Category Advanced Econometrics Takeshi Amemiya

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 same in case he or she does not join the union. Whether or not the worker joins the union is determined by the sign of the variable

Подпись: (10.10.3)Уи = УІі-УІ + *іа + і>і-

Because we observe only y2i if the worker joins the union and y3i if the worker does not, the logarithm of the observed wage, denoted yt, is defined by

Подпись:Уі = Уи if Тн>0

= y’ti if j/J) s 0.

Lee assumed that x2 and x3 (the independent variables in the y* and y3 equations) include the individual characteristics of firms and workers such as regional location, city size, education, experience, race, sex, and health, whereas z includes certain other individual characteristics and...

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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 applications by Balestra and Nerlove (1966). However, because in the Balestra-Nerlove model a lagged endogenous variable is included among regressors, which causes certain additional statistical problems, we shall discuss it separately in Section 6.6.3.

In matrix notation, 2ECM is defined by

y = X0 + u

(6.6.18)

and

u = 1^ + e.

(6.6.19)

The covariance matrix ft = £iiu’ is given by

ft = a* A + oJIjvt,

(6.6.20)

and its inverse by

n~l=±(lNT-yl A),

(6.6.21)

where у, = а*/{а + Та*) as before.

We can define GLS and FGLS as in Section 6.6.1...

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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 MIN x2 estimator can be computed by a direct method. Berkson (1955,1957) did so for the logit and the probit model, respectively, and found the exact mean squared error of the MIN x2 estimator to be smaller in all the examples considered.

Amemiya (1980b) obtained the formulae for the bias to the order of л~1 and the mean squared error to the order of л-2 of MLE and MIN x2 in a general logit model.3 The method employed in this study is as follows: Using (9.2.19) and the sampling scheme described in Section 9.2.5, the normal equation

(9.2.8) is reduced to

We can regard (9.2.40) as defining the MLE fi implicitly as a function of Pi, P2,...

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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 was actually the ratio of total durable goods expenditure to disposable income and the inde­pendent variables were the age of the head of the household and the ratio of liquid assets to disposable income.

Since then, and especially since the early 1970s, numerous applications of the standard Tobit model have appeared in economic journals, encompassing a wide range of fields in economics. A brief list of recent representative refer­ences, with a description of the dependent variable (y) and the main indepen­dent variables (x), is presented in Table 10.1...

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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 in the preceding subsec­tion. We shall also consider certain new theoretical problems that are likely to be encountered in practice.

Suppose we assume a homogeneous stationary first-order Markov model and estimate the Markov matrix (the matrix of transition probabilities) by. the MLE PJk given by (11.1.22). Let Pfk be a transition probability of lag two; that is, Pfk denotes the probability a person is in state к at time t given that he or she

Подпись:was in state j at time t — 2. If we define п? Лі) = 2fLiyj(t — 2)y‘k(t) and n(? =

-………………………………. (11.1.25) where P2 = PP...

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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, be estimated simultaneously with the parameters of the remaining equations by FIML, we shall consider simpler estimators that do not require the estimation of the parameters of the other structural equa­tions.

We shall assume for simplicity that a researcher wishes to estimate the parameters of the first structural equation

y^Y. y + Xj/H-UjC-Zje + u,), (7.3.1)

where we have omitted the subscript 1 from y, ft, and a. We shall not specify the remaining structural equations; instead, we shall merely specify the re­duced form equations for Y,,

Y,=Xn, + V,, (7.3.2)

which is a subset of (7.1.2)...

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