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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.

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Economics 310

Lecture 13

Heteroscedasticity Continued

• 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.

Group 1

(n-c)/2=(20-4)/2=8 obs

C=4

Group 2

(n-c)/2=(20-4)/2=8 obs

• 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.

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

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

• 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.

* 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

|_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

• 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

|_?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.

• 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*R2 is chi-squared variable

?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

|_?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

• 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.

|_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