# 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).

## Leave a reply