Introductory Econometrics: A Modern Approach 6th Edition by Jeffrey M Wooldridge

Exercise 12

Consider the multiple regression model containing three independent variables, under Assumptions MLR.1 through MLR.4:

y = ?0 + ?1x1 + ?2x2 + ?3x3 + u.

You are interested in estimating the sum of the parameters on x1 and x2; call this ?1 = ?1 + ?2

(i) Show that is an unbiased estimator of ?1.

(ii) Find Var() in terms of Var(), Var(), and Corr(,).

Step 1 of 2

Consider the multiple regression model containing three independent variables such that the model is given by:

Also consider that

(i)

Consider the four multiple-linear regression assumptions

1) The model of the population is such that the dependent variable is expressed as the linear combination of explanatory variables and the error term. This is what is exhibited by the given population model. The expression is

2) The observations relating to dependent and explanatory variables are random observation. This is assumed to hold as true in the given case

3) The explanatory variables are independent of each other such that no exact linear relationship among the independent variables holds. That means there is no problem of multi-collinearity

4) The expected value of the disturbance or error term in the population model is zero. That means

Given such assumptions, it could be ensured that:

Since,

That implies,

This indicates that is an unbiased estimator of