Category Advanced Econometrics Takeshi Amemiya

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 assume that is a known nonsingular matrix. Although the limited informa­tion model can be written in this way by assuming that certain elements of V are identically equal to 0, the reader may regard these two equations in a more abstract sense and not necessarily as arising from the limited information model considered in Section 7.3.1.

Premultiplying (7.3.21) and (7.3.22) by *P-I/2, we obtain

l/2y = 4′-i/2za + ‘F"i/2u (7.3.23)

and

^-i/2Z = ip-i/2Xn + <ir-i/2V. (7.3.24)

We define the G2SLS estimator of a as the 2SLS estimator of a applied to

(7.3.23) and (7.3.24); that is,

Ob2S = [Z’4′-1X(X,4′-1X)-,X,4′-1Z]-1Z/4′-‘X(X’4′-1X)-1X’4′-,y.

(7.3.25)

Give...

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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, = 1) if his tolerance (yf) is less than 0. Assuming that yf ~ N(—0x, 1), where x = (1, Age)’, we have

F(y,= 1) = Ф(#х). (9.4.15)

They also supposed that a coal miner develops wheeze (y2 = 1) if his tolerance (yf) against wheeze is less than 0 and that yf ~ N(—fi’2x, 1). Then we have

P(y2 = 1) = Ф(Дх). (9.4.16)

Now that we have specified the marginal probabilities of Уі and y2, the multi­variate model is completed by specifying the joint probability Р(Уі = 1, у2 = 1), which in turn is determined if the joint distribution of yf and yf is specified...

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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 utility function U(C, X) subject to X = W°H + V and C + H = T, where C is time spent at home for childcare, X represents all other goods, T is total available time, and V is other income. Thus a housewife does not work if

image760(10.7.10)

image761

and works if the inequality in (10.7.10) is reversed. If she works, the hours of work H and the actual wage rate W must be such that

Gronau called the left-hand side of (10.7.10) the housewife’s value of time or, more commonly, the reservation wage, denoted W1.12

Assuming that both W° and Wx can be written as linear combinations of independent variables plus error terms, his...

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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 error vector u is partitioned into N nonoverlapping subvectors as u = (u’,, Uj,. . . , Uy)’ such that £u, uj = a2lTr We can estimate each a2 consistently from the least squares residuals provided that each T, goes to infinity with Г. Note that this model is a special case of Zellner’s SUR model discussed in Section 6.4, so that the asymptotic results given there also hold here. We shall consider a test of homoscedasticity and the exact moments of FGLS.

If we assume the normality of u, the hypothesis of — <r2 for all і can be tested by the likelihood ratio test in a straightforward manner...

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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 economics because many behavioral responses are qualitative in nature: A consumer decides whether to buy a car or not; a commuter chooses a particular mode of trans­portation from several available ones; a worker decides whether to take a job offer or not; and so on. A long list of empirical examples of QR models can be found in my recent survey (Amemiya, 1981).

Qualitative response models, also known as quantal, categorical, or discrete models, have been used in biometric applications longer than they have been used in economics...

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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 and Willis (1977). As is customary with a study of labor force participation, it is postulated that a person works if the offered wage exceeds the reservation wage (shadow price of time). The offered wage depends on the person’s characteristics such as her education, training, intelligence, and motivation, as well as on local labor market condi­tions. The reservation wage depends on the person’s assets, her husband’s wage rate, prices of goods, interest rates, expected value of future wages and prices, and the number and age of her children...

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