Full Information Methods
In this section, we change the notation somewhat and write the jth equation as
Vj = Yj в + Xj Yj + U = Zj 8j + U (6-16)
Here, у? is T x 1, Yj is T x (G? – 1), Xj is T x K?, and Z? is T x (G? + K? – 1).
We can further write the whole linear simultaneous system in "stacked" form as
V = Z8 + u, (6.17)
where, now, y, Z, 8, and u are defined differently from previous sections. If we start with Y and U as defined in Section 1, then y’ = (vec Y)’ = (y1, у2,…, yG), V j = jth column of Y, u’ = (vec U)’ = (u1, u2,…, u’G), u? = ith column of U, 8′ = (81, 82,…, 8G), 8′ = (P;’, Yj), Zj = (Yj, Xj), and Z = diagonal (Zj).
Premultiplying both sides of (6.16) by X’, we get
X V j = X Z?8 j + Xuj = 1, 2, …, G (6.18)
or, for the whole system,
(I ® X)V = (I ® X)Z8 + (I ® X)u. (6.19)
Recall that the application of generalized least squares to equation (6.18) separately for each j, with Xu? ~ (0, a jX X) produces the 2SLS estimator of 8?. On the other hand, feasible generalized least squares applied to the whole system in
(6.19) , with (I ® X’)u ~ (0, X ® XX), leads to the three-stage least squares (3SLS) estimator of 8:
83SLS = (Z'(X-1 ® Px)Z)-1(Z'(X-1 ® Px)v), (6.20)
where X is estimated from calculated 2SLS structural residuals.
In contrast, note that the 2SLS estimator of 8, obtained by applying 2SLS separately to each equation, is 82SLS = (Z'(I ® Px)Z)-1(Z'(I ® PX)y). Thus 2SLS is a special case of 3SLS where I is taken to be the estimate of X. Also, 2SLS and 3SLS would be exactly equivalent if X is diagonal.
Being a generalized least squares estimator, 3SLS also can be interpreted as an IV estimator of 8 in (6.17), where the instrument matrix for Z is
W3SLS = (X 1 ® PX)Z.
The (i, j)th block of W3SLS can be written further as
aij(PXYj, Xj) = o^XП’, Xj),
where П? is the unrestricted least squares estimate of П?.
As in the limited information case, instrumental variable procedures have been developed as alternatives to 3SLS. One method is the full information instrumental variables efficient estimator (FIVE). In this procedure, instead of (6.21), the instrument matrix for Z would be sij (XП’, Xj). As in the case of LIVE, Ц is the appropriate sub-matrix of the restricted reduced form estimate П = – В-1Г based on some preliminary consistent estimates B and Г, and sij is the (i, j)th element of the inverse of S = й’й/T = (YB’ + t Г’)'(YB’ + X Г’ )/T.
Note that FIVE utilizes all the restrictions in the system to construct the instrument matrix. 3SLS, on the other hand, does not, in the sense that no restrictions on П are imposed in the calculation of X П’ in (6.22).
Another system method is the full information maximum likelihood (FIML) estimator. This is obtained by maximizing the likelihood of y defined in (6.17) subject to all prior restrictions in the model.
Solving d log L/dX-1 = 0 yields the FIML estimate of X : X = (YB’ + XГ’)'(YB’ + XF’ )/T. Thus, the concentrated loglikelihood with respect to X is log LC = C + T log | B | – T/2 log|(YB’ + XГ’)'(YB’ + XГ’) |, where C = – GT/2(1 + log 2n). The FIML estimate of B and Г are then solutions to 0 = d log LC/d5 = f (XFIML), say. A Taylor series expansion of f (XFIML) around some consistent estimator 5 to get
0 = f (Xfiml) » f (5) + (H(5))(8 – 5) (6.23)
gives the linearized FIML estimator:
Xlfiml = 5 – (H(5))-1f (5), (6.24)
where H is the Hessian matrix (of second order partial derivatives) of log LC with respect to 5.
This linearized FIML estimator provides one way of numerically approximating XFIML. Such a procedure, as we shall see in the next section, will be asymptotically equivalent to FIML under certain regularity conditions. If carried through to convergence, linearized FIML, if it converges, will coincide exactly with FIML. Furthermore, at convergence, (H(5))-1 provides an estimate of the asymptotic covariance matrix of XFIML.
Other iterative procedures also have been devised for the numerical calculation of the FIML estimator. One such algorithm proceeds from a given estimate of 5, say X(p at the pth iteration:
6j(p) = (yi – ZiXi(p)) (yj – ZXj(p))/T and X(p) = (^ij(p)),
where Z(p) = diag(Z1, …, ZG), Zj = (Yj, Xj) and Yj comes from the solution values of the estimated system based on the estimate X( p).
The formula for Xp+1 follows the 3SLS formula in (6.20) with the following major differences:
1. X is updated through the iteration rounds.
2. The endogenous components Yj in Z(p) are constructed as solutions to the structural system (as in FIVE) and not as in 3SLS.
We end this section with some additional algebraic relationships among the estimators.
1. 3SLS reduces to 2SLS when X is diagonal or when all equations are just identified.
2. If some equations are overidentified and some are just identified, then
(a) the 3SLS estimates of the overidentified equations can be obtained by the application of 3SLS to the set of over identified equations, ignoring all just identified equations.
(b) the 3SLS estimate of each just identified equation differs from the 2SLS estimate by a vector which is a linear function of the 3SLS residuals of the overidentified equations. In particular, if we are dealing with a system consisting of one just identified equation and one overidentified, then 2SLS and 3SLS will be exactly equivalent for the overidentified but not for the just identified equation.
2 Large Sample Properties of Estimators
The statistical behavior of these estimators has been analyzed in finite samples and in the following alternative parameter sequences:
1. Sample size T ^ – the standard asymptotic analysis.
2. Concentration parameter ^ ^ with T fixed. For example, see Basmann (1961), Mariano (1975), Anderson (1977) and Staiger and Stock (1997). The concentration parameter is defined in Section 6.2 of this chapter.
3. Structural error variances go to zero – the so-called small-о asymptotics discussed in Kadane (1971), and Morimune (1978).
4. Number of instruments, L, goes to infinity with sample size such that L/T ^ a (0 < a < «>). See Anderson (1977), Kunitomo (1980), Morimune and Kunitomo (1980), Morimune (1983), and Bekker (1994).
5. Weak instrument asymptotics. Here L is fixed and the coefficients of instruments in the first stage regression in IV or modified 2SLS estimators are assumed to be O (T-1/2) as a way of representing weak correlation between the instruments and the endogenous explanatory variables – see Staiger and Stock (1997), and Bound, Jager and Baker (1995). Related discussion of structural testing, model diagnostics and recent applications is in Wang and Zivot (1998), Angrist (1998), Angrist and Krueger (1992), Donald and Newey (1999), and Hahn and Hausman (1999).
In this section, we summarize results under large sample asymptotics, then consider finite sample properties and end with a discussion of the practical implications of the analysis based on these alternative approaches.