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
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 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:
- Model Estimation by the Method of Maximum Likelihood
- Interpretation of the Results
- Overall Significance and Goodness of Fit Measures
- Predicting Probabilities
- Testing for Heteroskedasticity
References
Good textbook discussion is:
Chapter 19 of William Greene, Econometric Analysis, Fourth Edition, Prentice-Hall, 2000.References with more technical details are:Chapter 17 of Jeffrey M. Wooldridge, Introductory Econometrics: A Modern Approach, South-Western College Publishing, 2000.
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.
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
