The Kelejian and Stephan Model

The RCM analyzed by Kelejian and Stephan (1983) is a slight generalization of Hsiao’s model, which we shall discuss in the next subsection. Their model is defined by

Уі, = + K) + e*. (6-7л)

/=1,2,. . . , N and /=1,2,. . . , T. Note that we have separated out the nonstochastic part ft and the random partju, + A, of the regression coefficients. Using the symbols defined in Section 6.6.1 and two additional symbols, we can write (6.7.1) in vector notation as

у = XP + Хц + X*A + e, (6.7.2)

where we have defined X = diag(X,,X2,. . . , XN), X* = (Xf’, XJ’,. . . , X%’Y, where X? = diag (xj,, Xq, . . . , x’iT). It is assumed that Ц, A, and є have zero means, are uncorrelated with each other, and have covariance matrices given by Ещі’= IN®2^, ЕЛЛ’ = ІГ©ХЛ, and Eee’ = 2e, where 2M, 2A, and 2e are all nonsingular.

Kelejian and Stephan were concerned only with the probabilistic order of the GLS estimator of ft—an important and interesting topic previously over­looked in the literature. For this purpose we can assume that 2^, 2д, and 2t are known. We shall discuss the estimation of these parameters in Sections 6.7.2 and 6.7.3, where we shall consider models more specific than model (6.7.1). In these models 2* is specified to depend on a fixed finite number of parameters: most typically, 2* = c^W-

The probabilistic order offiG can be determined by deriving the order of the inverse of its covariance matrix, denoted simply as V. We have

V-^X’IXtfQS^X’ + Al^X, (6.7.3)

where Л = Х*(І7-®2я)Х*’ + 2е. Using Theorem 20 of Appendix 1, we obtain

[Х(Ідг ® XA)X’ + A]-1 (6.7.4)

= A-1 – A^XKIjv® 2;1) + X’A-’XT’X’A-‘. Therefore, noting X = X(1jv ® I*) and defining A = Х’Л – ‘X, we have

V"1 = (IlN ® I*)'{A – A[(I„ ®2~’) + A]’‘A}(V ® h)- (6.7.5)

Finally, using Theorem 19 (ii) of Appendix 1, we can simplify the (6.7.5) as V-1 = (1* © I*)’® 2,) + A-4-41* ® I*) (6.7.6)

or as

V-1 = NS.;1 – (tw® 2;‘)'[(IW® X;1) + АГЧідг® X;1). (6.7.7)

Equation (6.7.7) is identical with Eq. (11) of Kelejian and Stephan (1983, p. 252).

Now we can determine the order of V-1. If we write the /, j’th block subma­trix of [(1^© 2^)+А]-1, /,У= 1,2,. . . , N, as GiJ, the second term of the right-hand side of (6.7.7) can be written as Therefore

the order of this term is N2/T. Therefore, if T goes to » at a rate equal to or faster than N, the order of V~1 is N. But, because our model is symmetric in / and t, we can conclude that if У goes to 00 at a rate equal to or faster than T, the order of V-1 is T. Combining the two, we can state the order of V-1 is min (N, T) or that the probabilistic order of is max (N~1/2, T~l/2).

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