# Category Advanced Econometrics Takeshi Amemiya

## Two-Step Estimator of Heckman

Heckman (1976a) proposed the two-step estimator of the reduced-form pa­rameters (which we discussed in Section 10.8.1); but he also reestimated the labor supply model of Heckman (1974) using the structural equations version. Because (10.8.6) is a reduced-form as well as a structural equation, the estima­tion of fi2 is done in the same way as discussed in Section 10.8.1, namely, by applying least squares to the regression equation for E( W,Hf> 0) after esti­mating the argument of X (the hazard rate) by probit MLE. So we shall discuss only the estimation of (10.8.7), which we can rewrite as

Hi = y~lWt – г’,ау~1 ~ rlvt – (Ю.8.9)

By subtracting E(v,Hf > 0) from vt and adding the same, we can rewrite (10.8.9) further as

щ = ri Wi – Z’ay~‘ – olvo^rlK*uPJox)- Г’е„ (10.8.10)

where aX...

## Variance as an Exponential Function of Regressors

As we mentioned before, the linear specification, however simple, is more general than it appears. However, a researcher may explicitly specify the variance to be a certain nonlinear function of the regressors. The most natural choice is an exponential function because, unlike a linear specification, it has the attractive feature of ensuring positive variances. Harvey (1976), who as­sumed yt — exp (z’,a), proposed estimating a by LS applied to the

regression of log fi? on z, and showed that the estimator is consistent if 1.2704 is subtracted from the estimate of the constant term. Furthermore, the esti­mator has an asymptotic covariance matrix equal to 4.9348(Z’Z)-‘, more than double the asymptotic covariance matrix of MLE, which is 2(Z’Z)~I.

6.6 Error Components Models

Error component...

## Iterative Methods for Obtaining the Maximum Likelihood Estimator

The iterative methods we discussed in Section 4.4 can be used to calculate a root of Eq. (9.2.8). For the logit and probit models, iteration is simple because of the global concavity proved in the preceding section. Here we shall only discuss the method-of-scoring iteration and give it an interesting interpreta­tion.

As we noted in Section 4.4, the method-of-scoring iteration is defined by

dlogL

dfi

where fa is an initial estimator of Д, and p2 is the second-round estimator. The iteration is to be repeated until a sequence of estimators thus obtained con­verges. Using (9.2.8) and (9.2.12), we can write (9.2.24) as

where we have defined Ft = F{s.’fix) and fj =/(x’/?1).

An interesting interpretation of the iteration (9.2.25) is possible. From

(9.2.1) we obtain

У1 = Р(х’А) + щ, (...

## Tobit Models

10.1 Introduction

Tobit models refer to censored or truncated regression models in which the range of the dependent variable is constrained in some way. In economics, such a model was first suggested in a pioneering work by Tobin (1958). He analyzed household expenditure on durable goods using a regression model that specifically took account of the fact that the expediture (the dependent variable of his regression model) cannot be negative. Tobin called his model the model of limited dependent variables. It and its various generalizations are known popularly among economists as Tobit models, a phrase coined by Goldberger (1964), because of similarities to probit models. These models are also known as censored or truncated regression models...

## Markov Chain and Duration Models

We can use the term time series models in a broad sense to mean statistical models that specify how the distribution of random variables observed over time depends on their past observations. Thus defined, Markov chain models and duration models, as well as the models discussed in Chapter S, are special cases of time series models. However, time series models in a narrow sense refer to the models of Chapter 5, in which random variables take on continu­ous values and are observed at discrete times. Thus we may characterize the models of Chapter 5 as continuous-state, discrete-time models. Continuous – state, continuous-time models also constitute an important class of models, although we have not discussed them...

## Linear Simultaneous Equations Models

In this chapter we shall give only the basic facts concerning the estimation of the parameters in linear simultaneous equations. A major purpose of the chapter is to provide a basis for the discussion of nonlinear simultaneous equations to be given in the next chapter. Another purpose is to provide a rigorous derivation of the asymptotic properties of several commonly used estimators. For more detailed discussion of linear simultaneous equations, the reader is referred to textbooks by Christ (1966) and Malinvaud (1980).

7.1 Model and Identification

We can write the simultaneous equations model as

Yr = XB + U (7.1.1)

where Y is a TXN matrix of observable random variables (endogenous variables), X is a T X К matrix of known constants (exogenous variables), U is аГХ N matrix of unobservable r...