Category Advanced Econometrics Takeshi Amemiya

Other Applications of the Type 2 Tobit Model

Nelson (1977) noted that a Type 2 Tobit model arises if y0 in (10.2.1) is assumed to be a random variable with its mean equal to a linear combination of independent variables. He reestimated Gronau’s model by the MLE.

In the study of Westin and Gillen (1978), yrepresents the parking cost with x2 including zonal dummies, wage rate (as a proxy for value of walking time), and the square of wage rate. A researcher observes y* = y2 if y2 < C where C represents transit cost, which itself is a function of independent variables plus an error term.

10.8 Type 3 Tobit Model: P(y, < 0) • Р(ул, y2)

10.8.1 Definition and Estimation

The Type 3 Tobit model is defined as follows:

ytt-*ufii + uu 00.8.1)

Уіі = + M2<

Уи = У и if У* > 0 = 0 if у S 0

Уи = Узі if У*>0

= 0 if yf, S0, і" — 1,...

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General Parametric Heteroscedastidty

In this subsection we shall assume a] = gfa, fix) without specifying g, where fix is a subset (possibly whole) of the regression parameters fi and a is another vector of parameters unrelated to ft. In applications it is often assumed that glot, fii) == g(ot, The estimation of a and ft can be done in several steps.

In the first step, we obtain the LS estimator of fi, denoted ft. In the second step, a and can be estimated by minimizing — gfa, fii)]2, where fi, =

y, — x’,fi. The consistency and the asymptotic normality of the resulting esti­mators, denoted a and Д, have been proved by Jobson and^Fuller (1980). In the third step we have two main options: FGLS using gfa, fix) or MLE under normality using a and fi as the initial estimates in some iterative algorithm...

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Univariate Binary Models

9.1.1 Model Specification A univariate binary QR model is defined by

Подпись: (9.2.1)Р(Уі = 1) = F{x%), /=1,2,. . . , n,

where {y,} is a sequence of independent binary random variables taking the value 1 or 0, x, is a A-vector of known constants, fi0 is a A-vector of unknown parameters, and F is a certain known function.

It would be more general to specify the probability as F(x„ Д>), but the specification (9.2.1) is most common. As in the linear regression model, specifying the argument of Fas xJ/?0 is more general than it seems because the elements of x, can be transformations of the original independent variables. To the extent we can approximate a general nonlinear function of the original independent variables by x$0, the choice of Fis not critical as long as it is a distribution function...

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Models with Heterogeneity and True State Dependence

In this subsection we shall develop a generalization of model (9.7.2) that can incorporate both heterogeneity and true state dependence.

Following Heckman (1981a), we assume that there is an unobservable continuous variable y* that determines the outcome of a binary variable yit by the rule

У и= 1 if yt> 0 (9.7.10)

= 0 otherwise.

In a very general model, y* would depend on independent variables, lagged values of yg, lagged values of yt„ and an error term that can be variously specified. We shall analyze a model that is simple enough to be computation­ally feasible and yet general enough to contain most of the interesting features of this type of model: Let

Уи = х’иР + УУи-і + vu=Vi, + vtt, (9.7.11)

where for each i, vu is serially correlated in general...

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Multivariate Generalizations

By a multivariate generalization of Type 5, we mean a model in which y* and у fj in (10.10.1) are vectors, whereas y* is a scalar variable the sign of which is observed as before. Therefore the Fair-JafFee model with likelihood function characterized by (10.10.14) is an example of this type of model.

In Lee’s model (1977) the y*( equation is split into two equations and

T% = zb<*2 + v2, (10.10.16)

where CJ, and T*t denote the cost and the time incurred by the rth person traveling by a private mode of transportation, and, similarly, the cost and the time of traveling by a public mode are specified as

Lee assumed that C* and Т% are observed if the ith person uses a private mode and C% and T% are observed if he or she uses a public mode...

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Swamy’s Model

Swamy’s model (1970) is a special case of the Kelejian-Stephan model obtained by putting

2A — 0 and 2e = 2 ® Ir,

where 2 = diag {a,a, . . . , a%). It is more restrictive than Hsiao’s model in the sense that there is no time-specific component in Swamy’s model, but it is more general in the sense that Swamy assumes neither the diagonality of XM nor the homoscedasticity of є like Hsiao.

Swamy proposed estimating XM and 2 in the following steps:

Step 1. Estimate a} by af = y'[I — X,(X-X,)- 1X/’]y(/(T — K).

Step 2. Define b, = (x; Х()->Х;у(.

Step 3. Estimate 2„ by % = (N – ІГ’Х^А-ЛМЕ^А)*

(b, – AM 2f_A)’ – N-y^SjiX’Xi)-‘.

It is easy to show that aj and 2^ are unbiased estimators of a2 and XM, respectively.

Swamy proved that the FGLS estimator of fi using a2 and 2^ is ...

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