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Logit Analysis

(2004-12-02 16:17:50) 下一个

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 beta.

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 95READ (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=BETASTOP

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:

Chapter 19 of William Greene, Econometric Analysis, Fourth Edition, Prentice-Hall, 2000.

Chapter 17 of Jeffrey M. Wooldridge, Introductory Econometrics: A Modern Approach, South-Western College Publishing, 2000.

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 &|  LOGINC PTCON YESVMUNIT 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|_GENR INCOME=EXP(LOGINC)/1000|_* 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 ITERATIONSCONVERGENCE TOLERANCE =0.00100LOG OF LIKELIHOOD WITH CONSTANT TERM ONLY =    -63.037BINOMIAL  ESTIMATE = 0.6211ITERATION  0      LOG OF LIKELIHOOD FUNCTION =   -63.037ITERATION  1 ESTIMATES 0.54133     0.97999     0.39823    -0.23810    -0.28618E-01  1.1845 0.49110E-01 -1.6498     0.68486ITERATION  1      LOG OF LIKELIHOOD FUNCTION =   -55.958ITERATION  2 ESTIMATES 0.61000      1.1179     0.44480    -0.30742    -0.31099E-01  1.7144 0.63240E-01 -2.0213     0.75025ITERATION  2      LOG OF LIKELIHOOD FUNCTION =   -55.560ITERATION  3 ESTIMATES 0.62370      1.1363     0.44904    -0.31404    -0.31469E-01  1.8634 0.65039E-01 -2.0686     0.75393ITERATION  3      LOG OF LIKELIHOOD FUNCTION =   -55.548ITERATION  4 ESTIMATES 0.62413      1.1368     0.44921    -0.31413    -0.31480E-01  1.8724 0.65077E-01 -2.0696     0.75389                                ASYMPTOTIC                         WEIGHTEDVARIABLE    ESTIMATED      STANDARD     T-RATIO    ELASTICITY      AGGREGATE  NAME     COEFFICIENT       ERROR                  AT MEANS      ELASTICITYPUB12         0.62413      0.66847      0.93366      0.10588      0.10248PUB34          1.1368      0.74861       1.5185      0.12577      0.10148PUB5          0.44921       1.2500      0.35937      0.66268E-02  0.61577E-02PRIV         -0.31413      0.77985     -0.40281     -0.11585E-01 -0.11295E-01YEARS        -0.31480E-01  0.26096E-01  -1.2063     -0.93925E-01 -0.88468E-01SCHOOL         1.8724       1.1255       1.6636      0.75959E-01  0.27663E-01INCOME        0.65077E-01  0.35634E-01   1.8263      0.52655      0.48027TAX           -2.0696       1.0383      -1.9932     -0.78308     -0.73375CONSTANT      0.75389       1.1352      0.66411      0.26413      0.24491SCALE FACTOR =   0.22761VARIABLE      MARGINAL      ----- PROBABILITIES FOR A TYPICAL CASE -----  NAME         EFFECT        CASE         X=0          X=1        MARGINAL                            VALUES                                 EFFECTPUB12         0.14206       0.0000      0.43871      0.59333      0.15462PUB34         0.25874       0.0000      0.43871      0.70897      0.27026PUB5          0.10224       0.0000      0.43871      0.55053      0.11182PRIV         -0.71499E-01   0.0000      0.43871      0.36342     -0.75286E-01YEARS        -0.71652E-02   8.5158SCHOOL        0.42617       0.0000      0.43871      0.83562      0.39691INCOME        0.14812E-01   23.094TAX          -0.47105       1.0800LOG-LIKELIHOOD FUNCTION =  -55.548LOG-LIKELIHOOD(0)  =   -63.037LIKELIHOOD RATIO TEST  =    14.9788    WITH     8  D.F.   