Introductory Econometrics 4th Edition by Jeffrey Wooldridge

Exercise 16

Suppose that the population model determining y is
y = 0 + 1 x 1 + 2 X 2 + 3 X 3 + 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 + 1 xi 1 + 2 xi 2 + 3 xi 3 + u. into this equation. After some algebra, take the expectation treating xi3 and ri 1 as nonrandom.]