Suppose that the population model determining y isy = β0 + β1x1 + β2X2 +β3X3+ u,and this m...

Suppose that the population model determining y is

y = β0 + β1x1 + β2X2 +β3X3+ u,

and this model satisifies Assumptions MLR.1 through MLR.4. However, we estimate the model that omits x3. Let β0, β1 and β2 be the OLS estimators from the regression of y on x1 and x2. Show that the expected value of β1 (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 β1 comes from equation. Plug y. = β0 + β1xi1 + β2xi2 + β3xi3 + u. into this equation. After some algebra, take the expectation treating xi3 and ri 1 as nonrandom.]