The Classical Simultaneous Equations Model

We now turn to identification in the classical simultaneous equations model. Estimation in this model is comprehensively reviewed in chapter 6 by Mariano. The model is

Подпись: (7.9)By + Гх = Z,

where y is an m-vector of endogenous variables, x is a stochastic k-vector of exogenous variables, and Z is an m-vector of disturbances. It is assumed that E(Z) = 0 and E(xZ’) = 0, so that x and Z are uncorrelated, and Ex = E(xx’) is nonsingular. The coefficient matrices B and Г are of order m x m and m x k, respectively, and B is nonsingular.

Подпись: (y, x') = (Z, X) image202 Подпись: -1 0 Ik _ ,

Under normality, the distribution of the observations (y; x) is uniquely deter­mined by the first two moments. Since model (7.9) can be rewritten as

image13 image14 Подпись: B' г Подпись: -1 0 Ik _ Подпись: (7.10)

the moment equations are given by

Подпись:(7.11)

Thus, equations (7.10) and (7.11) contain all observational information with regard to the structural parameter matrices B, Г and ^z = E(ZZ’).

However, for the identification of the structural parameter matrices we need only consider a subset of the equations in (7.10) and (7.11). The equations in (7.11) can be separated into

Подпись: 1 M << M ■fi i ~B'" _X xy X x _ _Г'_ (X xy, X x )
Подпись: (7.12) (7.13)
Подпись: B' Г
Подпись: = 0’

and, in fact, these equations contain all observational information relevant for the identification of B, Г and X?. That is to say, since the equations (7.10) and (7.11) are satisfied by the true parameter point, we find that (qy, q’x) = qX^X (Xxy, Xx) so that, if any matrices B and Г satisfy (7.13), they will also satisfy (7.10). The first moment equations do not provide additional information.

Now let prior information be given by a set of restrictions on the coefficient matrices B and Г:

p(B, Г) = 0, (7.14)

where p(B, Г) is a vector function of the matrices B and Г. The parameter matrix X? is assumed to be unrestricted. In that case all information relevant for the identification of B and Г is given by equation (7.13), which after stacking in a vector can be written as

Подпись: = d(g(B, Г); p(B, Г)) d(vec'(B'), vec'(r')) Подпись: (Im ® Xxy, Im ® Xx) RBr

°(B, Г) = (Im ® Xxy)vec(B’) + (Im <8> X>ec(F) = 0, and by equation (7.14). The Jacobian matrix is

where RBT – is defined implicitly and is assumed to be of full row rank. Post­multiplication of the Jacobian by a conveniently chosen nonsingular matrix:

0

I ® x

Im ® B’

0 "

=

Im ® B’

"0 "

Im ® Г’

Imk

Rвr

I ® Г’

L m J

_Imk _

J(B, Г)

image216 Подпись: (7.15)

shows that J(B, Г) has full column rank if and only if

has full column rank. Furthermore J(B, Г) and 4(B, Г) share regular points. Thus, by Theorem 7 we have the following result.

Theorem 10. Let H = {(B, Г )|(B, Г) Є Rm2+mk, p(B, Г) = 0} and let (B, Г )0 be a regular point of J(B, Г )| H Then (B, Г )0 is locally identified if and only if J(B°, Г0) has full row rank m2.

This result corresponds to the classical condition for identification in a simultane­ous equation model. If the restrictions on (B, Г) are linear, i. e. when p(B, Г) is a linear function, for example when separate elements of (B, Г) are restricted to be fixed, then both sets of identifying equations (7.13) and (7.14) are linear so that J(B, Г) is constant over the parameter space. In that case local identification im­plies global identification and we have the following corollary.

Corollary 2. Let p(B, Г) be linear, then (B, Г )0 is globally identified if and only if rank{J(B, Г)} = m2.

This constitutes the well-known rank condition for identification in simultaneous equations models, as developed in the early work of the Cowles Commission, e. g. Koopmans, Rubin, and Leipnik (1950), Koopmans (1953), and Koopmans and Hood (1953). Johansen (1995, theorem 3) gives an elegant formulation of the identification of the coefficients of a single equation.

4 Concluding Remarks

In this chapter we have presented a rigorous, self-contained treatment of identi­fication in parametric models with iid observations. The material is essentially from the book by Bekker, Merckens, and Wansbeek (1994); see Rigdon (1997) for an extended review. The reader is referred to this book for a number of further topics. For example, it discusses identification of two extensions of the classical simultaneous equations model in two directions, viz. restrictions on the covari­ance matrix of the disturbances, and the measurement error in the regressors. It also discusses local identification of the equally classical factor analysis model.4 These two models have been integrated in the literature through the hugely popular Lisrel model, which however often confronts researchers with identi­fication problems which are hard to tackle analytically since, in the rank con­dition for identification, inverse matrices cannot be eliminated as in the classical simultaneous equations model. The book tackles this issue by parameterizing the restrictions on the reduced form induced by the restrictions on the structural form.

A distinctive feature of the book is its use of symbolic manipulation of alge­braic structures by the computer. Essentially, all identification and equivalence results are couched in terms of ranks of structured matrices containing unknown parameters. To assess such ranks has become practically feasible by using com­puter algebra. The book contains a diskette with a set of computer algebra pro­grams that can be used for rank evaluation of parameterized matrices for the models discussed.

Notes

* This chapter is largely based on material adapted from P. A. Bekker, A. Merckens, and T. J. Wansbeek, Identification, Equivalent Models and Computer Algebra (Orlando: Academic Press, 1994). Reproduced by kind permission of the publisher. We are grateful to Badi Baltagi, Bart Boon, and an anonymous referee for their useful comments.

1 Identification in nonparametric models is a much different field, see, e. g., Prakasa Rao (1992).

2 Of course, it may be the case that exact knowledge of P(y, 00) is sufficient to derive bounds on 00k (see, e. g., Bekker et al., 1987; Manski, 1989; Manski, 1995). In such a case the sample information can be used to increase knowledge about 0 k even though this parameter is not locally identified.

3 However, if one uses a "natural" parameter sequence, it may happen that 0 0k is identi­fied, whereas no estimator converges in probability to 0 0k. For example, Gabrielsen (1978) discussed the model yt = Pr1 + ui, i = 1,. .., n, where the ui are iid N(0, 1), r is known and | r | < 1, and P is an unknown parameter. Here the OLS estimator S ~ N(P, (1 – r2)/(r2(1 – r2n))) is unbiased, so clearly P is identified, but it is not con­sistent in the natural sequence defined by the model where n ^ ~. Since S is efficient, there does not exist a consistent estimator.

4 For a discussion of global identification in factor analysis see Bekker and ten Berge (1997).

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