Methods Seminar: Heteroskedasticity & Autocorrelation

1. Methods Seminar: Heteroskedasticity & Autocorrelation Kira R. Fabrizio Fall 2008

2. Today’s Agenda • Introduce Heteroskedasticity (H) and Autocorrelation (AC). • For each: • What is it? • Why do we care? • Why does it occur? • What can we do about it? • Application: Data exercise for next class (Nov. 17th)

3. What are H & AC? • Basic OLS model: • We assume: (or) (or)

4. What are H & AC? • Basic OLS model: • We assume: Heteroskedasticity (or) (or)

5. What are H & AC? • Basic OLS model: • We assume: (or) (or) Autocorrelation

6. Why do we care? • OLS estimator no longer BLU • Heteroskedasticity: • OLS not the best estimator • Unbiased, but inefficient • Estimated SEs biased (too small if error variance increases w/ X) • Autocorrelation: • OLS not the best estimator • Unbiased, but inefficient • Will underestimate the true variance • SEs smaller than they should be • Reject H0 when you should not • BOTH: Invalid hypothesis testing.

7. When is H likely? • Heteroskedasticity: • When variance of dependent variable varies across observations. • Ex: Savings as a function of income. • Average savings increases with income. • But variability of savings may also increase with income. • Omitted variable not correlated with included variables, but have differing order of magnitude for different (groups of) observations. • Ex: Cross-sectional data on units of different size (e.g. states, cities). Omitted variables may be larger for more populous states / cities.

8. Example of H: Professor Pay • Data on 222 professors from 7 schools • Years of experience • Salary • Graph of salary versus years of experience • Graph of ln(salary) versus years of experience

9. Example of H: Professor Pay • Estimate equation: Y = ln(salary) X= work experience Regression output Graph

10. What can we do about it? • In general • Assume nothing is wrong, run OLS model • Examine the residuals • Plot squared residuals against the independent variable(s) – any evidence of a relationship? Noise?

11. Dealing w/ Heteroskedasticity • Tests: • Breusch-Pagan test / White test • Glejser test • Harvey-Godfrey test • Corrections: • When σ is known • Whenσ is not known

12. Dealing w/ Heteroskedasticity • Tests: • Breusch-Pagan test • Glejser test • Harvey-Godfrey test

13. Test Procedure • Estimate by OLS, obtain residuals ei, i=1,…,n • Estimate linear regression of ei2(BP), |ei| (G), or ln ei2 (HG) on constant and vector of Z’s and compute R2 • Compute test statistic LM=nR2 for H0:α2=0,…, αL=0. [Chi-squared dist w/ L-1 df]

14. Test Procedure • Example of BP test Z2=years, Z3=years squared • Example of White test Add Z4=years*years squared • In stata: estat hettest

15. Dealing w/ Heteroskedasticity • Corrections when σi is known • Transform the model to get rid of the heteroskedasticity • Weighted least squares (GLS): • give less weight to the observations with greater variance and more weight to the observations with less variance. • Generates BLU estimators

16. Dealing w/ Heteroskedasticity • Corrections when σi is not known • Run OLS, but correct the SE estimates: • White heteroskedasticity-consistent SEs • Corrects the var-cov matrix for differences in variance • Use “robust” option command in stata • If suspect variance based on groups (e.g. states, firms), use “cluster(var)” option in stata • Example comparison of results

17. What are H & AC? • Basic OLS model: • We assume: (or) (or) Autocorrelation

18. When is AC likely? • Autocorrelation: • Omitted variable • Mis-specified functional form (e.g. straight line fitted where curve should be) • Spatial or time pattern to the data • Ex: Observation at t correlated with t-1

19. What can we do about it? • In general • Assume nothing is wrong, run OLS model • Examine the residuals • Plot residuals over time [or against the independent variable(s)] – any evidence of a relationship? Noise?

20. Example: Patent data • Panel data, 546 technology classes for the 21 years 1980-2000 (11,466 obs). • # patents in class-year (#Patsk,t) • # university patents in class-year (#UnivPatk,t) • Average # citations to “science” in class-year (#Sciencek,t) • What is the relationship between the number of citations to “science” and the number of university patents in a tech class?

21. Example: Patent data • Panel data: tsset the data in stata • Tech class fixed effects model • Examine residuals • Scatter1 • Scatter2 • Scatter3

22. Regression Result • . xi: xtreg NumUnivPats NumPats avgscience i.appyear, fe i(ipc_num) • i.appyear _Iappyear_1980-2000 (naturally coded; _Iappyear_1980 omitted) • Fixed-effects (within) regression Number of obs = 11466 • Group variable (i): ipc_num Number of groups = 546 • R-sq: within = 0.1369 Obs per group: min = 21 • between = 0.1743 avg = 21.0 • overall = 0.1213 max = 21 • F(22,10898) = 78.58 • corr(u_i, Xb) = -0.4119 Prob > F = 0.0000 • ------------------------------------------------------------------------------ • NumUnivPats | Coef. Std. Err. t P>|t| [95% Conf. Interval] • -------------+---------------------------------------------------------------- • NumPats | .0092758 .0002828 32.80 0.000 .0087214 .0098301 • avgscience | -.9008765 .0640646 -14.06 0.000 -1.026455 -.7752982 • _cons | -.7809801 .2910149 -2.68 0.007 -1.351422 -.2105379 • -------------+---------------------------------------------------------------- • sigma_u | 3.6099338 • sigma_e | 6.7349259 • rho | .22317918 (fraction of variance due to u_i) • ------------------------------------------------------------------------------ • F test that all u_i=0: F(545, 10898) = 3.92 Prob > F = 0.0000 • **Year variables excluded from output in the interest of space.

