Spatial Lag Dependence

An alternative popular model for spatial lag dependence considered in Section 9.9 is given by: y = pWy + Хв + є

where є ~ IIN(0, a2), see Anselin (1988). Here yi may denote output in region i which is affected by output of its neighbors through the spatial coefficient p and the weight matrix W. Recall from section 9.9, W is a known weight matrix with zero elements along its diagonal. It could be a contiguity matrix having elements 1 if its a neighboring region and zero otherwise. Usually this is normalized such that each row sums to 1. Alternatively, W could be based on distances from neighbors again normalized such that each row sums to 1. It is clear that the presence of Wy as a regressor introduces endogeneity. Assuming (In — pW) nonsingular, one can solve for the reduced form model:

y = (In — pW )-1Хв + є*

where є* = (In — pW)-1є has mean zero and variance covariance matrix which has the same form as (9.38), i. e.,

£ = Е(є*є*’) = а2П = a2 (In — pW )-1(In — pW’)-1 For p < 1, one obtains

(In — pW )-1 = In + pW + p2W2 + p3W3 + …

Hence

E(y/X) = (In — pW )-1 Хв = Хв + pWXp + p2 W 2Хв + p3 W 3Хв + …

This also means that

EWy/Х) = W (In — pW )-1Хв = WXв + pW 2Хв + p2W 3Хв + p3W4Xв + …

Based on this last expression, Kelejian and Robinson (1993) and Kelejian and Prucha (1998) suggest the use of a subset of the following instrumental variables:

{Х, WX, W2Х, W3Х, W4X,…}

Lee (2003) suggested using the optimal instrument matrix:

{Х, W(In — pW)-1 Хв}

where the values for p and f3 are obtained from a first stage IV estimator, using {Х^Х} as instruments, possibly augmented with W2Х. Note that Lee’s (2003) instruments involve inverting a matrix of dimension n. Kelejian, et al. (2004) suggest an approximation based upon:

{Х, р^+1Хв}

s=0

where r, the highest order of this approximation depends upon the sample size, with r = o(n1/2). In their Monte Carlo experiments, they set r = nc where c = 0.25,0.35, and 0.45. This is a natural application of 2SLS to deal with the problem of spatial lag dependence.

11.3 System Estimation: Three-Stage Least Squares

If the entire simultaneous equations model is to be estimated, then one should consider system estimators rather than single equation estimators. System estimators take into account the zero restrictions in every equation as well as the variance-covariance matrix of the disturbances of the whole system. One such system estimator is Three-Stage Least Squares (3SLS) where the structural equations are stacked on top of each other, just like a set of SUR equations,

y = Z6 + u (11.43)

where

" yi

‘ Zi

0.

. 0

‘ 6i "

ui

y =

y2

; z =

0

Z2 .

. 0

; 6 =

62

; u =

u2

. yG _

0

Zg _

_ 6g _

ug

and u has zero mean and variance-covariance matrix £ О It, indicating the possible correlation among the disturbances of the different structural equations. £ = [aj], with E(uuj) = ajIt, for i, j = 1,2,…,G. This О notation was used in Chapter 9 and defined in the Appendix to Chapter 7. Problem 4 shows that premultiplying the i-th structural equation by X’ and performing GLS on the transformed equation results in 2SLS. For the system given in (11.43), the analogy is obtained by premultiplying by (IgOX’), i. e., each equation by X’, and performing GLS on the whole system. The transformed error (Ig О X’)u has a zero mean and variance – covariance matrix £ О (X’X). Hence, GLS on the entire system obtains

£GLS = {Z'(Ig О X)[£-i О (X’X)-1](Ig О X’)Z}-1

{Z'(Ig О X)[£-1 О (X’X)-1](Ig О X’)y (11.44)

which upon simplifying yields

£gls = {Z'[£-1 О Px]Z}-1{Z'[£-1 О Px]y} (11.45)

£ has to be estimated to make this estimator operational. Zellner and Theil (1962), suggest getting the 2SLS residuals for the i-th equation, say £ = yi — Zi6i,2SLS and estimating £ by £ = [£ij ] where

£ij = [uiuj/(T — gi — ki)1/2(T — gj — kj )1/2] for i, j = 1,2,…,G.

If £ is substituted for £ in (11.45), the resulting estimator is called 3SLS:

£3SLS = {Z'[£-1 О Px]Z}-1{Z'[£-1 О Px]y} (11.46)

The asymptotic variance-covariance matrix of 63SLS can be estimated by {Z'[£-1 О Px]Z}-1. If the system of equations (11.43) is properly specified, 3SLS is more efficient than 2SLS. But if say, the second equation is improperly specified while the first equation is properly specified, then a system estimator like 3SLS will be contaminated by this misspecification whereas a single equation estimator like 2SLS on the first equation is not. So, if the first equation is of interest it does not pay to go to a system estimator in this case.

Two sufficient conditions exist for the equivalence of 2SLS and 3SLS, these are the following: (i) £ is diagonal, and (ii) every equation is just identified. Problem 5 leads you step by step through these results. It is also easy to show, see problem 5, that a necessary and sufficient condition for 3SLS to be equivalent to 2SLS on each equation is given by

aij Zp>Sj =0 for i, j = 1,2,…,G

where Zi = PXZi, see Baltagi (1989). This is similar to the condition derived in the seemingly unrelated regressions case except it involves the set of second stage regressors of 2SLS. One can easily see that besides the two sufficient conditions given above, Z^Pg = 0 states that the set of second stage regressors of the i-th equation have to be a perfect linear combination of those in the j-th equation and vice versa. A similar condition was derived by Kapteyn and Fiebig (1981). If some equations in the system are over-identified while others are just-identified, the 3SLS estimates of the over-identified equations can be obtained by running 3SLS ignoring the just-identified equations. The 3SLS estimates of each just-identified equation differ from those of 2SLS by a vector which is a linear function of the 3SLS residuals of the over-identified equations, see Theil (1971) and problem 17.

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