SHAZAM Logit

## Logit and Probit Analysis

When the dependent variable is a 0-1 binary variable the logit or probit model estimation methods can be used. In SHAZAM, these methods are implemented with the `LOGIT` and `PROBIT` commands. The logit model is discussed and illustrated here. The probit model can be implemented in a similar style.

For the `LOGIT` command, the general command format is:

 `LOGIT depvar indeps / options `

where depvar is a 0-1 binary dependent variable, indeps is a list of the explanatory variables and options is a list of desired options. The list of options is described in the SHAZAM User's Reference Manual.

The logit model assumes that the response probability has the form:

An equivalent form can be stated by noting that:

The function guarantees probabilities in the (0,1) range. The logit form also gives a plausible shape for the marginal effects. That is, for a continuous variable Xk, at relatively high values, a marginal change will give a relatively smaller change in the probability of a success (Y=1).

The estimation problem is to find estimates of the unknown parameters .

#### Example

A data set on voting decisions for a school budget is available. The question of interest is: what factors influence the probability of a yes vote ? This question can be answered by interpreting the estimation results from a logit model. SHAZAM commands are given below.

 ```SAMPLE 1 95 READ (school.txt) PUB12 PUB34 PUB5 PRIV YEARS SCHOOL & LOGINC PTCON YESVM * The income and tax variables are in logarithms -- take anti-logs * to express the variables in thousands of \$. * Income GENR INCOME=EXP(LOGINC)/1000 * Property taxes GENR TAX=EXP(PTCON)/1000 * LOGIT estimation. LOGIT YESVM PUB12 PUB34 PUB5 PRIV YEARS SCHOOL INCOME TAX * Now use the log transformed form of income and taxes. LOGIT YESVM PUB12 PUB34 PUB5 PRIV YEARS SCHOOL LOGINC PTCON * Use the LOG option to compute elasticities and marginal effects * assuming log-transformed variables. LOGIT YESVM PUB12 PUB34 PUB5 PRIV YEARS SCHOOL LOGINC PTCON / LOG COEF=BETA STOP ```

The first model estimation includes the income and property tax variables in levels. The second model estimation includes log transformations of the income and property tax variables. Rubinfeld (1977, p. 35) comments: "The inclusion of logarithmic income and price terms resulted in a better fit than the inclusion of linear forms of the variables".

The SHAZAM output can be viewed. The results are discussed in the following sections:

#### References

Good textbook discussion is:

William Greene, Econometric Analysis.

Jeffrey M. Wooldridge, Introductory Econometrics: A Modern Approach.

References with more technical details are:

R. Davidson and J.G. MacKinnon, "Convenient Specification Tests for Logit and Probit Models", Journal of Econometrics, Vol 25, 1984, pp. 241-262.

D. A. Hensher and L. W. Johnson, Applied Discrete-Choice Modelling, John Wiley & Sons, 1981.

G. S. Maddala, Limited-dependent and Qualitative Variables in Econometrics, Cambridge University Press, 1983.

Kenneth Train, Qualitative Choice Analysis: Theory, Econometrics and an Application to Automobile Demand, MIT Press, 1986.

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#### SHAZAM output

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

|_* The income and tax variables are in logarithms -- take anti-logs
|_* to express the variables in thousands of \$.
|_* Income
|_* Property taxes
|_GENR TAX=EXP(PTCON)/1000
|_* LOGIT estimation.
|_LOGIT YESVM PUB12 PUB34 PUB5 PRIV YEARS SCHOOL INCOME TAX

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
0.54133     0.97999     0.39823    -0.23810    -0.28618E-01  1.1845
0.49110E-01 -1.6498     0.68486
ITERATION  1      LOG OF LIKELIHOOD FUNCTION =   -55.958

ITERATION  2 ESTIMATES
0.61000      1.1179     0.44480    -0.30742    -0.31099E-01  1.7144
0.63240E-01 -2.0213     0.75025
ITERATION  2      LOG OF LIKELIHOOD FUNCTION =   -55.560

ITERATION  3 ESTIMATES
0.62370      1.1363     0.44904    -0.31404    -0.31469E-01  1.8634
0.65039E-01 -2.0686     0.75393
ITERATION  3      LOG OF LIKELIHOOD FUNCTION =   -55.548

ITERATION  4 ESTIMATES
0.62413      1.1368     0.44921    -0.31413    -0.31480E-01  1.8724
0.65077E-01 -2.0696     0.75389

ASYMPTOTIC                         WEIGHTED
VARIABLE    ESTIMATED      STANDARD     T-RATIO    ELASTICITY      AGGREGATE
NAME     COEFFICIENT       ERROR                  AT MEANS      ELASTICITY
