mathematical proof whether 55032 | [30,61 ) = 55([32 | [90 ) . [5 points] d) The data analyst used statistical tests derived from sums of squares to determine whether x1 or x2 should be excluded to come up with the final regression model. That is, the final model may be a subset model of M1. 1) Discuss with mathematical proof whether the ordinary least-squares estimator(s) for the remaining regressor(s) in the final model is biased. [10 points] 2) Discuss with mathematical proof whether the ordinary least-squares estimator(s} for the remaining regressor(s) in the final model has a smaller variance than the respective ordinary least-squares estimator(s) in fitting the model M1. [10 points] 3) The analyst also explored addition of another regressor x3; that is, Model M1 could be extended to include x3. Discuss with mathematical proof whether the ordinary least- squares estimator for [91, say, based on the extended model is biased for [31 if the true model is M1. [10 points] 3. In the multiple regression model M1 given in Problem 1, it is suspected that the regressors may have an interaction to affect the response variable. Construct a statistical test to address the question of whether the potential interaction deserves attention. [10 points] 4. The data values (31;, x11- ,xZi), i = 1, ..., 15, are thought to satisfy the model (M2) given by: y=o+31x1+zx2 +5: where the response variable y is continuous, the regressor vector X = (x1,xz)' has mean vector Qu, my and positive definite variance-covariance matrix 2x , the random errors 85 conditional on Xi are statistically independent and normally distributed with mean zero and variance 02 which does not depend on X . The analysis gave R2 = 0.95, 37 = 50. a) Test statistical significance of the regression at a = 0.05. [5 points] b} What is the smallest value of R2 that would lead to a conclusion of statistical significance of the regression at a = 0.05? [5 points]