Category Advanced Econometrics Takeshi Amemiya

Box-Cox Transformation

Подпись:Подпись:Box and Cox (1964) proposed the model3
z,(A) = x# + u„

where, forj>, > 0,

— 1

Z,(A)== )T if ХФ0

= log y, if A = 0.

Note that because lim^oCy? — 1 )/A = log y,, z,(A) is continuous at A = 0. It is assumed that (и,) are i. i.d. with Eu, = 0 and Vu, = a2.

The transformation z,(A) is attractive because it contains y, and log y, as special cases, and therefore the choice between y, and log y, can be made within the framework of classical statistical inference.

Box and Cox proposed estimating A, fi, and a1 by the method of maximum likelihood assuming the normality of ut. However, u, cannot be normally distributed unless A = 0 because z,(A) is subject to the following bounds:

z,(A)S-I if A>0 (8.1.14)

if A <0.

A

Later we shall discuss their implications on the properties of the Box – Cox M...

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Results of Cosslett: Part I

Cosslett (1981a) proved the consistency and the asymptotic normality of CBMLE in the model where both /and Q are unknown and also proved that CBMLE asymptotically attains the Cramer-Rao lower bound. These results require much ingenuity and deep analysis because maximizing a likelihood function with respect to a density function /as well as parameters fi creates a new and difficult problem that cannot be handled by the standard asymptotic theory of MLE. As Cosslett noted, his model does not even satisfy the condi­tions of Kiefer and Wolfowitz (1956) for consistency of MLE in the presence of infinitely many nuisance parameters.

Cosslett’s sampling scheme is a generalization of the choice-based sampling we have hitherto considered...

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Amemiya’s Least Squares and Generalized Least Squares Estimators

Amemiya (1978c, 1979) proposed a general method of obtaining the esti­mates of the structural parameters from given reduced-form parameter esti­mates in general Tobit-type models and derived the asymptotic distribution. Suppose that a structural equation and the corresponding reduced-form equations are given by

y = Yy + X,)? + u – (10.8.11)

[y> Y] = X[jr, П] + V,

where X, is a subset of X. Then the structural parameters у and fi are related to the reduced-form parameters n and П in the following way:

п = Пу + ЗР, (10.8.12)

where J is a known matrix consisting of only ones and zeros. If, for example, Xj constitutes the first Kx columns of X (K = Kx+ K2), then we have J = (I, 0)’, where I is the identity matrix of size Kx and 0 is the K2 X Kx matrix of zeros...

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Three Error Components Models

6.6.1 Three error components models are defined by

= л. i= 1, 2,. . . ,N, (6.6.1)

t= 1,2,. . . ,Г, and

ий = а + Л, + €й, (6.6.2)

where Ці and A, are the cross-section-specific and time-specific components mentioned earlier. Assume that the sequence {pi), (A,), and {ей} are i. i.d. random variables with zero mean and are mutually independent with the variances a *, a, and a, respectively. In addition, assume that хй is a Af-vector of known constants, the first element of which is 1 for all i and t.

We shall write (6.6.1) and (6.6.2) in matrix notation by defining several symbols. Define y, u, €, and X as matrices of size NTX 1, NTX 1, NTX 1, and NTXK, respectively, such that their [(г — 1)Г + r]th rows are yit, ии, €й, and хй, respectively. Also define ц = (цх, ц2, • ...

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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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