23. Dealing w/ AC • Tests: • Durbin-Watson Statistic (estat dwatson) for 1st order serial correlation. • Breusch-Godfret test (estat bgodfrey) for higher order serial correlation. • In panel data: Wooldridge test for serial correlation (xtserial).

24. Dealing w/ AC • Test: Durbin-Watson Statistic When N large, 1st 2 terms in numerator almost =, so • For strong positive AC, ρ=1 and DW=0 • For strong negative AC, ρ=-1 and DW=4 • For no autocorrelation, ρ=0 and DW=2

25. Dealing w/ AC • Test for Panel Data: Wooldridge test for serial correlation (xtserial). Stata Output Wooldridge test for autocorrelation in panel data H0: no first order autocorrelation F( 1, 545) = 1096.592 Prob > F = 0.0000

26. Dealing w/ AC • Corrections: • Transform model with estimated ρ, run FGLS • Durbin-Watson method • Cochran-Orcutt method

27. Dealing w/ AC • Corrections: Transform model with estimated ρ • Durbin-Watson method: • Cochran-Orcutt method: with OLS, then obtain

28. Dealing w/ AC • Corrections: • Prais Winston (GLS) estimation (prais in stata), with consistent errors in the presence of AR(1) serial correlation. • Has option for Cochrane-Orcutt option

29. Prais Winston Regression • . prais NumUnivPats NumPats avgscience appyear* • Number of gaps in sample: 545 (gap count includes panel changes) • (note: computations for rho restarted at each gap) • Prais-Winsten AR(1) regression -- iterated estimates • Source | SS df MS Number of obs = 11466 • -------------+------------------------------ F( 22, 11443) = 43.46 • Model | 12933.8561 22 587.902549 Prob > F = 0.0000 • Residual | 154801.107 11443 13.5280177 R-squared = 0.0771 • -------------+------------------------------ Adj R-squared = 0.0753 • Total | 167734.963 11465 14.6301756 Root MSE = 3.678 • ------------------------------------------------------------------------------ • NumUnivPats | Coef. Std. Err. t P>|t| [95% Conf. Interval] • -------------+---------------------------------------------------------------- • NumPats | .0075549 .0003296 22.92 0.000 .0069088 .008201 • avgscience | -.3079257 .0327598 -9.40 0.000 -.3721406 -.2437108 • _cons | -.7945959 .7144436 -1.11 0.266 -2.195028 .6058359 • -------------+---------------------------------------------------------------- • rho | .9753461 • ------------------------------------------------------------------------------ • Durbin-Watson statistic (original) 0.272618 • Durbin-Watson statistic (transformed) 1.378562

30. Prais Winston Regression • . prais DNumUnivPats DNumPats Davgscience Dappyear* • Number of gaps in sample: 545 (gap count includes panel changes) • (note: computations for rho restarted at each gap) • Prais-Winsten AR(1) regression -- iterated estimates • Source | SS df MS Number of obs = 11466 • -------------+------------------------------ F( 22, 11443) = 43.45 • Model | 12833.4183 22 583.337194 Prob > F = 0.0000 • Residual | 153628.16 11443 13.4255143 R-squared = 0.0771 • -------------+------------------------------ Adj R-squared = 0.0753 • Total | 166461.578 11465 14.5191085 Root MSE = 3.6641 • ------------------------------------------------------------------------------ • DNumUnivPats | Coef. Std. Err. t P>|t| [95% Conf. Interval] • -------------+---------------------------------------------------------------- • DNumPats | .0074327 .0003337 22.27 0.000 .0067785 .0080868 • Davgscience | -.3304883 .0333152 -9.92 0.000 -.3957917 -.2651849 • _cons | -2.61e-09 .3323198 -0.00 1.000 -.6514038 .6514038 • -------------+---------------------------------------------------------------- • rho | .9272602 • ------------------------------------------------------------------------------ • Durbin-Watson statistic (original) 0.323533 • Durbin-Watson statistic (transformed) 1.343161

31. Application: Data • Panel data: Annual (1981-1999) data on 278 US electricity generating plants (5282 obs). • Inputs: fuel, employees • Output: Megawatt hours (Mwhs) of electricity • Plant characteristics: MW size, fuel (gas, coal, etc) • Goal: Estimate labor productivity (Mwhs per employee) at the plant, controlling for plant characteristics (MW size and fuel type) and year effects.

32. Application: Data • Regression: • Expectations?

33. Application: Assignment • Get to know the structure of the data. Do you expect heteroskedasticity and / or autocorrelation in this data? Why or why not? • TEST for the presence of each – what do your tests indicate? • Run regression (w/o correction) • Assuming H and/or AC are present, what impact are they having on your results? • Modify regression to deal with H / AC • How do results change? • How do you know whether you solved the problem(s)?

34. Next Class • Please bring • Written notes from assignment • Print out of “log” from stata & graphs • THOUGHTS about what you did • QUESTIONS about this and other applications