Economics 310 Lecture 13 Heteroscedasticity Continued

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.

Goldfeld-Quandt Test

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

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.

Shazam Commands sample 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

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 CHARGE E E CONSTANT E |_diagnos / chowone=589 REQUIRED MEMORY IS PAR= 123 CURRENT PAR= 500 DEPENDENT VARIABLE = LOS 859 OBSERVATIONS REGRESSION COEFFICIENTS E SEQUENTIAL CHOW AND GOLDFELD-QUANDT TESTS N1 N2 SSE1 SSE2 CHOW PVALUE G-Q DF1 DF2 PVALUE

Breusch-Pagan-Godfrey Test

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.

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

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 E E MCPH CONSTANT |_gen1 ess=$ssr..NOTE..CURRENT VALUE OF $SSR = |_gen1 pbg=ess/2 |_Print pbg PBG

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

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 E HETEROSKEDASTICITY TESTS E**2 ON YHAT: CHI-SQUARE = WITH 1 D.F. E**2 ON YHAT**2: CHI-SQUARE = WITH 1 D.F. E**2 ON LOG(YHAT**2): CHI-SQUARE = WITH 1 D.F. E**2 ON X (B-P-G) TEST: CHI-SQUARE = WITH 2 D.F. E**2 ON LAG(E**2) ARCH TEST: CHI-SQUARE = WITH 1 D.F. LOG(E**2) ON X (HARVEY) TEST: CHI-SQUARE = WITH 2 D.F. ABS(E) ON X (GLEJSER) TEST: CHI-SQUARE = WITH 2 D.F.

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

White Test

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

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 $R2 = |_gen1 numb=$n..NOTE..CURRENT VALUE OF $N = |_gen1 white=numb*rsqaux |_print white WHITE Note: Critical chi-square 5 df. =

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.

Covariance Matrix OLS

OB Example with White Correction |_ols los rate charge / hetcov USING HETEROSKEDASTICITY-CONSISTENT COVARIANCE MATRIX R-SQUARE = R-SQUARE ADJUSTED = VARIANCE OF THE ESTIMATE-SIGMA**2 = STANDARD ERROR OF THE ESTIMATE-SIGMA = SUM OF SQUARED ERRORS-SSE= MEAN OF DEPENDENT VARIABLE = LOG OF THE LIKELIHOOD FUNCTION = VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY NAME COEFFICIENT ERROR 856 DF P-VALUE CORR. COEFFICIENT AT MEANS RATE CHARGE E E CONSTANT

Estimated Generalized Least- Squares

Possible variance forms