Question

In the file pubexp.dat there are data on public expenditure on education (EE), gross domestic product (GDP), and population (P) for 34 countries in the year 1980. It is hypothesized that per capita expenditure on education is linearly related to per capita GDP. That is, yi ¼ b1 þ b2xi þ ei where yi ¼ EEi Pi and xi ¼ GDPi Pi It is suspected that ei may be heteroskedastic with a variance related to xi.

(a) Why might the suspicion about heteroskedasticity be reasonable?

(b) Estimate the equation using least squares; plot the least squares function and the residuals. Is there any evidence of heteroskedasticity?

(c) Test for the existence of heteroskedasticity using a White test.

(d) Use White’s formula for least squares variance estimates to find some alternative standard errors for the least squares estimates obtained in part (b). Use these standard errors and those obtained in part (b) to construct two alternative 95% confidence intervals for b2. What can you say about the confidence interval that ignores the heteroskedasticity?

(e) Reestimate the equation under the assumption that var(ei)= alpha^2*xi. Report the results. Construct a 95% confidence interval for b2. Comment on its width relative to that of the confidence intervals found in part (d)

Answer 1

3.1 Since, GDP is the total expenditure and it also includes public expenditures for education(EE) and as the variation in the GDP depends upon many other factors which are not necessarily smooth or similar across different values hence the GDP will vary differently for a range of values of public expenditures for education(EE) this is expected to result in heteroscedasticity.

3.2

We are interested in finding the linear relationship between per capita expenditures on education(Y) and per capita GDP(X). The p-value corresponding to X is 0.000 that means these two have highly significant linear realtionship and R-Squared value suggests that X is capable of explaining 85.75% variability of Y. Also, the standard error of regression(S.E. of regression) is quite low hence it is an acceptable model to predict Y from X.