Using annual data on output and labor and capital outputs for the United States

Question

Using annual data on output and labor and capital outputs for the United States for the period 1929-1967, the production function, Ink,-+A In Lah, +B,Cap, +1, has been estimated by employing ordinary least square method and results obtained are given below In Y, =3.938 + 1.451 In Lab, + 0.384 In Cap' se (0.237 ) (0.083) (0.048) R2:0.9946 =0.9943 RSS=0.0434 2=0.001205 cov(B,,A,)=-0.001 552 n=39 (i) Using an appropriate test statistic test whether labor and capital elasticities of production are statistically significant, clearly stating appropriate null and alternative hypotheses. 5 (i) Test the null hypothesis that the sum of the labor and capital elasticities are equal to unity, -p2=1, and interpret your result. 5 (ii) Test the significance of R2, clearly stating your null and alternative hypotheses. 5 iv) Suppose you have re-estimated the production function for the two periods 1929-1948 and 1949-1967 separately and got the following results. Test whether the estimated values of the intercept coefficient and labor and capital elasticities are stable over the sample period. 5 1929 -1948 In Y, =-4.058 + 1.617 In lab , t 0.220 In Cap , se (0.357) (0.209) (0.230) R2=0.9759 RSS= 0.03555 N=20 949 -1967 In Y, =-2.498 + 1.009 In Lab , + 0.579 In Cap se (0.531) (0.144) (0.055) R2=0.9958 RSS= 0.0336 N=19

Explanation / Answer

Answer:

(i) The statistical significance of the coefficients can be determined by the t-statistics or the p-value of the estimate. the The t statistic is determined by the ratio of the coefficient and its standard error.

The t-statistic for Labour is = (1.451/0.083) = 17.482

The t-statistic for Captial is = (0.384/0.048) = 8.00

According to 5 percent significance level if the t-statistics value is greater than the critical value, we reject the null hypothesis. The null hypothesis is that the coefficents is zero whereas the alternative is that the coefficients are significant. Given the number of observations to be 39, the degree of freedom is (39-2 = 37). The critcal value for this 1.687 which is less than the calculated t-statistics. Hence we reject the null hypothesis and accpet the alternative hypothesis.

(ii) For testing the null hypothesis that the sum of the labor and captial elasticities is equal to unity, we perform F-test. F-test is the ratio between the difference between the restricted RSS and unrestricted dividied by the total restriction impose on unrestricted RSS divided by the degree of freedom. As in the above question the restricted RSS cannot be computed as data is not given hence the F-test cannot be calculated.

(iii) For testing whether the R square is signiifcant, we perform t- test. the null hypothesis is that r value is significantly different from zero. So the alternative hypothesis becomes that the Rsquare is not different from zero. If t-test value is greater than 2 (as per rule of thumb) we reject the null hypothesis.

(iv) We use Chow test for testing the stability of the coefficients in the two regressions. The formula is:

F = ((RSS1 - RSS2)/(k+1))/(RSS2/(n-2k-2)), where k standard for the no. of independent variable used in the regression.

= ((0.036-0.034)/(2+1))/(0.034/(19-4-2)) = 0.26

According to the critical value the F-test is 3.41. The critical value is greater than the calculated value hence we accpet the null hyopthesis that the the parameters are stable.

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