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Uniform Approximations

LIML Asymptotic Distribution In this section we show that the LIML estimator is asymptotically equivalent to the 2SLS estimator. We recommend, however, a di§erent covariance matrix estimator based on the IV representation. We start by deriving the asymptotic distribution. Recall that the LIML estimator has several representations, including b liml = X0 (In bMZ) X 1 X0 (In bMZ) y  where b = min 0Y 0M1Y 0Y 0MZY : For the distribution theory, it is useful to rewrite this as b liml = X0P ZX bX0MZX 1 X0P Zy bX0MZy  where b = b 1 = min 0Y 0M1Z2 (Z 0 2M1Z2) 1 Z 0 2M1Y 0Y 0MZY : This second equality holds since the span of Z = [Z1; Z2] equals the span of [Z1;M1Z2]. This implies P Z = Z Z 0Z 1 Z 0 = Z1 Z 0 1Z1 1 Z 0 1 + M1Z2 Z 0 2M1Z2 1 Z 0 2M1: We now show that nb = Op(1). The reduced form (12.35) implies that Y = Z11 + Z22 + a: It will be important to note that 2 = [2;22] = [22 2 ;22] using (12.18). It follows that 2 = 0 for = (1; 0 2 ) 0 : Note u = e. Then MZY = MZe and M1Y = M1e. Hence nb = min 0Y 0M1Z2 (Z 0 2M1Z2) 1 Z 0 2M1Y 0 1 n Y 0MZY   p 1 n e 0M1Z2  1 nZ 0 2M1Z2 1  p 1 n Z 0 2M1e  1 n e 0MZe = Op(1): It follows that p n  b liml  =  1 n X0P ZX b 1 n X0MZX 1  1 p n X0P Ze p nb 1 n X0MZe  =  1 n X0P ZX op(1)1  1 p n X0P Ze op(1) = p n  b 2sls  + op(1) which means that LIML and 2SLS have the same asymptotic distribution. This holds under the same assumptions as for 2SLS, and in particular does not CHAPTER 12. INSTRUMENTAL VARIABLES 429 Consequently, one method to obtain an asymptotically valid covariance estimate for LIML is to use the same formula as for 2SLS. However, this is not the best choice. Rather, consider the IV representation for LIML b liml =  Xf0 X 1  Xf0 y  where Xf =  X1 X2 bUb 2  and Ub 2 = MZX2 . The asymptotic covariance matrix formula for an IV estimator is Vb =  1 n Xf0 X 1 b  1 n X0Xf 1 (12.44) where b = 1 n Xn i=1 xeixeieb 2 i ebi = yi x 0 i b liml: This simpliÖes to the 2SLS formula when b = 1 but otherwise di§ers. The estimator (12.44) is a better choice than the 2SLS formula for covariance matrix estimation as it takes advantage of the LIML estimator struct