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

Exercise 22

Suppose that the population model determining y is

y = ?0 + ?1x1 + ?2X2 +?3X3+ u,

and this model satisifies Assumptions MLR.1 through MLR.4. However, we estimate the model that omits x3. Let ?0, ?1 and ?2 be the OLS estimators from the regression of y on x1 and x2. Show that the expected value of ?1 (given the values of the independent variables in the sample) is

where the rf1 are the OLS residuals from the regression of x1 on x2. [Hint: The formula for ?1 comes from equation. Plug y. = ?0 + ?1xi1 + ?2xi2 + ?3xi3 + u. into this equation. After some algebra, take the expectation treating xi3 and ri 1 as nonrandom.]

Step 1 of 2

It is known that:

…… (1)

The above formula ofis obtained from equation (3.22) of this textbook. Here are denoted as residuals as given in the question. Now, consider the below equation by making use of the given equation:

…… (2)