Macroeconomists are interested in factors that explain economic growth. An aggregate production function specification was studied by Duffy and Papageorgiou. \({ }^{19}\) The data are in the data file ces. They consist of cross-sectional data on 82 countries for 28 years, 1960-1987.

a. Estimate a Cobb-Douglas production function

\[L Y_{i t}=\beta_{1}+\beta_{2} L K_{i t}+\beta_{3} L L_{i t}+e_{i t}\]

where \(L Y\) is the \(\log\) of GDP, \(L K\) is the \(\log\) of capital, and \(L L\) is the log of labor. Interpret the coefficients on \(L K\) and \(L L\). Test the hypothesis that there are constant returns to scale, \(\beta_{2}+\beta_{3}=1\).

b. Add a time trend variable \(t=1,2, \ldots, 28\), to the specification in (a). Interpret the coefficient of this variable. Test its significance at the \(5 \%\) level. What effect does this addition have on the estimates of \(\beta_{2}\) and \(\beta_{3}\) ?

c. Assume \(\beta_{2}+\beta_{3}=1\). Solve for \(\beta_{3}\) and substitute this expression into the model in (b). Show that the resulting model is \(L Y L_{i t}=\beta_{1}+\beta_{2} L K L_{i t}+\lambda t+e_{i t}\) where \(L Y L\) is the \(\log\) of the output-labor ratio, and \(L K L\) is the \(\log\) of the capital-labor ratio. Estimate this restricted, constant returns to scale, version of the Cobb-Douglas production function. Compare the estimate of \(\beta_{2}\) from this specification to that in part (b).

d. Estimate the model in (b) using a fixed effects estimator. Test the hypothesis that there are no cross-country differences. Compare the estimates to those in part (b).

e. Using the results in (d), test the hypothesis that \(\beta_{2}+\beta_{3}=1\). What do you conclude about constant returns to scale?

f. Estimate the restricted version of the Cobb-Douglas model in (c) using the fixed effects estimator. Compare the results to those in part (c). Which specification do you prefer? Explain your choice.

g. Using the specification in (b), replace the time trend variable \(t\) with dummy variables \(D 2-D 28\). What is the effect of using this dummy variable specification rather than the single time trend variable?