PUB12         0.62413      0.66847      0.93366      0.10588      0.10248
PUB34          1.1368      0.74861       1.5185      0.12577      0.10148
PUB5          0.44921       1.2500      0.35937      0.66268E-02  0.61577E-02
PRIV         -0.31413      0.77985     -0.40281     -0.11585E-01 -0.11295E-01
YEARS        -0.31480E-01  0.26096E-01  -1.2063     -0.93925E-01 -0.88468E-01
SCHOOL         1.8724       1.1255       1.6636      0.75959E-01  0.27663E-01
INCOME        0.65077E-01  0.35634E-01   1.8263      0.52655      0.48027
TAX           -2.0696       1.0383      -1.9932     -0.78308     -0.73375
CONSTANT      0.75389       1.1352      0.66411      0.26413      0.24491

SCALE FACTOR =   0.22761

VARIABLE      MARGINAL      ----- PROBABILITIES FOR A TYPICAL CASE -----
NAME         EFFECT        CASE         X=0          X=1        MARGINAL
VALUES                                 EFFECT
PUB12         0.14206       0.0000      0.43871      0.59333      0.15462
PUB34         0.25874       0.0000      0.43871      0.70897      0.27026
PUB5          0.10224       0.0000      0.43871      0.55053      0.11182
PRIV         -0.71499E-01   0.0000      0.43871      0.36342     -0.75286E-01
YEARS        -0.71652E-02   8.5158
SCHOOL        0.42617       0.0000      0.43871      0.83562      0.39691
INCOME        0.14812E-01   23.094
TAX          -0.47105       1.0800

LOG-LIKELIHOOD FUNCTION =  -55.548
LOG-LIKELIHOOD(0)  =   -63.037
LIKELIHOOD RATIO TEST  =    14.9788    WITH     8  D.F.   P-VALUE= 0.05956

ESTRELLA R-SQUARE           0.15452
CRAGG-UHLER R-SQUARE        0.19853
ADJUSTED FOR DEGREES OF FREEDOM        0.36838E-01
APPROXIMATELY F-DISTRIBUTED    0.15168      WITH        8  AND     9  D.F.
CHOW R-SQUARE               0.13244

PREDICTION SUCCESS TABLE
ACTUAL
0             1
0     14.            6.
PREDICTED 1     22.           53.

NUMBER OF RIGHT PREDICTIONS =        67.0
PERCENTAGE OF RIGHT PREDICTIONS =    0.70526
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" =           19.397
WEIGHTED SUM OF SQUARED "RESIDUALS" =     89.109

HENSHER-JOHNSON PREDICTION SUCCESS TABLE
OBSERVED    OBSERVED
PREDICTED  CHOICE        COUNT       SHARE
ACTUAL           0          1
0           16.718     19.282     36.000      0.379
1           19.282     39.718     59.000      0.621

PREDICTED COUNT        36.000     59.000     95.000      1.000
PREDICTED SHARE         0.379      0.621      1.000
PROP. SUCCESSFUL        0.464      0.673      0.594
SUCCESS INDEX           0.085      0.052      0.065
PROPORTIONAL ERROR      0.000      0.000
NORMALIZED SUCCESS INDEX                      0.138

|_* Now use the log transformed form of income and taxes.
|_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
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
CRAGG-UHLER R-SQUARE        0.25218
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

|_* Use the LOG option to compute elasticities and marginal effects
|_* assuming log-transformed variables.
|_LOGIT YESVM PUB12 PUB34 PUB5 PRIV YEARS SCHOOL LOGINC PTCON / LOG

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

ELASTICITIES ASSUME LOG-TRANSFORMED VARIABLES

ASYMPTOTIC                         WEIGHTED
VARIABLE    ESTIMATED      STANDARD     T-RATIO    ELASTICITY      AGGREGATE
NAME     COEFFICIENT       ERROR                  AT MEANS      ELASTICITY
PUB12         0.58364      0.68778      0.84858      0.19410      0.18107
PUB34          1.1261      0.76820       1.4659      0.37451      0.34937
PUB5          0.52606       1.2693      0.41445      0.17495      0.16321
PRIV         -0.34142      0.78299     -0.43605     -0.11355     -0.10592
YEARS        -0.26127E-01  0.26934E-01 -0.97006     -0.86893E-02 -0.81059E-02
SCHOOL         2.6250       1.4101       1.8616      0.87301      0.81439
LOGINC         2.1872      0.78781       2.7763      0.72739      0.67856
PTCON         -2.3945       1.0813      -2.2145     -0.79633     -0.74287
CONSTANT      -5.2014       7.5503     -0.68890      -1.7298      -1.6137

SCALE FACTOR =   0.22197

MARGINAL EFFECTS ASSUME ALL VARIABLES ARE LOG-TRANSFORMED
(EXCEPT DUMMY VARIABLES)

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        -0.28859E-21   8.5158
SCHOOL        0.58267       0.0000      0.44231      0.91631      0.47400
PTCON        -0.49214E-03   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
CRAGG-UHLER R-SQUARE        0.25218
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
|_STOP
```

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