Using 28 annual observations on output \((Y)\), capital \((K)\), labor \((L)\) and intermediate materials \((M)\) for the U.S manufacturing sector, to estimate the Cobb-Douglas production function

gave the following results

The standard deviations of the explanatory variables are \(s_{\ln (K)}=0.28108, s_{\ln (L)}=0.17203\), and \(s_{\ln (M)}=0.27505\). The sums of squared errors from running auxiliary regressions on the explanatory variables are (the subscript refers to the dependent variable in the auxiliary regression)

a. Find (i) the standard errors for \(b_{2}, b_{3}\), and \(b_{4}\), (ii) the \(R^{2}\) 's for each of the auxiliary regressions, and (iii) the variance inflation factors for \(b_{2}, b_{3}\), and \(b_{4}\).

b. Test the significance of \(b_{2}, b_{3}\), and \(b_{4}\) using a \(5 \%\) level of significance.

c. Use a \(5 \%\) level of significance to test the following hypotheses: (i) \(H_{0}: \beta_{2}=0, \beta_{3}=0\), (ii) \(H_{0}: \beta_{2}=0, \beta_{4}=0\), (iii) \(H_{0}: \beta_{3}=0, \beta_{4}=0\), and (iv) \(H_{0}: \beta_{2}=0, \beta_{3}=0, \beta_{4}=0\). The restricted sums of squared errors for the first three hypotheses are (i) \(S S E_{R}=0.0551\), (ii) \(S S E_{R}=0.08357\) and (iii) \(S S E_{R}=0.12064\).

d. Comment on the presence and impact of collinearity.

Transcribed Image Text:

In(Y) B, B In(K) + 3 In(L) + B4 In(M) + e