Heteroskedasticity

1. Heteroskedasticity • Definition • Consequences of heteroscedasticity • Testing for Heteroskedasticity • Breusch-Pagan • White test (2 forms) • Fixing the problem • Robust standard errors • Weighted Least Squares • Fixing heteroskedasticity in LPM model.

2. Heteroskedasticity • Consequences of heteroskedasticity for OLS • OLS still unbiased and consistent under heteroskedastictiy • Interpretation of R-squared unchanged. • Heteroskedasticity invalidates variance formulas for OLS estimators • The usual F tests and t tests are not valid with heteroskedasticity • Under heteroskedasticity, OLS is no longer the best linear unbiased estimator (BLUE); there may be more efficient linear estimators Unconditional error variance is unaffected by heteroskedasticity (which refers to the conditional error variance)

3. Heteroskedasticity • Heteroskedasticity-robust inference after OLS estimation • Formulas for OLS standard errors and related statistics have been developed that are robust to heteroskedasticity of unknown form • All formulas are only valid in large samples • Formula for heteroskedasticity-robust OLS standard error • Using these formulas for standard errors, the usual t test is valid asymptotically valid with heteroskedasticity • The usual F statistic does not work under heteroskedasticity, but heteroskedasticity robust versions are available in Stata • To obtain heteroskedasticity robust standard errors and F-stats in Stata, reg y x, robustCoefficients are unchanged by robust option, but standard errors, t-statistics, and F-statistics change. Also called White/Huber/Eicker standard errors.

4. Heteroskedasticity • Example: Hourly wage equation Heteroskedasticity robust standard errors may be larger or smaller than their nonrobust counterparts. The differencesareoftensmall in practice. • F statistics are also often similar. If there is strong heteroskedasticity, differences may be larger. Tobe on thesafeside, itisadvisabletoalwayscompute robust standarderrors.

5. Heteroskedasticity • Testing for heteroskedasticity • Even with robust standard errors, it may still be useful to test whether there is heteroskedasticity because then OLS may not be the most efficient linear estimator anymore • Breusch-Pagan test for heteroskedasticity Under MLR.4 The meanof u2 must not varywith x1, x2, …, xk

6. Heteroskedasticity • Breusch-Pagan test for heteroskedasticity (cont.)Regress squared residuals on all expla-natory variables and test whether this regression has explanatory power. A large teststatistic (= a high R-squared) isevidenceagainstthe null hypothesis. Alternative teststatistic (= Lagrange multiplierstatistic, LM). Again, highvaluesoftheteststatistic (= high R-squared) leadtorejectionofthe null hypothesisthattheexpectedvalueof u2isunrelatedtotheexplanatory variables.

7. Heteroskedasticity • Example: Heteroskedasticity in housing price equations Homoskedasticity rejected homoskedasticity not rejected

8. Heteroskedasticity • The White test for heteroskedasticity • Disadvantage of this form of the White test • Including all squares and interactions leads to a large number of estimated parameters (e.g. k=6 leads to 27 parameters to be estimated) Regress squared residuals on all expla-natory variables, theirsquares, and in-teractions (here: examplefor k=3) The White test detects more general deviations from heteroskedasticity than the Breusch-Pagan test

9. Heteroskedasticity • Alternative form of the White test • Example: Heteroskedasticity in (log) housing price equations Thisregressionindirectlyteststhedependenceofthesquaredresiduals on theexplanatory variables, theirsquares, andinteractions, becausethepredictedvalueof y anditssquareimplicitlycontain all oftheseterms.

10. Heteroskedasticity • Weighted least squares estimation • Heteroskedasticity is known up to a multiplicative constant The functional form of the heteroskedasticity is known Transformed model

11. Heteroskedasticity • Example: Savings and income • The transformed model is homoskedastic • If the other Gauss-Markov assumptions hold as well, OLS applied to the transformed model is the best linear unbiased estimator Note thatthisregression model hasnointercept

12. Heteroskedasticity • OLS in the transformed model is weighted least squares (WLS) • WLS is more efficient than OLS because less weight is placed on observations with a large variance • In Stata, suppose estimate WLS with • gen inv_inc=1/inc • reg sav inc [aw=inv_inc] where inv_inc=(1/inc) • aw=analytic weights that are inversely proportional to variance of the residual Observationswith a large varianceget a smallerweight in theoptimizationproblem

13. Heteroskedasticity Example: Financial wealth equation Net financial wealth (in 1000s) Assumed form of heteroskedasticity Stata: reg nettfa inc age_25_sq male e401k [aw=1/inc]Code for estimation in g:\eco\evenwe\wls_with_401k Participation in 401K pension plan

14. Heteroskedasticity • Special cases of heteroskedasticity • If the observations are reported as averages at the city/county/state/-country/firm level, they should be weighted by the size of the unit • If observations are reported as aggregates, weight by inverse of size. Averagecontributiontopension plan in firm i Averageearningsandage in firm i Percentage firm contributesto plan Heteroskedastic error term Error variance if errors are homoskedastic at the individual-level If errors are homoskedastic at the individual-level, WLS with weights equal to firm size mi should be used. If the assumption of homoskedasticity at the individual-level is not exactly right, one can calculate robust standard errors after WLS (i.e. forthetransformed model).

15. Heteroskedasticity • Skip sections on • feasbile GLS (8-4b) • prediction intervals with heteroskedasticity (8-4d)

16. Heteroskedasticity • WLS in the linear probability model • Infeasible if LPM predictions are below zero or greater than one • If such cases are rare, they may be adjusted to values such as .01/.99 • Otherwise, it is probably better to use OLS with robust standard errors In the LPM, the exact form of heteroskedasticity is known Use inverse valuesasweights in WLS

17. Summary of Issues with Heteroskedasticity • If heteroscedasticity exists and not corrected for • Standard errors, t-statistics, and F-statistics are wrong • Coefficient estimates are still unbiased, but inefficient • Several tests for heteroscedasticity available • Breusch-Pagan • White (2 forms) • Corrections for heteroscedasticity • Heteroskedasticity robust standard errors • No change in coefficients (still inefficient like OLS), but standard errors are correct • Weighted Least Squares • More efficient than OLS and correct standard errors • Requires knowledge of functional form for heteroscedasticity • Linear probability model requires correction for heteroscedasticity • Robust standard errors are simple fix • WLS is more efficient, but creates problems if predicted probabilities are outside unit interval.