Category Advanced Econometrics Takeshi Amemiya

Berkson’s Minimum Chi-Square Method

There are many variations of the minimum chi-square (MIN x1) method, one of which is Berkson’s method. For example, the feasible generalized least squares (FGLS) estimator defined in Section 6.2 is a MIN x2 estimator. An­other example is the Barankin-Gurland estimator mentioned in Section 4.2.4. A common feature of these estimators is that the minimand evaluated at the estimator is asymptotically distributed as chi-square, from which the name is derived.

The MIN x2 method in the context of the QR model was first proposed by Berkson (1944) for the logit model but can be used for any QR model...

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Standard Tobit Model (Type 1 Tobit Model)

Tobin (1958) noted that the observed relationship between household ex­penditures on a durable good and household incomes looks like Figure 10.1,

Подпись: ОIncome

Figure 10.1 An example of censored data where each dot represents an observation for a particular household. An important characteristic of the data is that there are several observations where the expenditure is 0. This feature destroys the linearity assumption so that the least squares method is clearly inappropriate. Should we fit a nonlin­ear relationship? First, we must determine a statistical model that can generate the kind of data depicted in Figure 10.1...

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Markov Chain Models

11.1.1 Basic Theory

Define a sequence of binary random variables

yjit) = 1 if /th person is in state j at time t (11.1. l)

= 0 otherwise,

/=1,2,. . . ,N, /=1,2,. . . , Г,

j — 1, 2,—- M.

Markov chain models specify the probability distribution of yj(Z) as a func – tionofy^(j), k= 1, 2,. . . , Л/ands = t — l, t — 2,. . . , as well as (possi­bly) of exogenous variables.

Markov chain models can be regarded as generalizations of qualitative response models. As noted in Section 9.7, Markov chain models reduce to QR models if y‘j(t) are independent over t. In fact, we have already discussed one type of Markov models in Section 9.7.2.

Models where the distribution of y](t) depends on y’k(t — 1) but not on y’k(t — 2), yk{t — 3),. . .are called first-order Markov models...

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Full Information Maximum Likelihood Estimator

In this section we shall define the maximum likelihood estimator of the parameters of model (7.1.1) obtained by assuming the normality of U, and we shall derive its asymptotic properties without assuming normality. We attach the term full information (FIML) to distinguish it from the limited information maximum likelihood (LIML) estimator, which we shall discuss later.

The logarithmic likelihood function under the normality assumption is given by

NT T

log L = – ~ log (2к) + Tlog ІІГІІ – – log |2| (7.2.1)

– j tr 2-‘(УГ – XB)'(YT – XB),

where ІІГІІ denotes the absolute value of the determinant of Г. We define the FIML estimators as the values of 2, Г, and В that maximize log L subject to the normalization on Г and the zero restrictions on Г and B...

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Universal Logit Model

In Section 9.2.1 we stated that for a binary QR model a given probability function G(xf, в) can be approximated by F[H(xf, в)] by choosing an appro­priate H(xf, в) for a given F. When F is chosen to be the logistic function A, such a model is called a universal logit model. A similar fact holds in the multinomial case as well. Consider the trichotomous case of Example 9.3.2. A universal logit model is defined by

Подпись:, exp (go)

° 1 + exp (&,) + exp (ga) ’

,_________ exp (g„)

n 1 +exp (gn) + exp (gay

and

Pi0 1 + exp (gn) + exp (ga) ’ (9’3’71)

where gn and ga are functions of all the explanatory variables of the model— z№, zn, za, and w,. Any arbitrary trichotomous model can be approximated by this model by choosing the functions gn and ga appropriately...

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Serial Correlation

Robinson (1982a) proved the strong consistency and the asymptotic normal­ity of the Tobit MLE under very general assumptions about ut (normality is presupposed) and obtained its asymptotic variance-covariance matrix, which is complicated and therefore not reproduced here. His assumptions are slightly stronger than the stationarity assumption but are weaker than the assumption that щ possesses a continuous spectral density (see Section 5.1.3). His results are especially useful because the full MLE that takes account of even a simple type of serial correlation seems computationally intractable. The autocorrelations of u, need not be estimated to compute the Tobit MLE but must be estimated to estimate its asymptotic variance-covariance matrix...

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