Estimation in a System of Equations
Define a system of N nonlinear simultaneous equations by
fu(y,, <*i) = ua, і = 1, 2,. . . , N, t = 1, 2,. . . , T,
where y, is an N-vector of endogenous variables, x, is a vector of exogenous variables, and at is a vector of unknown parameters. We assume that the
N-vector u, = (uu, u2t,. . . , uNt)’ is an i. i.d. vector random variable with zero mean and variance-covariance matrix 2. Not all of the elements of vectors yf and x( may actually appear in the arguments of each f,. We assume that each equation has its own vector of parameters a, and that there are no constraints among a, but the subsequent results can easily be modified if each at can be parametrically expressed as where the number of elements in в is less than 2fLt K,.
Strictly speaking, (8.2.1) is not a complete model by itself because there is no guarantee that a unique solution for y, exists for every possible value of uit unless some stringent assumptions are made on the form of fit. Therefore we assume either that fit satisfies such assumptions or that if there is more than one solution for yt, there is some additional mechanism by which a unique solution is chosen.4
We shall not discuss the problem of identification in the model (8.2.1). There are not many useful results in the literature beyond the basic discussion of Fisher (1966), as summarized by Goldfeld and Quandt (1972,p.221), and a recent extension by Brown (1983). Nevertheless, we want to point out that nonlinearity generally helps rather than hampers identification, so that, for example, in a nonlinear model the number of excluded exogenous variables in a given equation need not be greater than or equal to the number of parameters of the same equation. We should also point out that we have actually given one sufficient condition for identifiability—that plim T_I(GoP»XJo) the
conclusion of Theorem 8.1.2 is nonsingular.
Definition of the symbols used in the following sections may facilitate the discussion.
a = (a’1,ai,. . . ,а^)’
Л = 2 ® I, where ® is the Kronecker product ft, =/й(У„ x„ a,)
f, = an N-vector, the Zth element of which is fit f(i) = a Г-vector, the Zth element is/ft. f = (fо), f(2), ■ • • , f(Ao)r> an А^Г-vector F = (f(i>, f(2)> • • • , f(Af)), a Г X TV matrix
gи = XT’ a ^/-vector
G, = -3, a ГХ Kj matrix, the Zth row is g’,
G = diag(G,, G2,. . . , G*}, an NTX(2£.! К,) block diagonal matrix