P-VALUE= 0.05956ESTRELLA R-SQUARE           0.15452MADDALA R-SQUARE            0.14587CRAGG-UHLER R-SQUARE        0.19853MCFADDEN R-SQUARE           0.11881     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 =        67.0PERCENTAGE OF RIGHT PREDICTIONS =    0.70526NAIVE MODEL PERCENTAGE OF RIGHT PREDICTIONS =    0.62105EXPECTED OBSERVATIONS AT 0  =         36.0   OBSERVED =     36.0EXPECTED OBSERVATIONS AT 1  =         59.0   OBSERVED =     59.0SUM OF SQUARED "RESIDUALS" =           19.397WEIGHTED SUM OF SQUARED "RESIDUALS" =     89.109HENSHER-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      0.621PREDICTED COUNT        36.000     59.000     95.000      1.000PREDICTED SHARE         0.379      0.621      1.000PROP. SUCCESSFUL        0.464      0.673      0.594SUCCESS INDEX           0.085      0.052      0.065PROPORTIONAL ERROR      0.000      0.000NORMALIZED 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 ITERATIONSCONVERGENCE TOLERANCE =0.00100LOG OF LIKELIHOOD WITH CONSTANT TERM ONLY =    -63.037BINOMIAL  ESTIMATE = 0.6211ITERATION  0      LOG OF LIKELIHOOD FUNCTION =   -63.037ITERATION  1 ESTIMATES 0.45375     0.92076     0.43035    -0.28835    -0.23416E-01  1.3330  1.6059     -1.7546     -3.7958ITERATION  1      LOG OF LIKELIHOOD FUNCTION =   -54.139ITERATION  2 ESTIMATES 0.55298      1.0944     0.50979    -0.32984    -0.25855E-01  2.1655  2.0427     -2.2551     -4.7103ITERATION  2      LOG OF LIKELIHOOD FUNCTION =   -53.370ITERATION  3 ESTIMATES 0.58166      1.1250     0.52500    -0.33987    -0.26178E-01  2.5635  2.1706     -2.3799     -5.1361ITERATION  3      LOG OF LIKELIHOOD FUNCTION =   -53.304ITERATION  4 ESTIMATES 0.58362      1.1261     0.52605    -0.34139    -0.26129E-01  2.6239  2.1869     -2.3942     -5.2003ITERATION  4      LOG OF LIKELIHOOD FUNCTION =   -53.303ITERATION  5 ESTIMATES 0.58364      1.1261     0.52606    -0.34142    -0.26127E-01  2.6250  2.1872     -2.3945     -5.2014                                ASYMPTOTIC                         WEIGHTEDVARIABLE    ESTIMATED      STANDARD     T-RATIO    ELASTICITY      AGGREGATE  NAME     COEFFICIENT       ERROR                  AT MEANS      ELASTICITYPUB12         0.58364      0.68778      0.84858      0.93986E-01  0.91051E-01PUB34          1.1261      0.76820       1.4659      0.11827      0.96460E-01PUB5          0.52606       1.2693      0.41445      0.73664E-02  0.69375E-02PRIV         -0.34142      0.78299     -0.43605     -0.11952E-01 -0.12037E-01YEARS        -0.26127E-01  0.26934E-01 -0.97006     -0.73996E-01 -0.68592E-01SCHOOL         2.6250       1.4101       1.8616      0.10108      0.28999E-01LOGINC         2.1872      0.78781       2.7763       7.2529       6.7561PTCON         -2.3945       1.0813      -2.2145      -5.5262      -5.1745CONSTANT      -5.2014       7.5503     -0.68890      -1.7298      -1.6137SCALE FACTOR =   0.22197VARIABLE      MARGINAL      ----- PROBABILITIES FOR A TYPICAL CASE -----  NAME         EFFECT        CASE         X=0          X=1        MARGINAL                            VALUES                                 EFFECTPUB12         0.12955       0.0000      0.44231      0.58706      0.14476PUB34         0.24996       0.0000      0.44231      0.70978      0.26747PUB5          0.11677       0.0000      0.44231      0.57304      0.13073PRIV         -0.75785E-01   0.0000      0.44231      0.36049     -0.81814E-01YEARS        -0.57995E-02   8.5158SCHOOL        0.58267       0.0000      0.44231      0.91631      0.47400LOGINC        0.48548       9.9711PTCON        -0.53150       6.9395LOG-LIKELIHOOD FUNCTION =  -53.303LOG-LIKELIHOOD(0)  =   -63.037LIKELIHOOD RATIO TEST  =    19.4681    WITH     8  D.F.   P-VALUE= 0.01255ESTRELLA R-SQUARE           0.19956MADDALA R-SQUARE            0.18529CRAGG-UHLER R-SQUARE        0.25218MCFADDEN 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 =        70.0PERCENTAGE OF RIGHT PREDICTIONS =    0.73684NAIVE MODEL PERCENTAGE OF RIGHT PREDICTIONS =    0.62105EXPECTED OBSERVATIONS AT 0  =         36.0   OBSERVED =     36.0EXPECTED OBSERVATIONS AT 1  =         59.0   OBSERVED =     59.0SUM OF SQUARED "RESIDUALS" =           18.513WEIGHTED SUM OF SQUARED "RESIDUALS" =     86.839HENSHER-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      0.621PREDICTED COUNT        36.000     59.000     95.000      