Prof. George Karras

Homework #4 uses the White and Breusch-Pagan procedures to test for the presence of heteroskedasticity in the empirical Neoclassical growth model, and to correct using Weighted Least Squares (WLS).

0. THEORY. As in the previous Homeworks, 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 the previous Homeworks, consider the MRW data for the 98 countries of the non-oil sample.

2. OLS ESTIMATION. As in the last two Homeworks, 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. Using OLS, estimate equation (1) and recover the squared estimated residuals. In RATS, writing

linreg growth / resid

# constant lny1960 lns lnpop lnschool

will run the regression and save the residuals ^ as the new variable resid. Create the squared residuals series, residsq, with the instruction

set residsq = resid**2

3. DIAGNOSIS OF HETEROSKEDASTICITY. We want to investigate whether the error term is homoskedastic. Implement the following two tests.

(a) The White Test. First, construct the products and cross-products of the explanatory variables. In RATS, specify

set lny1960sq = lny1960**2

set lnssq = lns**2

set lnpopsq = lnpop**2

set lnschoolsq = lnschool**2

set lny1960lns = lny1960*lns

set lny1960lnpop = lny1960*lnpop

set lny1960lnschool = lny1960*lnschool

set lnslnpop = lns*lnpop

set lnslnschool = lns*lnschool

set lnpoplnschool = lnpop*lnschool

Next, regress ^ ² on these products and cross-products (including those of the constant term):

^_i ² = ₀ + ₁lny1960_i + ₂lns_i + ₃lnpop_i + ₄lnschool_i

+ ₅lny1960 _i² + ₆lns _i² + ₇lnpop _i² + ₈lnschool _i²

+ ₉lny1960_ilns_i + ₁₀ln1960_ilnpop_i + ... + ₁₄lnpop_ilnschool_i + v_i, i=1,...,98. (2)

Using the R² from this auxiliary regression calculate White's statistic and test the null hypothesis of homoskedasticity.

(b) The Breusch-Pagan Test. You somehow obtain the information that _i² = E( _i²), the variance of _i, may have the following exact structure:

_i² = h[₀ + ₁(lnpop_i)²], (3)

where the function h takes only positive values. Estimate the auxiliary regression of ^_i ² on a constant and (lnpop_i)², and use the regression's R² to calculate the Breusch-Pagan statistic for the null hypothesis of ₁=0 (i.e., there is homoskedasticity).

4. WEIGHTED LEAST SQUARES (WLS) ESTIMATION. Suppose ² _i is indeed proportional to (lnpop_i)². Apply WLS estimation as follows. First transform the variables by dividing by lnpop_i. In RATS, write

set trgrowth = growth/lnpop

set trconstant = 1/lnpop

set trlny1960 = lny1960/lnpop

set trlns = lns/lnpop

set trlnpop = lnpop/lnpop

set trlnschool = lnschool/lnpop

Next, implement WLS by estimating the transformed model with OLS. Compare the OLS and WLS results. Is the neoclassical model supported by the WLS results?