Heteroscedasticity 1

Transcription

1 Heteroscedasticity 1 Pierre Nguimkeu BUEC 333 Summer Based on P. Lavergne, Lectures notes

2 Outline Pure Versus Impure Heteroscedasticity Consequences and Detection Remedies

3 Pure Heteroscedasticity Homoscedasticity The variance of the error term is constant Heteroscedasticity The variance of the error terms varies, that is Var ε i = σ 2 i i = 1,..., n Violates Classical Assumption 5, which states that Var ε i = σ 2 i = 1,..., n. Pure heteroscedasticity The model is well specified, i.e. Classical Assumptions 1,2,3 holds, but there is heteroscedasticity. Occurs in particular In cross-section, when there is large variation in the dependent variable. In time-series, when there is large variation in the dependent variable over time. When the quality of data collection changes a lot across the sample.

4 Heteroscedasticity: Examples R i = rent of renter i, I i = income of renter i. R i = β 0 + β 1 I i + ε i Seems sensible to expect that not only mean of rent increases with income, but also that variance (or s.d.) of rent increases with income. W i = β 0 + β 1 E i + β 3 X i + ε i W i = wage of worker i, E i = education level of worker i, X i = experience level of worker i. Mean wage increases with education and experience, but wage dispersion also increases with education and experience.

5 Impure Heteroscedasticity Caused by misspecification in the model, i.e. Classical Assumption 1 does not hold. If Y i = β 0 + β 1 X 1i + β 2 X 2i + ε i but we omit X 2i, then Y i = β 0 + β 1 X 1i + ε i where ε i = ε i + β 2 X 2i If X 2 is relevant, i.e. β 2 0, Var ε i = f (X 2i ). We should write Y i = β 0 + β 1 X 1i + ε i ε i = ε i + β 0 β 0 + (β 1 β 1 ) X 1i + β 2 X 2i because nothing ensures the coefficients are the same in the two equations. If Y i = β 0 + β 1 X 1i + β 2 X1i 2 + ε i, but we specify a linear equation, then Y i = β 0 + β 1 X 1i + ε i ε i = ε i + f (X 1i ). ε i depends on X 1i, so its s.d.

6 Consequences Estimates remain unbiased OLS is not BLUE in general, then OLS has not minimum variance The standard errors are biased t-scores don t have a t-distribution, so confidence intervals and tests are unreliable. t-scores are often too large.

7 Preliminary Checks Are there any obvious specification errors? Delay testing for heteroscedasticity until you are confident with your specification. Is the dependent variable likely afflicted with heteroscedasticity? Range of dependent variable, previous studies,... Is there any likely factor of heteroscedasticity? Graph the residuals against this variable.

8 Rent versus Income Rent of renter Income of Renter Dependent Variable: RENT Method: Least Squares Date: 11/09/09 Time: 17:38 Sample: Included observations: 108 C INCOME R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid 1.16E+09 Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) RESID Income of Renter

9 The Park Test Assume that Var ε i = σ 2 Zi 2 i = 1,..., n for some proportionality factor Z you can observe. Then ln IEε 2 = ln σ ln Z i The same reasonning applies if Var ε i = σ 2 Z α i i = 1,..., n. 1. Estimate your equation by OLS and get the residuals e i 2. Run the OLS auxiliary regression ln e 2 i = α 0 + α 1 ln Z i + u i 3. Test the significance of α 1 with a t-test. The Park test assumes that there is only one proportionality factor and you know which one. We look at whether squared residuals are related to Z (all in logs).

10 Rent versus Income LOGRESIDSQ LOGRESIDSQ vs. Log Income of Renter Income of Renter Dependent Variable: LOG(RESID01^2) Method: Least Squares Date: 03/12/08 Time: 21:30 Sample: Included observations: 108 C LOG(INCOME) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) What is the outcome of the test?

