Econometrics II

Econometrics IIProblem Set 21. (Wooldridge 8.2) Consider a linear model to explain monthly beer consumption:beer = β 0 + β 1 inc + β 2 price + β 3 educ + β 4 female + uE(u|inc, price, educ, female) = 0V ar(u|inc, price, educ, female) = σ 2 inc 2Write the transformed equation that has a homoskedastic error term.2. (Wooldridge 8.3) True or False: WLS is preferred to OLS, when an important variable hasbeen omitted from the model.3. (Wooldridge 8.6) There are different ways to combine features of the Breusch-Pagan andWhite tests for heteroskedasticity. One possibility not covered in the text is to run theregressionû 2 i on x i1 , x i2 , ..., x ik , ŷ 2 i , i = 1, ..., nwhere the û i are the OLS residuals and the ŷ i are the OLS fitted values. Then, we wouldtest joint significance of x i1 , x i2 , ..., x ik and ŷi 2 . (Of course, we always include an interceptin this regression.)(a) What are the df associated with the proposed F test for heteroskedasticity?(b) Explain why the R-squared from the regression above will always be at least as largeas the R-squareds for the BP regression and the special case of the White test.(c) Does part (b) imply that the new test always delivers a smaller p-value than either theBP or special case of the White statistic? Explain.(d) Suppose someone suggests also adding ŷ i to the newly proposed test. What do youthink of this idea?4. Consider a model at the employee level,y i,e = β 0 + β 1 x i,e,1 + β 2 x i,e,2 + ... + β k x i,e,k + f i + v i,e ,1

where the unobserved variable f i is a ’firm effect’ to each employee at a given firm i. Theerror term v i,e is specific to employee e at firm i. The composite error is u i,e = f i + v i,e ,such as in equation (8.28).(a) Assume that V ar(f i ) = σ 2 f , V ar(v i,e) = σ 2 v, and f i and v i,e are uncorrelated. Showthat V ar(u i,e ) = σ 2 f + σ2 v; call this σ 2 .(b) Now suppose that for e ≠ g, v i,e and v i,g are uncorrelated. Show that Cov(u i,e , u i,g ) =σ 2 f .(c) Let ū i = m −1ithat V ar(ū i ) = σ 2 f + σ2 v/m i .∑ mii=1 u i,e be the average of the composite errors within a firm. Show(d) Discuss the relevance of part (c) for WLS estimation using data averaged at the firmlevel, where the weight used for observation i is the usual firm size.5. Wooldridge C8.22