Question

Murti and Sastri (Econometrica, v.25/#2) investigated the production characteristics of various Indian industries, including cotton and sugar. They specified Cobb-Douglas production functions for output (Q) as a double-log function of labor (L), and capital (K):                         

lnQ_i = β₀ + β₁lnLi + β₂lnKi + ɛ_i

and obtained the following estimates (standard errors in parentheses):

Industry	β 0^	β 1^	β 2^	R2
Cotton	0.97	0.92	0.12	.98
		(0.03)	(0.04)
Sugar	2.70	0.59	0.33	.80
		(0.14)	(0.17)

a. Rewrite the equation with a time subscript (t substituted for i) for a single product (rice) with annual data for the period 1970-2010.

b.   How would apply the F-test to test the hypothesis of constant return to scale. Write out your maintained and alternative hypotheses and specify the particular form of the F-statistic you would employ for this test. Can you test the return- to-scale hypothesis with a t-test? Explain.

c.  Write out your maintained and alternative hypotheses and specify the particular form of the F-statistic you would employ to test for the all-over significanceof your equation.

d.   Suppose you were told that a genetically-modified, high yielding variety was introduced extensively into the rice producing industry in t=1990. You wish to test if the slope estimates were significantly different after 1990 compared to those before 1990. Write out your maintained and alternative hypotheses and specify the particular form of the Chow test you would employ for this purpose.  Make sure you fully explain the different components in your test-statistic.  

Answer 1

(a) Equation for Cotton Industry : lnQ_i = 0.97 + 0.92lnL_t + 0.12lnK_t + ɛ_t

Equation for Sugar Industry : lnQ_i = 2.70+ 0.59lnL_t + 0.33nK_t + ɛ_t

(b) F-test for testing the hypothesis for constant returns to scale would be :

Maintained Hypothesis , H_o : β 1^ = β 2^ = 1

Alternative Hypothesis , H_o : β 1^ + β 2^ $\neq$ 1

Returns to scale cannot be tested through t-test it involves combines testing of β 1^ & β 2^ but t-test is done for testing effect of single coefficient.

(c)  F-statistic to test for the all-over significance of your equation is as follows :

Maintained Hypothesis , H_o : β 0^ =  β 1^ = β 2^ = 0

Alternative Hypothesis , H_o : β 0^ =  β 1^ = β 2^ $\neq$  0