Nonlinear Full Information Maximum Likelihood Estimator
In this subsection we shall consider the maximum likelihood estimator of model (8.2.1) under the normality assumption of uu. To do so we must assume that (8.2.1) defines a one-to-one …
Maximum Score Estimator—A Multinomial Case
The multinomial QR model considered by Manski has the following structure. The utility of the і th person when he or she chooses the jth alternative is given by Uij^XvPo …
Model of Nelson and Olson
The empirical model actually estimated by Nelson and Olson (1978) is more general than Type 4 and is a general simultaneous equations Tobit model (10.9.4). The Nelson-Olson empirical model involves …
Two Error Components Model with Endogenous Regressors
6.6.5 The two error components model with endogenous regressors is defined by У! = Xifi + Zy + YlS + n + el (6.6.40) У2 = X2)? + Zy + …
Aggregate Prediction
We shall consider the problem of predicting the aggregate proportion r = і yt in the QR model (9.2.1). This is often an important practical problem for a policymaker. For …
Least Squares Estimator
From Figure 10.1 it is clear that the least squares regression of expenditure on income using all the observations including zero expenditures yields biased estimates. Although it is not so …
Number of Completed Spells
The likelihood function (11.2.5) depends on the observed durations ti, t2, ■ ■ ■ , tr only through r and T. In other words, r and Tconstitute the sufficient statistics. …
Exact Distributions of the Limited Information Maximum Likelihood Estimator and the Two-Stage Least Squares Estimator
The exact finite sample distributions of the two estimators differ. We shall discuss briefly the main results about these distributions as well as their approximations. The discussion is very brief …
Multivariate Nested Logit Model
The model to be discussed in this subsection is identical to the nested logit model discussed in Section 9.3.5. We shall merely give an example of its use in a …
Type 2 Tobit Model: P(y1 < 0) • P(y, > 0, y2)
10.7.1 Definition and Estimation The Type 2 Tobit model is defined as follows: у*и = хА + Щі (Ю.7.1) У 21 = *2ifi2 + U2i У2і = У*і if ^>0 …
Time Series Analysis
1. We are using the approximation sign s* to mean that most elements of the matrices of both sides are equal. 2. The subscript p denotes the order of the …
Tests of Hypotheses, Prediction, and Computation
8.3.1 Tests of Hypotheses Suppose we want to test a hypothesis of the form h(a) = 0 in model (8.1.1), where h is a ^-vector of nonlinear functions. Because we …
Generalized Maximum Likelihood Estimator
Cosslett (1983) proposed maximizing the log likelihood function (9.2.7) of a binary QR model with respct to P and F, subject to the condition that F is a distribution function. …
Model of Tomes
Tomes (1981) studied a simultaneous relationship between inheritance and the recipient’s income. Although it is not stated explicitly, Tomes’ model can be defined by .У?/ = УіУ2і + x'ufii + …
Random Coefficients Models
Random coefficients models (RCM) are models in which the regression coefficients are random variables, the means, variances, and covariances of which are unknown parameters to estimate. The Hildreth and Houck …
Multinomial Models
9.3.1 Statistical Inference In this section we shall define a general multinomial QR model and shall discuss maximum likelihood and minimum chi-square estimation of this model, along with the associated …
Heckman’s Two-Step Estimator
Heckman (1976a) proposed a two-step estimator in a two-equation generalization of the Tobit model, which we shall call the Type 3 Tobit model. But his estimator can also be used …
Durations as Dependent Variables of a Regression Equation
Suppose that each individual experiences one complete spell. Then the likelihood function is N L = J"JA, exp (—A^j). (11.2.25) /-і The case of a person having more than one …
Interpretations of the Two-Stage Least Squares Estimator
There are many interpretations of 2SLS. They are useful when we attempt to generalize 2SLS to situations in which some of the standard assumptions of the model are violated. Different …
Log-Linear Model
A log-linear model refers to a particular parameterization of a multivariate model. We shall discuss it in the context of the 2 X 2 model given in Table 9.1. For …
A Special Case of Independence
Dudley and Montmarquette (1976) analyzed whether or not the United States gives foreign aid to a particular country and, if it does, how much foreign aid it gives using a …
Generalized Least Squares Theory
1. If rank (ЩХ) = K, fiis uniquely determined by (6.1.2). 2. Farebrother (1980) presented the relevant tables for the case in which there is no intercept. 3. Breusch (1978) …
Prediction
Bianchi and Calzolari (1980) proposed a method by which we can calculate the mean squared prediction error matrix of a vector predictor based on any estimator of the nonlinear simultaneous …
Panel Data QR Models
Panel data consist of observations made on individuals over time. We shall consider models in which these observations are discrete. If observations are independent both over individuals and over time, …
Type 5 Tobit Model: P(yі < 0, ya) • P(y, > 0, ya)
10.10.1 Definition and Estimation The Type 5 Tobit model is obtained from the Type 4 model (10.9.1) by omitting the equation for yu. We merely observe the sign of yf,. …
The Kelejian and Stephan Model
The RCM analyzed by Kelejian and Stephan (1983) is a slight generalization of Hsiao’s model, which we shall discuss in the next subsection. Their model is defined by Уі, = …
Multinomial Logit Model
In this and the subsequent subsections we shall present various types of unordered multinomial QR models. The multinomial logit model is defined by ptj = [Д exp (x'*)?)j exp (9.3.34) …
Nonlinear Least Squares and Nonlinear Weighted Least Squares Estimators
In this subsection we shall consider four estimators: the NLLS and NLWLS estimators applied to (10.4.11), denoted yN and ftw, respectively, and the NLLS and NLWLS estimators applied to (10.4.23), …
Discrete Observations
In the analysis presented in the preceding three subsections, we assumed that an individual is continuously observed and his or her complete event history during the sample period is provided. …
Consider regression equations
у = Za + u (7.3.21) Z = ХП + V, (7.3.22) where Ем = О, ЕУ = 0, Em' = T*, and u and V are possibly correlated. We …
Multivariate Probit Model
A multivariate probit model was first proposed by Ashford and Sowden (1970) and applied to a bivariate set of data. They supposed that a coal miner develops breathlessness (y, = …
Gronau’s Model
Gronau (1973) assumed that the offered wage W° is given to each housewife independently of hours worked (H), rather than as a schedule W°(H). Given W°, a housewife maximizes her …
Constant Variance in a Subset of the Sample
The heteroscedastic model we shall consider in this subsection represents the simplest way to restrict the number of estimable parameters to a finite and manageable size. We assume that the …
Qualitative Response Models
9.1 Introduction Qualitative response models (henceforth to be abbreviated as QR models) are regression models in which dependent (or endogenous) variables take discrete values. These models have numerous applications in …
Models with Heterogeneity
We shall study the problem of specifying a panel data QR model by considering a concrete example of sequential labor force participation by married women, following the analysis of Heckman …
