Category Advanced Econometrics Takeshi Amemiya

Seemingly Unreleted Regression Model

The seemingly unrelated regression (SUR) model proposed by Zellner (1962) consists of the following N regression equations, each of which satisfies the assumptions of the standard regression model (Model 1):

y.-X/A+u,, 1=1,2______ ,N, (6.4.1)

where y, and u, are T-vectors, X, is a T X Kt matrix, and fit is a A, -vector. Let utt bethetthelementofthevectoru,-.Thenweassumethat(uu, Цц,. . . , uM)is an i. i.d. random vector with Eu„ = 0 and Cov (ий, Uj,) = a0. Defining у = (УЇ, У2. • • • ,У*)’> Р^іРиРг,- • • ,0’nY, u = (ui, u^,. . . ,ui,)’, and X = diag (Xj, X2,. . . , ХД we can write (6.4.1) as

у = Xfi + u. (6.4.2)

This is clearly a special case of Model 6, where the covariance matrix of u is given by

£uu’ = ft = 2 ® Iy, (6.4.3)

where 2 = {(Ту) and ® denotes the ...

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Classical Least Squares Theory

In this chapter we shall consider the basic results of statistical inference in the classical linear regression model—the model in which the regressors are inde­pendent of the error terrn and the error term is serially uncorrelated and has a constant variance. This model is the starting point of the study; the models to be examined in later chapters are modifications of this one.

1.1 Linear Regression Model

In this section let us look at the reasons for studying the linear regression model and the method of specifying it. We shall start by defining Model 1, to be considered throughout the chapter.

1.1.1 Introduction

Consider a sequence of К random variables (y„ x^, x3t,. . . , x^,), f = 1, 2,. . . ,T. Define a Г-vector у = (у,, y2> ■ • • .Уг)’, a (K-iy vector xf = (x2l, x3(). . ...

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Stein’s Estimator: Heteroscedastic Case

Assume model (2.2.5), where Л is a general positive definite diagonal matrix. Two estimators for this case can be defined.

Ridge estimator: a* = (Л 4- yI)_1Aa,

af = (1 – Bi)dt where B, =

(Note: The transformation a = H’fi translates this estimator into the ridge estimator (2.2.12). у is either a constant or a function of the sample.)

Generalized ridge estimator: a* = (A + Г)-1Ла where Г is

diagonal,

af = (1 — 2?,)a, where

R – Уі

‘ ^ + У/’

fi* = (X’X + HTHT’X’y.

Other ridge and generalized ridge estimators have been proposed by various authors. In the three following ridge estimators, у is a positive quantity that does not depend on A,; therefore Bt is inversely proportional to A,...

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Time Series Analysis

Because there are many books concerned solely with time series analysis, this chapter is brief; only the most essential topics are considered. The reader who wishes to study this topic further should consult Doob (1953) for a rigorous probabilistic foundation of time series analysis; Anderson (1971) or Fuller (1976) for estimation and large sample theory; Nerlove, Grether, and Car­valho (1979) and Harvey (1981 a, b), for practical aspects of fitting time series by autoregressive and moving-average models; Whittle (1983) for the theory of prediction; Granger and Newbold (1977) for the more practical aspects of prediction; and Brillinger (1975) for the estimation of the spectral density.

In Section 5...

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