1. You have run a regression (n=48) to estimate a Cobb-Douglas production function. The estimated slopes on capital and labor, respectively, are 0.3 and 0.75. Their estimated variances are 0.0001 and 0.0004, respectively, and estimated correlation is –0.25. The sum of these two coefficients, 1.05 is a measure of economies of scale, and for constant returns to scale equals unity (1). Set up the hypothesis test using the t-stat that constant returns to scale exists. Solve for the t-stat (you will have to calculate standard error), and tell me if you accept or reject the null hypothesis?
s= standard deviation = sqrt (VAR)
s(X) = sqrt (.0004) = .02
s(Y)= sqrt (.0001) = .01
stdev(X+Y) = sqrt( VAR(X) +VAR(Y) + 2COV (X,Y))
= sqrt(.0004 + .0001 + (2* -.00005)
= sqrt(.0004 + .0001 - .0001)
Standard Error = SE (X,Y)= stdev(X+Y) / Sqrt (n)
= .028284 / 6.9282
T-stat = -.25 / .0040824
T-Critical (.05 2 tail test/ df=94)= 2.00
The Null Hypothesis is accepted because the t-critical value is further from Zero than the T-stat value.