SHAZAM Logit - Test for Heteroskedasticity

Logit and Probit Models - Testing for Heteroskedasticity


Davidson and MacKinnon (1984) propose test statistics for heteroskedasticity in logit and probit models. It is assumed that the heteroskedasiticity is a function of variables Z. The Z variables are typically chosen from the X variables that are included in the logit or probit model. Test statistics are based on the Lagrange multiplier (LM) principle. The estimation results from a logit or probit model are used to construct an artificial regression designed to test for heteroskedasticity. A test statistic is the explained sum of squares from the artificial regression.

Sampling experiments were used to compare the properties of alternative forms of the LM test statistics. Davidson and MacKinnon (1984, p. 259) concluded that the test statistic named LM2 "tends to be the most reliable test under the null, but not the most powerful".

The SHAZAM procedure TESTHET calculates the Davidson and MacKinnon LM2 test statistic for heteroskedasticity in a logit or probit model. A SHAZAM procedure is a set of SHAZAM commands that perform a specific task. The set of procedure commands can be maintained in a file in the same folder as the SHAZAM command file. The commands are executed with a SHAZAM EXEC command.

The general format for using the TESTHET procedure is as follows.

FILE PROC   filename of the TESTHET procedure file
MODEL: LOGIT
DEPVAR: Dependent variable (0-1 binary variable)
X: List of explanatory variables in the model (a constant term is assumed)
Z: List of variables in the error variance equation
EXEC TESTHET

The MODEL: LOGIT line can be replaced by MODEL: PROBIT for the probit model. This line must be entered exactly as stated - that is, one blank space followed by the name LOGIT or PROBIT in upper case.

Warning: The TESTHET procedure assumes the SAMPLE command starts at observation 1 and there are no SKIPIF commands or missing values. These variations require appropriate modification to the procedure commands.

SHAZAM commands for applying tests of heteroskedasticity following logit estimation for the school budget voting model are below.

SAMPLE 1 95
READ (school.txt) PUB12 PUB34 PUB5 PRIV YEARS SCHOOL &
 LOGINC PTCON YESVM

LOGIT YESVM PUB12 PUB34 PUB5 PRIV YEARS SCHOOL LOGINC PTCON 

* Test for heteroskedasticity 
FILE PROC TESTHET
MODEL: LOGIT
*   Dependent variable
DEPVAR: YESVM
*   List of explanatory variables (a constant term is assumed)
X: PUB12 PUB34 PUB5 PRIV YEARS SCHOOL LOGINC PTCON
*   List of variables in the error variance equation
*   Include all the explanatory variables in the model.
Z: PUB12 PUB34 PUB5 PRIV YEARS SCHOOL LOGINC PTCON

*   Get the LM test statistic for heteroskedasticity
EXEC TESTHET

*   Now assume a different form for the heteroskedasticity.
*   Test that the error variance is a function of the SCHOOL variable.
Z: SCHOOL
EXEC TESTHET
STOP

The SHAZAM output can be viewed.

The first test considered that the heteroskedasiticity was a function of all the explanatory variables in the logit model. The calculated test statistic was 5.72. A comparison with the chi-square distribution with 8 degrees of freedom gives a p-value of 0.68. Therefore, there is no evidence of heteroskedasiticity at any of the usual significance levels.

The second test for heteroskedasticity considered the possibility of a different error variance for school teachers and individuals in occupations other than school teaching. (It can be noted that, for an OLS regression, the Goldfeld-Quandt test is designed for testing for different error variances in two groups of observations). For this test, the calculated test statistic was 1.96. The p-value of 0.16 again suggests no evidence of heteroskedasticity.


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TESTHET

SHAZAM procedure for testing for heteroskedasticity in logit and probit models

=SET NOECHO
PROC TESTHET
* Logit and Probit Models - Test for heteroskedasticity
* Reference: R. Davidson and J.G. MacKinnon, "Convenient Specification
*   Tests for Logit and Probit Models", Journal of Econometrics,
*   Vol 25, 1984, pp. 241-262.
SET NODOECHO NOOUTPUT
GEN1 TYPE_="[MODEL]"  
* Check that the model type is valid
FORMAT(' ERROR: Model must be either PROBIT or LOGIT')
IF ((TYPE_.NE." LOGIT").AND.(TYPE_.NE." PROBIT"))   
  PRINT / FORMAT 
IF ((TYPE_.NE." LOGIT").AND.(TYPE_.NE." PROBIT"))
  STOP

* Model estimation
[MODEL] [DEPVAR] [X] / INDEX=XBETA_ PREDICT=CDF_

IF (TYPE_.EQ." LOGIT") 
 GENR PDF_=(1+EXP(-XBETA_))/((1+EXP(-XBETA_))**2)
IF (TYPE_.EQ." PROBIT")  
 DISTRIB XBETA_ / TYPE=NORMAL PDF=PDF_ 

