The following regression equation is estimated as a production function for q based on a sample size of 30 observations
In (Q)=1.37 + 0.632 in(Ki)+0.452 in(Li)+uhati
(0.257) (0.219)
R2=0.98 Cov(Bkhat,Bhat L)=0.055
And standard errors are in parenthesis,where bhatk and bhatl are capital and labour elsaticities with respect to output,respectively
a. Test the hypothesis that capital(K) AND Labour(L) elasticities of output are identical at 5% level of Significance
b. Test the hypothesis that there are constant returns to scale at 5% level of significance
a. To test the hypothesis that the capital (K) and labor (L) elasticities of output are identical, we can use the t-test. The null hypothesis (H0) is that the elasticities are identical, and the alternative hypothesis (Ha) is that they are different.
1. Calculate the t-statistic for testing the equality of the elasticities:
t = (bhatk - bhatl) / sqrt(se(bhatk)^2 + se(bhatl)^2)
where bhatk and bhatl are the estimated coefficients for capital and labor elasticities, and se(bhatk) and se(bhatl) are the standard errors for these coefficients.
2. Look up the critical value from the t-distribution table at the desired significance level (5% in this case). The degrees of freedom for the t-distribution are equal to the sample size minus the number of parameters estimated in the model (30 - 3 = 27).
3. Compare the calculated t-statistic with the critical value to make a decision. If the absolute value of the t-statistic is larger than the critical value, reject the null hypothesis (H0) and conclude that the capital and labor elasticities are different. Otherwise, fail to reject the null hypothesis and conclude that there is not enough evidence to suggest that the elasticities are different.
b. To test the hypothesis of constant returns to scale, we will use the F-test. The null hypothesis (H0) is that there are constant returns to scale (i.e., the sum of the capital and labor elasticities equals 1), and the alternative hypothesis (Ha) is that there are not constant returns to scale.
1. Calculate the F-statistic for testing constant returns to scale:
F = [(sum of squared residuals restricted model - sum of squared residuals full model) / (number of restrictions)] / [sum of squared residuals full model / (number of observations - number of parameters)]
The restricted model assumes constant returns to scale, where the sum of the elasticities (bhatk + bhatl) equals 1. The full model is the original model.
2. Determine the number of restrictions. In this case, there is only one restriction because we are testing if the sum of the elasticities equals 1.
3. Look up the critical value from the F-distribution table at the desired significance level (5% in this case). The degrees of freedom for the F-distribution are the number of restrictions and the difference in the degrees of freedom between the restricted and full models.
4. Compare the calculated F-statistic with the critical value to make a decision. If the calculated F-statistic is larger than the critical value, reject the null hypothesis (H0) and conclude that there are not constant returns to scale. Otherwise, fail to reject the null hypothesis and conclude that there is not enough evidence to suggest that there are not constant returns to scale.