Prof. George Karras

Homework #3 tests structural stability in the empirical Neoclassical growth model. We are interested in whether the model's coefficients are the same in two subsamples, the African and non-African economies.

0. THEORY. As in Homework #2, the theoretical model is

ln(y_T) - ln(y₀) = (1-e^-T)[(1-)^-1ln(s) - (1-)^-1ln(n) + (1-)^-1ln(h) - ln(y₀)], (0)

where variables and parameters are defined as in Homework #2.

1. DATA. As in Homework #2, we consider the MRW data for the 98 countries of the non-oil sample. Note that the African sample consists of the first 38 observations (i=1,...,38), whereas the non-African sample of the last 60 observations (i=39,...,98).

2. RESTRICTED ESTIMATION. Begin by assuming that the model's coefficients are the same for African and non-African economies. As in Homework #2, rewrite equation (0) above in linear regression form:

growth_i = ₀ + ₁lny1960_i + ₂lns_i + ₃lnpop_i + ₄lnschool_i + _i, i=1,...,98, (1)

where the variables and parameters are defined as in Homework #2. Suppose satisfies the classical assumptions (E_i=0 and E²_i=², for all i; and E_ij=0, for i=/j), and that _i is normally distributed. Using OLS, estimate equation (1). [Note: This is identical to part 2 of Homework #2.]

3. UNRESTRICTED ESTIMATION. Now allow the coefficients to differ between the African and non-African economies and test whether there is structural difference between the two samples. There are two ways to conduct this test.

(a) A Chow Test. Estimate equation (1) separately for the African and non-African samples:

growth_i = ₀ + ₁lny1960_i + ₂lns_i + ₃lnpop_i + ₄lnschool_i + v_i, i=1,...,38, (2)

and

growth_i = ₀ + ₁lny1960_i + ₂lns_i + ₃lnpop_i + ₄lnschool_i + v_i, i=39,..,98, (3)

where the v_is satisfy the classical assumptions with normality. Note that (2) and (3) constitute the unrestricted model. Calculate the "Chow" F statistic and test the null hypothesis H₀:=, where =[_{0 1 2 3 4}]' and =[_{0 1 2 3 4}]'.

(b) A Dummy-Variable Test. Construct an "Africa" dummy variable, dum, which takes the value 1 for the African sample and 0 for the non-African sample. More formally, dum_i=1 for 1i38 and dum_i=0 for 39i98. Multiply the dummy variable by the explanatory variables to construct the "interaction" terms. The RATS code should look like this:

set dum = t<=38

set dumlny1960 = dum*lny1960

set dumlns = dum*lns

set dumlnpop = dum*lnpop

set dumlnschool = dum*lnschool

Write the unrestricted model as:

growth_i = ₀ + ₀dum_i + ₁lny1960_i + ₁dumlny1960_i + ₂lns_i + ₂dumlns_i +

₃lnpop_i + ₃dumlnpop_i + ₄lnschool_i + ₄dumlnschool_i + w_i, i=1,...,98. (4)

Assume the w_is satisfy the classical assumptions with normality, and estimate (4) with OLS. Test the null hypothesis H₁:₀=₁=₂=₃=₄=0 using a Wald F-test. How does this differ from the "Chow" F-test? How does H₁ differ from H₀?