Disequilibrium Models
Disequilibrium models constitute an extensive area of research, about which numerous papers have been written. Some of the early econometric models have been surveyed by Maddala and Nelson (1974). A …
Hsiao’s Model
Hsiao’s model (1974, 1975) is obtained as a special case of the model of the preceding subsection by assuming 2,, and 2* are diagonal and putting 2€ = Hsiao (1975) …
Multinomial Discriminant Analysis
The DA model of Section 9.2.8 can be generalized to yield a multinomial DA model defined by X? l(y< =j) ~ Mjij, 2;) (9.3.46) and Р(Уі =j) = Qj (9.3.47) …
Tobit Maximum Likelihood Estimator
The Tobit MLE maximizes the likelihood function (10.2.5). Under the assumptions given after (10.2.4), Amemiya (1973c) proved its consistency and asymptotic normality. If we define 0 = (/?', a2)', the …
Nonstationary Models
So far we have assumed X)k(t) = Ajfc for all t (constant hazard rate). Now we shall remove this assumption. Such models are called nonstationary or semi - Markov. Suppose …
Three-Stage Least Squares Estimator
In this section we shall again consider the full information model defined by (7.1.1) . The 3SLS estimator of a in (7.1.5) can be defined as a special case of …
Choice-Based Sampling
9.5.1 Introduction Consider the multinominal QR model (9.3.1) or its special case (9.3.4). Up until now we have specified only the conditional probabilities of alternatives j = 0, 1,. . …
Other Applications of the Type 2 Tobit Model
Nelson (1977) noted that a Type 2 Tobit model arises if y0 in (10.2.1) is assumed to be a random variable with its mean equal to a linear combination of …
General Parametric Heteroscedastidty
In this subsection we shall assume a] = gfa, fix) without specifying g, where fix is a subset (possibly whole) of the regression parameters fi and a is another vector …
Univariate Binary Models
9.1.1 Model Specification A univariate binary QR model is defined by Р(Уі = 1) = F{x%), /=1,2,. . . , n, where {y,} is a sequence of independent binary random …
Models with Heterogeneity and True State Dependence
In this subsection we shall develop a generalization of model (9.7.2) that can incorporate both heterogeneity and true state dependence. Following Heckman (1981a), we assume that there is an unobservable …
Multivariate Generalizations
By a multivariate generalization of Type 5, we mean a model in which y* and у fj in (10.10.1) are vectors, whereas y* is a scalar variable the sign of …
Swamy’s Model
Swamy’s model (1970) is a special case of the Kelejian-Stephan model obtained by putting 2A — 0 and 2e = 2 ® Ir, where 2 = diag {a,a, . . …
Nested Logit Model
In Section 9.3.3 we defined the multinomial logit model and pointed out its weakness when some of the alternatives are similar. In this section we shall discuss the nested (or …
The EM Algorithm
The EM algorithm is a general iterative method for obtaining the MLE; it was first proposed by Hartley (1958) and was generalized by Dempster, Laird, and Rubin (1977) to a …
Problem of Left-Censoring
In this subsection we shall consider the problem of left-censoring in models for unemployment duration. When every individual is observed at the start of his or her unemployment spell, there …
Further Topics
The following topics have not been discussed in this chapter but many important results in these areas have appeared in the literature over the past several years: (1) lagged endogenous …
Results of Manski and Lerman
Manski and Lerman (1977) considered the choice-based sampling scheme represented by the likelihood function (9.5.3)—the case where Qis known— and proposed the estimator (WMLE), denoted pw, which maximizes 5л=І w(/)logP(/|x,.,A …
Heckman’s Model
Heckman’s model (Heckman, 1974) differs from Gronau’s model (10.7.11) in that Heckman included the determination of hours worked (H) in his model.16 Like Gronau, Heckman assumes that the offered wage …
Variance as a Linear Function of Regressors
In this subsection we shall consider the model <rj = z,'a, where z, is a 6-vector of known constants and a is a 6-vector of unknown parameters unrelated to the …
Global Concavity of the Likelihood Function in the Logit and Probit Models
Global concavity means that d2 log L/dfldf}' is a negative definite matrix for fi Є В. Because we have by a Taylor expansion (P~P)- where P* lies between P and …
One-Factor Models
The individual likelihood function (9.7.12) involves a Г-tuple normal integral, and therefore its estimation is computationally infeasible for large Г (say, greater than 5). For this reason we shah consider …
Multinomial Generalizations
In all the models we have considered so far in Section 10.10, the sign of у ft determined two basic categories of observations, such as union members versus nonunion members, …
Other Models
In the preceding subsections we have discussed models in which cross-section-specific components are independent across individuals and time-specific components are independent over time periods. We shall cite a few references …
Higher-Level Nested Logit Model
The nested logit model defined in the preceding section can be regarded as implying two levels of nesting because the responses are classified into S groups and each group is …
Properties of the Tobit Maximum Likelihood Estimator under Nonstandard Assumptions
In this section we shall discuss the properties of the Tobit MLE—the estimator that maximizes (10.2.5)—under various types of nonstandard assumptions: heteroscedasticity, serial correlation, and nonnormality. It will be shown …
Useful Theorems in Matrix Analysis
The theorems listed in this appendix are the ones especially useful in econometrics. All matrices are assumed to be real. Proofs for many of these theorems can be found in …
Nonlinear Simultaneous Equations Models
In this chapter we shall develop the theory of statistical inference for nonlinear simultaneous equations models. The main results are taken from the author’s recent contributions (especially, Amemiya, 1974c, 1975a, …
Results of Manski and McFadden
Manski and McFadden (1981) presented a comprehensive summary of all the types of models and estimators under choice-based sampling, including the results of the other papers discussed elsewhere in Section …
Two-Step Estimator of Heckman
Heckman (1976a) proposed the two-step estimator of the reduced-form parameters (which we discussed in Section 10.8.1); but he also reestimated the labor supply model of Heckman (1974) using the structural …
Variance as an Exponential Function of Regressors
As we mentioned before, the linear specification, however simple, is more general than it appears. However, a researcher may explicitly specify the variance to be a certain nonlinear function of …
Iterative Methods for Obtaining the Maximum Likelihood Estimator
The iterative methods we discussed in Section 4.4 can be used to calculate a root of Eq. (9.2.8). For the logit and probit models, iteration is simple because of the …
Tobit Models
10.1 Introduction Tobit models refer to censored or truncated regression models in which the range of the dependent variable is constrained in some way. In economics, such a model was …
Markov Chain and Duration Models
We can use the term time series models in a broad sense to mean statistical models that specify how the distribution of random variables observed over time depends on their …
Linear Simultaneous Equations Models
In this chapter we shall give only the basic facts concerning the estimation of the parameters in linear simultaneous equations. A major purpose of the chapter is to provide a …
