Question

(i) Consider a simple linear regression

yi = β0 + β1xi + ui

Write down the formula for the estimated standard error of the OLS estimator and the formula for the White heteroskedasticity-robust standard error on the estimated coefficient bβ1. (ii) What is the intuition for the White test for heteroskedasticity? (You do not need to describe how the test is implemented in practice.)

Answer 1

formula for the estimated standard error of the OLS estimator is as

estimated regression equation will be as

and now estimated standard error of the OLS estimator will be denoted by s.

where numerator is the sum of square of residuals computed from the estimation equation and given observations of x and y.

denomenator is the degrees of freedom where n is total observations and one is the intercept estimate which is excluded.

Now formula for white heteroskedasticity robust standard error on the slope coefficient would be firstly given is the variance of the OLS estimator for simple linear regression when population variance is known.

now when population variance is not known then we can replace it with corresponding least square error terms which would aid in deriving consistent estimator of the standard error.

white heteroskedasticity robust standard error is as

This is commonly used for large sample test.

The intuition for white test for heteroscedasticity is that it is an asymptotic test that estimated value tends to true parameters as sample size becomes large and that the ols estimates are consistent.