11 1. Estimate your equation The White Test Y i = β 0 + β 1 X 1i + β 2 X 2i + ε i by OLS and get the residuals e i 2. Run the OLS auxiliary regression e 2 i = α 0 + α 1 X 1i + α 2 X 2i + α 3 X 2 1i + α 4 X 2i + α 5 X 1i X 2i + u i That is regress the squared residuals on all the independent variables, their squares and their cross-products. 3. Test the significance of all coefficients but α 0 with an F-test. H 0 : α 1 = α 2 =... = α 5 = 0 against H A : at least one is not 0 Beware of perfect multicollinearity: If the equation is Y i = β 0 + β 1 X 1i + β 2 X 2 1i + ε i regress squared residuals on an intercept, X 1i, X 2 1i, X 3 1i and X 4 1i.

12 Rent versus Income Dependent Variable: RESID01^2 Method: Least Squares Date: 11/09/09 Time: 17:58 Sample: Included observations: 108 C INCOME INCOME^ R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid 9.53E+16 Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) What is the outcome of the test?

13 Weighted Least-Squares Y i = β 0 + β 1 X 1i + β 2 X 2i + ε i and Var ε i = σ 2 Z 2 i. Then Y i Z i = β 0 1 Z i + β 1 X 1i Z i If Z i = X 1i, Now we can use OLS! But careful + β 2 X 2i Z i + u i Y i 1 X 2i = β 0 + β 1 + β 2 + u i X 1i X 1i X 1i is such that Var u i = Var ε i Z i = σ 2 There may be no intercept in the equation. The transformation is only to get OLS estimates, but interpretation relies on the original equation R i = β 0 + β 1 I i + ε i R i 1 = β 0 + β 1 + ε i I i I i β 1 : marginal effect of income on rent.

14 Rent versus Income.9 RATIO INVINCOME Dependent Variable: RENT/INCOME Method: Least Squares Date: 11/09/09 Time: 18:05 Sample: Included observations: 108 1/INCOME C R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) What is the marginal effect of income on rent?

15 Redefining the Model C i = expenditure in city i, Y i = income in city i, POP i = population in city i, W i = average wage in city i. C i = β 0 + β 1 Y i + β 2 POP i + β 3 W i + ε i When estimated by OLS, this formulation gives a large weight to the large cities. See Figure It makes sense to consider a specification that redefine the variables with respect to the size of the city, i.e. C i Y i = α 0 + α 1 + α 2 W i + u i POP i POP i This is a new formulation that relates per capita consumption to per capita income. There may still be heteroscedasticity.

16 Heteroscedasticity-Corrected Standard Errors In place of another estimation method or another model, we can use OLS (unbiased and consistent) and correct the standard errors. Heteroscedasticity-robust standard errors (White standard errors) Estimate the standard deviation of the OLS coefficients whether there is heteroscedasticity or not Are often larger than the OLS standard errors Can be used to construct tests and confidence intervals in the usual way Works well in large samples Are given by Eviews, see Options/Heteroscedasticty consistent coefficient covariance.

17 Rent versus Income Dependent Variable: RENT Method: Least Squares Date: 11/09/09 Time: 17:38 Sample: Included observations: 108 C INCOME R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid 1.16E+09 Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) Dependent Variable: RENT White Heteroskedasticity-Consistent Standard Errors & Covariance C INCOME R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid 1.16E+09 Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) OK, the difference is small here, but not always! Id the sample size large enough?

18 Log Hourly Wage versus Educ and Age Dependent Variable: LWAGE Method: Least Squares Date: 03/12/08 Time: 21:39 Sample: Included observations: 340 C EDUC AGE R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) Dependent Variable: RESID01^2 Method: Least Squares Date: 11/09/09 Time: 18:31 Sample: Included observations: 340 C EDUC AGE EDUC^ AGE^ EDUC*AGE R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) Seems like there is heteroscedasticity!

19 Log Hourly Wage versus Educ and Age Dependent Variable: LWAGE Method: Least Squares Date: 11/09/09 Time: 18:29 Sample: Included observations: 340 C EDUC AGE AGE^ R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) Dependent Variable: RESID02^2 Method: Least Squares Date: 11/09/09 Time: 18:32 Sample: Included observations: 340 C EDUC AGE EDUC^ AGE^ EDUC*AGE R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) It was likely impure!