申論題

4. (Total 40%) Consider a multiple linear regression model as

whereβo is the intercept, βjis the slope parameter associated withXj (j = 1,2, ..., k).

Given n observationson Y, ) : i = 1.2.... ,n}.Let

=0,1,... . , k, be the ordinary least squares (OLS) estimates for the parameters

of this model, denote the residual associated with the i-th observation, and t_jde-

note the t-ratio for B_j j = 0,1

(3) For i =, be the standardized version of the data, where are the sample mean and sample standard deviation for y over these i observations, and are the sample mean and sample standard deviation for tj. j = 1.2. . k. Now we regress y* on 1 and xj, j = 1,2, , k, to yield the estimated model as
 where, j = O, 1. . ., k, are the associated OLS estimates.

 a. (10%) Find the OLS estimates= 0, 1, 

b. (10%) Find the t-ratio for b_j in terms of