1. Economics 310 Lecture 13 Heteroscedasticity Continued

2. Tests to be Discussed Goldfeld-Quandt Test Assumes variance monotonically associated with some variable. Breusch-Pagan-Godfrey Test Variance linear function of set of variables or function of a linear combination of variables. White General Heteroscedasticity Test Source unknown, but may exist.

3. Goldfeld-Quandt Test

4. Data Organization Group 1 (n-c)/2=(20-4)/2=8 obs C=4 Group 2 (n-c)/2=(20-4)/2=8 obs

5. Obstetrics Example Data from 800+ hospitals. Dependent variable is the average length of stay in maternity ward. Explanatory variables is the charge per day and % of deliveries that are c-sections. Expect greater variability in length of stay at hospitals that are not subject to high managed care.

6. Shazam Commands sample 1 859 read (d:\econom~1\classe~1\ob1_het.txt) cases rate los cost billed neo mcph mcpm genr charge=billed/los ols los rate charge diagnos / chowone=589

7. Shazam Output for Goldfeld- Quandt Test VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY NAME COEFFICIENT ERROR 856 DF P-VALUE CORR. COEFFICIENT AT MEANS RATE 4.1647 0.2426 17.17 0.000 0.506 0.4940 0.4056 CHARGE -0.43644E-03 0.2652E-04 -16.46 0.000-0.490 -0.4737 -0.3229 CONSTANT 2.1049 0.6367E-01 33.06 0.000 0.749 0.0000 0.9173 |_diagnos / chowone=589 REQUIRED MEMORY IS PAR= 123 CURRENT PAR= 500 DEPENDENT VARIABLE = LOS 859 OBSERVATIONS REGRESSION COEFFICIENTS 4.16471785492 -0.436436399782E-03 2.10491763784 SEQUENTIAL CHOW AND GOLDFELD-QUANDT TESTS N1 N2 SSE1 SSE2 CHOW PVALUE G-Q DF1 DF2 PVALUE 589 270 238.20 21.357 40.564 0.000 5.082 586 267 0.000

8. Breusch-Pagan-Godfrey Test

9. Example of BPG Test using OB Data Null hypothesis is homoscedasticity Let the Z’s be (1) the number of of OB cases per year and (2) whether the hospital is under high managed care Expect variance to be negatively related to both variables.

10. Shazam Code for OB Example * performing Breusch-Pagan Test on cases and mcph ?ols los rate charge / resid=e dn anova gen1 sigsq=$sig2 genr esq=e*e genr p=esq/sigsq ols p cases mcph / anova gen1 ess=$ssr gen1 pbg=ess/2 Print pbg

11. Shazam Output for OB Example |_ols p cases mcph / anova VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY NAME COEFFICIENT ERROR 856 DF P-VALUE CORR. COEFFICIENT AT MEANS CASES -0.37767E-03 0.2353E-03 -1.605 0.109 -0.055 -0.0551 -0.5201 MCPH -0.52494 0.6677 -0.7861 0.432 -0.027 -0.0270 -0.1650 CONSTANT 1.6851 0.4774 3.529 0.000 0.120 0.0000 1.6851 |_gen1 ess=$ssr..NOTE..CURRENT VALUE OF $SSR = 288.00 |_gen1 pbg=ess/2 |_Print pbg PBG 144.0017

12. Built in BPG Test in Shazam Shazam has a built in BPG test. Uses the explanatory variables as the Zs. Invoked by using the command “DIAGNOS” with the option “HET” right after the “OLS” command. i.e. ols y x1 x2 diagnos / het

13. Using HET on OB example |_?ols los rate charge |_diagnos / het REQUIRED MEMORY IS PAR= 123 CURRENT PAR= 500 DEPENDENT VARIABLE = LOS 859 OBSERVATIONS REGRESSION COEFFICIENTS 4.16471785492 -0.436436399782E-03 2.10491763784 HETEROSKEDASTICITY TESTS E**2 ON YHAT: CHI-SQUARE = 19.852 WITH 1 D.F. E**2 ON YHAT**2: CHI-SQUARE = 80.223 WITH 1 D.F. E**2 ON LOG(YHAT**2): CHI-SQUARE = 1.018 WITH 1 D.F. E**2 ON X (B-P-G) TEST: CHI-SQUARE = 35.644 WITH 2 D.F. E**2 ON LAG(E**2) ARCH TEST: CHI-SQUARE = 0.027 WITH 1 D.F. LOG(E**2) ON X (HARVEY) TEST: CHI-SQUARE = 5.259 WITH 2 D.F. ABS(E) ON X (GLEJSER) TEST: CHI-SQUARE = 62.292 WITH 2 D.F.

14. White General Test for Heteroscedasticity This is a general test. No preconception of cause of heteroscedasticity Is a Lagrange-Multiplier Test Regress squared residuals on explanatory variables, their squares and their cross products. n*R 2 is chi-squared variable

15. White Test

16. Shazam code for White test for OB example ?ols los rate charge / resid=e genr esq=e*e genr rate2=rate*rate genr charge2=charge*charge genr charrate=charge*rate ?ols esq rate charge rate2 charge2 charrate gen1 rsqaux=$r2 gen1 numb=$n gen1 white=numb*rsqaux print white

17. Results of White’s test for OB example |_?ols los rate charge / resid=e |_genr esq=e*e |_genr rate2=rate*rate |_genr charge2=charge*charge |_genr charrate=charge*rate |_?ols esq rate charge rate2 charge2 charrate |_gen1 rsqaux=$r2..NOTE..CURRENT VALUE OF $R2 = 0.44959 |_gen1 numb=$n..NOTE..CURRENT VALUE OF $N = 859.00 |_gen1 white=numb*rsqaux |_print white WHITE 386.2005 Note: Critical chi-square 5 df. = 11.0705

18. White Correction Do not know the source of heteroscedasticity. Forced to use OLS estimates. Consistent estimate of true variance- covariance matrix of OLS estimators. Gives test of hypothesis that are asymptotically unbiased.

19. Covariance Matrix OLS

20. OB Example with White Correction |_ols los rate charge / hetcov USING HETEROSKEDASTICITY-CONSISTENT COVARIANCE MATRIX R-SQUARE = 0.3424 R-SQUARE ADJUSTED = 0.3409 VARIANCE OF THE ESTIMATE-SIGMA**2 = 0.34648 STANDARD ERROR OF THE ESTIMATE-SIGMA = 0.58863 SUM OF SQUARED ERRORS-SSE= 296.59 MEAN OF DEPENDENT VARIABLE = 2.2946 LOG OF THE LIKELIHOOD FUNCTION = -762.125 VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY NAME COEFFICIENT ERROR 856 DF P-VALUE CORR. COEFFICIENT AT MEANS RATE 4.1647 1.189 3.501 0.000 0.119 0.4940 0.4056 CHARGE -0.43644E-03 0.5672E-04 -7.694 0.000-0.254 -0.4737 -0.3229 CONSTANT 2.1049 0.1944 10.83 0.000 0.347 0.0000 0.9173

21. Estimated Generalized Least- Squares

22. Possible variance forms