COPY [Z] Z_
MATRIX Z_=Z_
GEN1 DF_=$COLS
* Equation (26), p. 247.  
GENR ONE_=1 
COPY [X] ONE_ X_
DO #=1,DF_
 MATRIX ZZ_=Z_(0,#)
 GENR ZZ_=-XBETA_*ZZ_
 MATRIX Z_(0,#)=ZZ_
ENDO
MATRIX X_ = X_ | Z_
* Equations (16) and (17) , p. 245.     
GENR YAUX_=[DEPVAR]*SQRT((1-CDF_)/CDF_) + ([DEPVAR]-1)*SQRT(CDF_/(1-CDF_))
MATRIX R_=(PDF_/SQRT(CDF_*(1-CDF_)))*X_
* Artificial regression - Equation (18), p. 246.
OLS YAUX_ R_ / NOCONSTANT
* LM test statistic - explained sum of squares
GEN1 LM2=$ZSSR
* p-value
DISTRIB LM2 / TYPE=CHI DF=DF_
GEN1 pvalue_=1-$CDF
* Print results
PRINT MODEL / NONAME
FORMAT(' Test statistic for heteroskedasticity  LM2 ='/F15.5)
PRINT LM2 / NONAME FORMAT  
FORMAT(' chi-square degrees of freedom'/5X,F5.0)
PRINT DF_ / NONAME FORMAT  
FORMAT(' p-value'/5X,F10.5)
PRINT pvalue_ / NONAME FORMAT  
DELETE / ALL_
SET DOECHO OUTPUT 
PROCEND
SET ECHO


SHAZAM output

|_SAMPLE 1 95
|_READ (school.txt) PUB12 PUB34 PUB5 PRIV YEARS SCHOOL &
|  LOGINC PTCON YESVM
UNIT 88 IS NOW ASSIGNED TO: school.txt
   9 VARIABLES AND       95 OBSERVATIONS STARTING AT OBS       1

|_LOGIT YESVM PUB12 PUB34 PUB5 PRIV YEARS SCHOOL LOGINC PTCON

 LOGIT ANALYSIS     DEPENDENT VARIABLE =YESVM    CHOICES =  2
      95. TOTAL OBSERVATIONS
      59. OBSERVATIONS AT ONE
      36. OBSERVATIONS AT ZERO
  25 MAXIMUM ITERATIONS
CONVERGENCE TOLERANCE =0.00100

LOG OF LIKELIHOOD WITH CONSTANT TERM ONLY =    -63.037
BINOMIAL  ESTIMATE = 0.6211
ITERATION  0      LOG OF LIKELIHOOD FUNCTION =   -63.037

ITERATION  1 ESTIMATES
 0.45375     0.92076     0.43035    -0.28835    -0.23416E-01  1.3330
  1.6059     -1.7546     -3.7958
ITERATION  1      LOG OF LIKELIHOOD FUNCTION =   -54.139

ITERATION  2 ESTIMATES
 0.55298      1.0944     0.50979    -0.32984    -0.25855E-01  2.1655
  2.0427     -2.2551     -4.7103
ITERATION  2      LOG OF LIKELIHOOD FUNCTION =   -53.370

ITERATION  3 ESTIMATES
 0.58166      1.1250     0.52500    -0.33987    -0.26178E-01  2.5635
  2.1706     -2.3799     -5.1361
ITERATION  3      LOG OF LIKELIHOOD FUNCTION =   -53.304

ITERATION  4 ESTIMATES
 0.58362      1.1261     0.52605    -0.34139    -0.26129E-01  2.6239
  2.1869     -2.3942     -5.2003
ITERATION  4      LOG OF LIKELIHOOD FUNCTION =   -53.303

ITERATION  5 ESTIMATES
 0.58364      1.1261     0.52606    -0.34142    -0.26127E-01  2.6250
  2.1872     -2.3945     -5.2014

                                ASYMPTOTIC                         WEIGHTED
VARIABLE    ESTIMATED      STANDARD     T-RATIO    ELASTICITY      AGGREGATE
  NAME     COEFFICIENT       ERROR                  AT MEANS      ELASTICITY
PUB12         0.58364      0.68778      0.84858      0.93986E-01  0.91051E-01
PUB34          1.1261      0.76820       1.4659      0.11827      0.96460E-01
PUB5          0.52606       1.2693      0.41445      0.73664E-02  0.69375E-02
PRIV         -0.34142      0.78299     -0.43605     -0.11952E-01 -0.12037E-01
YEARS        -0.26127E-01  0.26934E-01 -0.97006     -0.73996E-01 -0.68592E-01
SCHOOL         2.6250       1.4101       1.8616      0.10108      0.28999E-01
LOGINC         2.1872      0.78781       2.7763       7.2529       6.7561
PTCON         -2.3945       1.0813      -2.2145      -5.5262      -5.1745
CONSTANT      -5.2014       7.5503     -0.68890      -1.7298      -1.6137