1.000PREDICTED SHARE         0.379      0.621      1.000PROP. SUCCESSFUL        0.489      0.688      0.612SUCCESS INDEX           0.110      0.067      0.083PROPORTIONAL ERROR      0.000      0.000NORMALIZED 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 ITERATIONSCONVERGENCE TOLERANCE =0.00100LOG OF LIKELIHOOD WITH CONSTANT TERM ONLY =    -63.037BINOMIAL  ESTIMATE = 0.6211ITERATION  0      LOG OF LIKELIHOOD FUNCTION =   -63.037ITERATION  1 ESTIMATES 0.45375     0.92076     0.43035    -0.28835    -0.23416E-01  1.3330  1.6059     -1.7546     -3.7958ITERATION  1      LOG OF LIKELIHOOD FUNCTION =   -54.139ITERATION  2 ESTIMATES 0.55298      1.0944     0.50979    -0.32984    -0.25855E-01  2.1655  2.0427     -2.2551     -4.7103ITERATION  2      LOG OF LIKELIHOOD FUNCTION =   -53.370ITERATION  3 ESTIMATES 0.58166      1.1250     0.52500    -0.33987    -0.26178E-01  2.5635  2.1706     -2.3799     -5.1361ITERATION  3      LOG OF LIKELIHOOD FUNCTION =   -53.304ITERATION  4 ESTIMATES 0.58362      1.1261     0.52605    -0.34139    -0.26129E-01  2.6239  2.1869     -2.3942     -5.2003ITERATION  4      LOG OF LIKELIHOOD FUNCTION =   -53.303ITERATION  5 ESTIMATES 0.58364      1.1261     0.52606    -0.34142    -0.26127E-01  2.6250  2.1872     -2.3945     -5.2014ELASTICITIES ASSUME LOG-TRANSFORMED VARIABLES                                ASYMPTOTIC                         WEIGHTEDVARIABLE    ESTIMATED      STANDARD     T-RATIO    ELASTICITY      AGGREGATE  NAME     COEFFICIENT       ERROR                  AT MEANS      ELASTICITYPUB12         0.58364      0.68778      0.84858      0.19410      0.18107PUB34          1.1261      0.76820       1.4659      0.37451      0.34937PUB5          0.52606       1.2693      0.41445      0.17495      0.16321PRIV         -0.34142      0.78299     -0.43605     -0.11355     -0.10592YEARS        -0.26127E-01  0.26934E-01 -0.97006     -0.86893E-02 -0.81059E-02SCHOOL         2.6250       1.4101       1.8616      0.87301      0.81439LOGINC         2.1872      0.78781       2.7763      0.72739      0.67856PTCON         -2.3945       1.0813      -2.2145     -0.79633     -0.74287CONSTANT      -5.2014       7.5503     -0.68890      -1.7298      -1.6137SCALE 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                                 EFFECTPUB12         0.12955       0.0000      0.44231      0.58706      0.14476PUB34         0.24996       0.0000      0.44231      0.70978      0.26747PUB5          0.11677       0.0000      0.44231      0.57304      0.13073PRIV         -0.75785E-01   0.0000      0.44231      0.36049     -0.81814E-01YEARS        -0.28859E-21   8.5158SCHOOL        0.58267       0.0000      0.44231      0.91631      0.47400LOGINC        0.21022E-04   9.9711PTCON        -0.49214E-03   6.9395LOG-LIKELIHOOD FUNCTION =  -53.303LOG-LIKELIHOOD(0)  =   -63.037LIKELIHOOD RATIO TEST  =    19.4681    WITH     8  D.F.   P-VALUE= 0.01255ESTRELLA R-SQUARE           0.19956MADDALA R-SQUARE            0.18529CRAGG-UHLER R-SQUARE        0.25218MCFADDEN 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 =        70.0PERCENTAGE OF RIGHT PREDICTIONS =    0.73684NAIVE MODEL PERCENTAGE OF RIGHT PREDICTIONS =    0.62105EXPECTED OBSERVATIONS AT 0  =         36.0   OBSERVED =     36.0EXPECTED OBSERVATIONS AT 1  =         59.0   OBSERVED =     59.0SUM OF SQUARED "RESIDUALS" =           18.513WEIGHTED SUM OF SQUARED "RESIDUALS" =     86.839HENSHER-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      0.621PREDICTED COUNT        36.000     59.000     95.000      1.000PREDICTED SHARE         0.379      0.621      1.000PROP. SUCCESSFUL        0.489      0.688      0.612SUCCESS INDEX           0.110      0.067      0.083PROPORTIONAL ERROR      0.000      0.000NORMALIZED SUCCESS INDEX                      0.177|_STOP
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