midterm1_2012_solutions

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a) (10) Test the hypothesis of the increasing return-to-scale, that is ε(L) + ε(K) > 1, if the estimate of ε(L) is 0.807 with a standard error of 0.151. The estimate of ε(K) is 0.233 with a standard error of 0.070. And the total variance-covariance matrix of (ε(K), ε(L)) is: ε(K) ε(L) ------------------------------------------------- ε(K) 0.00494695 ε(L) -0.00961159 0.02289165 ANSWER: H0: ε(L) + ε(K) = 1 Ha: ε(L) + ε(K) > 1 Significance level is 5% t-stat = (0.807 + 0.233 – 1)/ s.e. s.e = sqrt(Var(ε(L)) + Var(ε(K)) + 2cov(ε(L), ε(K))) = = sqrt(0.151^2 + 0.070^2 + 2*(-0.00961159)) = 0.092075078 t-stat = (0.807 + 0.233 – 1)/ 0.092075078 = 0.434428087 95% CI is [-∞, 1.64] T-stat is in CI, we fail to reject the H0. Thus, there is no enough statistical evidence to support the increasing return-to-scale hypothesis.

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b) (10) Test the hypothesis that “is we increase labor input twice and also capital input twice, then the output will also increase twice”. ANSWER: This is a constant return-to-scale hypothesis. H0: ε(L) + ε(K) = 1 Ha: ε(L) + ε(K) ≠ 1 Significance level is 5% t-stat = 0.434428087 (same) 95% CI is [-1.96, 1.96] T-stat is in CI, we fail to reject the H0. Thus, the data supports the constant return-to-scale hypothesis. c) (5) Construct 99% Confidence Interval for ε(L). [0.807 -2.57* 0.151, 0.807 + 2.57* 0.151].

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