Suppose that the data generating process is determined by the following Cobb-Douglas production function

Problem 1

Suppose that the data generating process is determined by the following Cobb-Douglas production function

 log Yi = α0 + α1 log L1i + α2 log L2i + α3 log Ki + ui

where Yi = output of firm i, L1i = production labor of firm i, L2i = non-production labor of firm i, and Ki = capital stock of firm i. Suppose you instead use OLS to estimate the following model:

using cross-section data on many firms.

1.     Will E[βˆ1] = α1 and E[βˆ2] = α3?  Explain.

2.     You think that the error term may be heteroskedastic so you use heteroskedasticity- robust  standard  errors  for  βˆ0,  βˆ1,  and  βˆ2.  Would  they  be  valid  for  testing  hypotheses about true values of α1 and α3? Explain.

3.     How will your answers to parts 1 and 2 change if it is known that α2 = 0, i.e. L2 is an irrelevant input in the production function? Explain.

4.     Assuming α2 = 0, explain how you would test for constant returns to scale.

Hint: See SW Problem 7.9c

 Problem 2

Suppose that the data generating process is now determined by the following Cobb-Douglas production function:

log Yi = β0 + β1 log L1i + β2 log Ki + ui    (1)

where Yi = output of firm i, L1i = production labor of firm i, and Ki = capital stock of the firm

Suppose you instead use OLS to estimate the following model:

 log Yi = α0 + α1 log L1i + α2 log L2i + α3 log Ki + ui   (2) where L2i = non-production labor of firm i using cross-section data on many firms.

1.     Will E[αˆ1] = β1 and E[αˆ3] = β2?  Explain.

2.     You think that the error term may be heterokedastic so you use heteroskedasticity- robust  standard  errors  for  αˆ0,  αˆ1,  αˆ2,  and  αˆ3.   Will  the  standard  errors  be  valid for testing hypotheses about true values of β0, β1 and β2? Explain.

3.     Suppose you estimate both models (1) and (2) by OLS using cross-sectional data onvery many firms. You test H0 : α1 = 1 against H1 : α11 using the t-statistics t =   αˆ1 − 1 s.e.(αˆ1) and H0 : β1 = 1 against H1 : β1 1 using t =βˆ1 − 1s.e.(βˆ1)and the decision rule “reject H0 if |t| > 1.96” (since 1.96 is the 5% critical value). Suppose H0 is true.   Are you more likely,  less likely,  or equally likely to reject H0 using the first t statistic (involving αˆ1) than using the second t statistic (involving βˆ1)?Explain.