SCALE FACTOR =   0.22197

VARIABLE      MARGINAL      ----- PROBABILITIES FOR A TYPICAL CASE -----
  NAME         EFFECT        CASE         X=0          X=1        MARGINAL
                            VALUES                                 EFFECT
PUB12         0.12955       0.0000      0.44231      0.58706      0.14476
PUB34         0.24996       0.0000      0.44231      0.70978      0.26747
PUB5          0.11677       0.0000      0.44231      0.57304      0.13073
PRIV         -0.75785E-01   0.0000      0.44231      0.36049     -0.81814E-01
YEARS        -0.57995E-02   8.5158
SCHOOL        0.58267       0.0000      0.44231      0.91631      0.47400
LOGINC        0.48548       9.9711
PTCON        -0.53150       6.9395

LOG-LIKELIHOOD FUNCTION =  -53.303
LOG-LIKELIHOOD(0)  =   -63.037
LIKELIHOOD RATIO TEST  =    19.4681    WITH     8  D.F.   P-VALUE= 0.01255

ESTRELLA R-SQUARE           0.19956
MADDALA R-SQUARE            0.18529
CRAGG-UHLER R-SQUARE        0.25218
MCFADDEN R-SQUARE           0.15442
     ADJUSTED FOR DEGREES OF FREEDOM        0.75759E-01
     APPROXIMATELY F-DISTRIBUTED    0.20544      WITH        8  AND     9  D.F.
CHOW R-SQUARE               0.17197

         PREDICTION SUCCESS TABLE
                       ACTUAL
                 0             1
          0     18.            7.
PREDICTED 1     18.           52.

NUMBER OF RIGHT PREDICTIONS =        70.0
PERCENTAGE OF RIGHT PREDICTIONS =    0.73684
NAIVE MODEL PERCENTAGE OF RIGHT PREDICTIONS =    0.62105

EXPECTED OBSERVATIONS AT 0  =         36.0   OBSERVED =     36.0
EXPECTED OBSERVATIONS AT 1  =         59.0   OBSERVED =     59.0
SUM OF SQUARED "RESIDUALS" =           18.513
WEIGHTED SUM OF SQUARED "RESIDUALS" =     86.839

HENSHER-JOHNSON PREDICTION SUCCESS TABLE
                                           OBSERVED    OBSERVED
                    PREDICTED  CHOICE        COUNT       SHARE
        ACTUAL           0          1
           0           17.591     18.409     36.000      0.379
           1           18.409     40.591     59.000      0.621

PREDICTED COUNT        36.000     59.000     95.000      1.000
PREDICTED SHARE         0.379      0.621      1.000
PROP. SUCCESSFUL        0.489      0.688      0.612
SUCCESS INDEX           0.110      0.067      0.083
PROPORTIONAL ERROR      0.000      0.000
NORMALIZED SUCCESS INDEX                      0.177

|_* Test for heteroskedasticity
|_FILE PROC TESTHET
UNIT 82 IS NOW ASSIGNED TO: TESTHET
|_MODEL: LOGIT
|_*   Dependent variable
|_DEPVAR: YESVM
|_*   List of explanatory variables (a constant term is assumed)
|_X: PUB12 PUB34 PUB5 PRIV YEARS SCHOOL LOGINC PTCON
|_*   List of variables in the error variance equation
|_*   Include all the explanatory variables in the model.
|_Z: PUB12 PUB34 PUB5 PRIV YEARS SCHOOL LOGINC PTCON
|_*   Get the LM test statistic for heteroskedasticity
|_EXEC TESTHET
 _PROC TESTHET
 _        SET NODOECHO NOOUTPUT
 LOGIT
Test statistic for heteroskedasticity  LM2 =
       5.72363
chi-square degrees of freedom
       8.
p-value
       0.67816
 _        PROCEND

|_*   Now assume a different form for the heteroskedasticity.
|_*   Test that the error variance is a function of the SCHOOL variable.
|_Z: SCHOOL
|_EXEC TESTHET
 _PROC TESTHET
 _        SET NODOECHO NOOUTPUT
 LOGIT
Test statistic for heteroskedasticity  LM2 =
       1.96123
chi-square degrees of freedom
       1.
p-value
       0.16138
 _        PROCEND
|_STOP


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