## Limited-Dependent and Qualitative Variables in EconometricsThis book presents the econometric analysis of single-equation and simultaneous-equation models in which the jointly dependent variables can be continuous, categorical, or truncated. Despite the traditional emphasis on continuous variables in econometrics, many of the economic variables encountered in practice are categorical (those for which a suitable category can be found but where no actual measurement exists) or truncated (those that can be observed only in certain ranges). Such variables are involved, for example, in models of occupational choice, choice of tenure in housing, and choice of type of schooling. Models with regulated prices and rationing, and models for program evaluation, also represent areas of application for the techniques presented by the author. |

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A classic.

### Contents

Introduction | 1 |

12 Censored regression models | 3 |

13 Dummy endogenous variables | 6 |

Discrete regression models | 13 |

22 The linear probability model | 15 |

23 The linear discriminant function | 16 |

24 Analogy with multiple regression and the linear probability model | 18 |

25 The probit and logit models | 22 |

69 Truncated regression models | 165 |

610 Endogenous stratification and truncated regression models | 170 |

611 Truncated and censored regression models with stochastic and unobserved thresholds | 174 |

heteroscedasticity | 178 |

613 Problems of aggregation | 182 |

614 Miscellaneous other problems | 185 |

615 A general specification test | 192 |

616 Mixtures of truncated and untruncated distributions | 194 |

26 Comparison of the logit model and normal discriminant analysis | 27 |

27 The twin linear probability model | 28 |

29 Illustrative examples with grouped data | 32 |

unordered variables | 34 |

211 Measures of goodness of fit | 37 |

212 Multinomial logit and McFaddens conditional logit | 41 |

orderedresponse models | 46 |

sequentialresponse models | 49 |

Poisson regression | 51 |

216 Estimation of logit models with randomized data | 54 |

217 Estimation of logit and pro hit models from panel data | 56 |

Probabilisticchoice models | 59 |

32 The Luce model | 61 |

33 The multinomial probit model | 62 |

34 The eliminationbyaspects model | 64 |

35 The hierarchical eliminationbyaspects model | 66 |

36 The nested multinomial logit model | 67 |

37 The generalized extremevalue model | 70 |

38 The relationship between the NMNL model and the GEV model | 72 |

39 Estimation methods | 73 |

310 Goodnessoffit measures | 76 |

311 Some tests for specification error | 77 |

312 Concluding remarks | 78 |

Discriminant analysis | 79 |

43 Prior probabilities and costs of misclassification | 80 |

44 Nonnormal data and logistic discrimination | 81 |

45 The case of several groups | 86 |

46 Bayesian methods | 88 |

47 Separatesample logistic discrimination | 90 |

Multivariate qualitative variables | 93 |

52 Some minimum chisquare methods for grouped data | 96 |

53 Loglinear models | 103 |

54 Conditional logistic models | 105 |

55 Recursive logistic models | 108 |

56 Some comments on LLM CLM RLM the conditional loglinear models and simultaneous equations | 113 |

some consistent and inconsistent models | 117 |

58 Heckmans model with structural shift and dummy endogenous variables | 125 |

59 Unobserved latent variables and dummy indicators | 138 |

510 Summary and conclusions | 147 |

Censored and truncated regression models | 149 |

63 The tobit censored regression model | 151 |

64 A reparametrization of the tobit model | 156 |

65 Twostage estimation of the tobit model | 158 |

66 Prediction in the tobit model | 159 |

67 The twolimit tobit model | 160 |

68 Models of friction | 162 |

Simultaneousequations models with truncated and censored variables | 197 |

72 Simultaneousequations models with truncation and or censoring | 199 |

73 Simultaneousequations models with probit and tobittype selectivity | 205 |

75 The question of logical consistency | 214 |

76 Summary and conclusions | 216 |

Twostage estimation methods | 221 |

83 Twostage methods for switching regression models | 223 |

84 Twostage estimation of censored models | 228 |

85 Twostage estimation of Heckmans model | 231 |

86 Twostage estimation of structural equations | 234 |

87 Probit twostage and tobit twostage methods | 240 |

88 Twostage methods for models with mixed qualitative truncated and continuous variables | 242 |

89 Some alternatives to the twostage methods | 247 |

810 Some final comments | 252 |

Models with selfselectivity | 257 |

92 Selfselection and evaluation of programs | 260 |

93 Selectivity bias with nonnormal distributions | 267 |

94 Some general transformations to normality | 272 |

95 Polychotomouschoice models and selectivity bias | 275 |

96 Multiple criteria for selectivity | 278 |

97 Endogenous switching models and mixturedistribution models | 283 |

98 When can the selection model be used but not the mixture model? | 288 |

99 Summary and conclusions | 289 |

Disequilibrium models | 291 |

102 The Fair and Jaffee model | 292 |

sample separation unknown | 296 |

sample separation known | 305 |

105 Some generalized disequilibrium models | 310 |

106 Price adjustment and disequilibrium | 319 |

107 Models with controlled prices | 326 |

108 Tests for disequilibrium | 335 |

109 Multimarketdisequilibrium models | 337 |

1010 Models for regulated markets and models for centrally planned economies | 341 |

1011 Summary and conclusions | 343 |

Some applications unions and wages | 347 |

112 The AshenfelterJohnson study | 348 |

113 The Schmidt and Strauss study | 349 |

114 Lees binarychoice model | 356 |

115 Alternative specifications of the unionismwages model | 359 |

116 The Abowd and Farber study | 362 |

117 Summary and conclusions | 364 |

Some results on truncated distributions | 365 |

373 | |

397 | |

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### Common terms and phrases

alternative Amemiya analysis assume assumption asymptotic covariance matrix bivariate censored regression model Chapter choice coefficients consistent estimates covariance matrix defined demand function denote density function dependent variable derived dichotomous variables discussed disequilibrium model distribution function dummy variable endogenous equa equilibrium esti example exogenous variables explanatory variables Fair and Jaffee follows formulation given by equations groups Heckman Hence heteroscedasticity individual iterative joint density latent variables least-squares likelihood function linear probability model log-linear model logit model Maddala maximized McFadden ML estimates model considered multinomial logit multinomial logit model multinomial probit normal distribution Note number of observations obtained otherwise preceding section price-adjustment equation probit method probit model problem procedure Quandt reduced forms regime residuals sample separation selectivity bias self-selection simultaneous-equations model specification standard normal structural equations suggested supply function tion tobit truncated regression model two-stage method values variance vector write zero