# Category Advanced Econometrics Takeshi Amemiya

## 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 Section 9.7.2. We shall also discuss an example of the model, attributed to Boskin and Nold (1975), to illustrate several important points.

This model is similar to a univariate binary QR model. To make the subse­quent discussion comparable to the discussion of Chapter 9, assume that/ = 0 or 1 rather than 1 or 2. Let yu = 1 if the tth person is in state 1 at time t and ya = 0 otherwise. (Note that yu is the same as the y[(t) we wrote earlier.) The model then can be written as

P{yit= lyi. t-i) = F(Fxit + a’xttyUt-l), (11.1.34)

where Fisa certain distribution function. Note that (11.1...

## 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 model to analyze the demand for natural gas in 36 states in the period 1950-1962.

All the asymptotic results stated earlier for 2ECM hold also for the Balestra- Nerlove model provided that both N and T go to <», as shown by Amemiya (1967). However, there are certain additional statistical problems caused by the presence of a lagged endogenous variable; we shall delineate these prob­lems in the following discussion.

First, the LS estimator ofobtained from (6.6.18) is always unbiased and is consistent if N goes to oo. However, if xft contain y/f/_!, LS is inconsistent even when both N and T go to °o...

## 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 can be tested using general methods discussed in Section 4.5.1. The problem of choosing a model among several alternatives can be solved either by the Akaike Information Criterion (Section 4.5.2) or by Cox’s test of nonnested hypotheses (Section 4.5.3). For other criteria for choosing models, see the article by Amemiya (1981).

Here we shall discuss only a chi-square test based on Berkson’s MIN /2 estimator as this is not a special case of the tests discussed in Section 4.5. The test statistic is the weighted sum of squared residuals (WSSR) from Eq.

(9.2.30) defined by

WSSR = 2 aJ2[F-Pt) – х’\$]г. (9.2.42)

i=i

In the...

## 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 maximum likelihood (ML), least squares (LS), Heckman’s two-step least squares, nonlinear least squares (NLLS), nonlinear weighted least squares (NLWLS), and Tobit ML estimators.

x

Income, marital status, number of children Years of schooling, working experience

Sex, age, number of years married, number of children, education, occupation, degree of religiousness Preprogram hours worked, change in the wage rate, family characteristics Ratio of social security benefits lost at time of full-time employment to full-time earnings Price of contributions, income

Wages of husbands and wives, education of husbands and wives, ...

## 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 likelihood function can be derived from (11.1.4) by speci­fying Pjk(t) as a function of exogenous variables and parameters. The equiva­lence of the NLWLS to the method of scoring iteration was discussed for the
general case in Section 11.1.1, and the minimum chi-square estimator defined for the two-state case in Section 11.1.3 can be straightforwardly generalized to the multistate case. Therefore it should be sufficient to discuss an empirical article by Toikka (1976) as an illustration of the NLWLS (which in his case is a linear WLS estimator because of his linear probability specification).

Toikka’s model is a three-...

## 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 Rubin (1949) proposed this estimator (without the particular normalization on Г we have adopted here) and obtained an explicit formula for the LIML estimator:4

and m, = і – x, (x; x, r1 x;.

7.3.2 Two-Stage Least Squares Estimator

The two-stage least squares (2SLS) estimator of a, proposed by Theil (1953),5 is defined by

d2S = (Z’1PZ1)-1Z’1Pyl,

where P = X(X’X)-1X.