Nonlinear Limited Information Maximum Likelihood Estimator
In the preceding section we assumed the model (8.1.1) without specifying the model for Y( or assuming the normality of u, and derived the asymptotic distribution of the class of …
Results of Cosslett: Part II
Cosslett (1981b) summarized results obtained elsewhere, especially from his earlier papers (Cosslett, 1978, 1981a). He also included a numerical evaluation of the asymptotic bias and variance of various estimators. We …
Other Examples of Type 3 Tobit Models
Roberts, Maddala, and Enholm (1978) estimated two types of simultaneous equations Tobit models to explain how utility rates are determined. One of their models has a reduced form that is …
Model of Lee
In the model of Lee (1978), y2i represents the logarithm of the wage rate of the rth worker in case he or she joins the union and y3i represents the …
Two Error Components Model
In 2ECM there is no time-specific error component. Thus the model is a special case of 3ECM obtained by putting o = 0. This model was first used in econometric …
Comparison of the Maximum Likelihood Estimator and the Minimum Chi-Square Estimator
In a simple model where the vector x( consists of 1 and a single independent variable and where T is small, the exact mean and variance of MLE and the …
Empirical Examples
Tobin (1938) obtained the maximum likelihood estimates of his model applied to data on 735 nonfarm households obtained from Surveys of Consumer Finances. The dependent variable of his estimated model …
Empirical Examples of Markov Models without Exogenous Variables
In this subsection we shall discuss several empirical articles, in which Markov chain models without exogenous variables are estimated, for the purpose of illustrating some of the theoretical points discussed …
Limited Information Model
7.3.1 Introduction In this section we shall consider situations in which a researcher wishes to estimate only the parameters of one structural equation. Although these pa rameters can, of course, …
Multinomial Probit Model
Let Uj, j = 0, 1, 2,. . ., m, be the stochastic utility associated with the yth alternative for a particular individual. By the multinomial probit model, we mean …
Tests for Normality
The fact that the Tobit MLE is generally inconsistent when the true distribution is nonnormal makes it important for a researcher to test whether the data are generated by a …
Recent Developments in Regression Analysis
1. A much more detailed account of this topic can be found in Amemiya (1980a). 2. We use the term estimator here, but all the definitions and the results of …
Estimation in a System of Equations
8.2.1 Introduction Define a system of N nonlinear simultaneous equations by fu(y,, <*i) = ua, і = 1, 2,. . . , N, t = 1, 2,. . . , …
Distribution-Free Methods
In this section we shall discuss two important articles by Manski (1975) and Cosslett (1983). Both articles are concerned with the distribution-free estimation of parameters in QR models—Manski for multinomial …
Type 4 Tobit Model: P(y, < 0, y3) • P(y,, y2)
10.9.1 Definition and Estimation The Type 4 Tobit model is defined as follows: У и = x'uA + Uu У*і = *2.7*2 + “2, Уіі — хЗіРз + U3i Уи …
Two-State Models with Exogenous Variables
We shall consider a two-state Markov model with exogenous variables, which accounts for the heterogeneity and nonstationarity of the data. This model is closely related to the models considered in …
Balestra-Nerlove Model
As we mentioned earlier, this is a generalization of 2ECM in the sense that a lagged endogenous variable yu_, is included among the regressors. Balestra and Nerlove (1966) used this …
Tests of Hypotheses
To test a hypothesis on a single parameter, we can perform a standard normal test using the asymptotic normality of either MLE or the MIN /2 estimator. A linear hypothesis …
Properties of Estimators under Standard Assumptions
In this section we shall discuss the properties of various estimators of the T obit model under the assumptions of the model. The estimators we shall consider are the probit …
Multistate Models with Exogenous Variables
Theoretically, not much need be said about this model beyond what we have discussed in Section 11.1.1 for the general case and in Section 11.1.3 for the two-state case. The …
Limited Information Maximum Likelihood Estimator
The LIML estimator is obtained by maximizing the joint density of y! and Y! under the normality assumption with respect to а, П, and X without any constraint. Anderson and …
Sequential Probit and Logit Models
When the choice decision is made sequentially, the estimation of multinomial models can be reduced to the successive estimation of models with fewer responses, and this results in computational economy. …
Powell’s Least Absolute Deviations Estimator
Powell (1981, 1983) proposed the least absolute deviations (LAD) estimator (see Section 4.6) for censored and truncated regression models, proved its consistency under general distributions, and derived its asymptotic distribution. …
Large Sample Theory
1. Representative textbooks are, in a roughly increasing order of difficulty, Hoel (1971); Freund (1971); Mood, Graybill, and Boes (1974); Cox and Hinkley (1974), and Bickel and Doksum (1977). 2. …
Nonlinear Three-Stage Least Squares Estimator
As a natural extension of the class of the NL2S estimators—the version that minimizes (8.1.22), Jorgenson and Laffont (1974) defined the class of nonlinear three-stage least squares (NL3S) estimators as …
Maximum Score Estimator—A Binary Case
Manski (1975) considered a multinomial QR model, but here we shall define his estimator for a binary QR model and shall prove its consistency. Our proof will be different from …
Model of Kenny, Lee, Maddala, and Trost
Kenny et al. (1979) tried to explain earnings differentials between those who went to college and those who did not. We shall explain their model using the variables appearing in …
Two Error Components Model with a Serially Correlated Error
In the subsection we shall discuss the 2ECM defined by Eqs. (6.6.18) and (6.6.19) in which e follows an AR(1) process, that is, in = УЧ,-1 + 4, where {<!;„} …
Discriminant Analysis
The purpose of discriminant analysis is to measure the characteristics of an individual or an object and, on the basis of the measurements, to classify the individual or the object …
Probit Maximum Likelihood Estimator
The Tobit likelihood function (10.2.5) can be trivially rewritten as т = П [І - Ф(х',0/о) П Ф(x',0/a) (10.4.1) 0 1 П Ф(х'іР/о)~1а~1Ф[(Уі ~ *іР)/о]. і Then the first two …
Duration Models
11.1.3 Stationary Models—Basic Theory We shall first explain a continuous-time Markov model as the limit of a discrete-time Markov model where the time distance between two adjacent time periods approaches …
Asymptotic Distribution of the Limited Information Maximum Likelihood Estimator and the Two-Stage Least Squares Estimator
The LIML and 2SLS estimators of a have the same asymptotic distribution. In this subsection we shall derive it without assuming the normality of the observations. We shall derive the …
Multivariate Models
9.3 9.4.1 Introduction A multivariate QR model specifies the joint probability distribution of two or more discrete dependent variables. For example, suppose there are two binary dependent variables yx and …
Generalized Tobit Models
As stated in Section 10.1, we can classify Tobit models into five common types according to similarities in the likelihood function. Type 1 is the standard Tobit model, which we …
Asymptotic Properties of Extremum Estimetors
1. In the proof of Theorem 4.1.1, continuity of QT<ff) is used only to imply continuity of Q(0) and to make certain the measurability of §T. Therefore we can modify …
