3. Consider the output below from the OLS regression for the following dual cost function:
InC? = ?? + ??lnw?? + ??lnw?? + ??lnq? + ??
where, for firm i, C is total cost, q is total output, w? is the price of labor, and w? is the price of capital.
\begin{bmatrix} 0.9659 & -0.0873 & 0.1628 & -0.0222 \\ -0.0873 & 0.0271 & 0.0010 & 0.0003 \\ 0.1628 & 0.0010 & 0.0802 & -0.0002 \\ -0.0222 & 0.0003 & -0.0002 & 0.0013 \end{bmatrix} (X'X)^{-1} , b = \begin{bmatrix} -1.928 \\ 0.184 \\ 0.519 \\ 0.937 \end{bmatrix} , R^2 = 0.9302 , SSE = 51.40 , n = 800
(a) What is the estimate for s²?
(b) The random variable \frac{(n-k)s^2}{?^2} follows what distribution?
Use the above results to test the following null hypotheses. In each case, construct a test statistic and use it to
test the null hypothesis. Showing all your work, tell me if you accept or reject the null hypothesis.
(c) H?: ?? = ?? = ?? = 0. (Note that this is all coefficients except ??.)
(d) H?: ?? = 1 (Note that ?? corresponds to the fourth variable, lnq.)
(e) H?: ?? + ?? = 1