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137 XY, = XY, t,V, = 0 (9A.6) We assume that the exogenous variables are independent of the errors so that plimXu = 0 and also that plim XX is a positive definite matrix (i.e., there are no linear dependencies among the xs and no degeneracies). These assumptions now enable us to prove that the instrumental variable (IV) estimator is consistent and also enable us to derive its asymptotic distribution. Define Z, = [Y,X,]. Then using the relations (9A.6) we can check that Z,Z, = ZJZ,. The IV estimator of 6 in (9A.4) is 6,v = (Z,Z,)-z;y = (Z,Z,)"Zi(Z,5 + u) = 5 + (Z,Z,)-Z,u plim S,v = 8 -b plimZ,Z, . plimZ,u = 5, since pHmZ,u (9A.7) and plim-ZjZ, j is finite. These relations follow from the assumptions that plimXu = 0 and plimXX is a positive definite matrix. Thus, as is expected, the IV estimator is consistent. The asymptotic covariance matrix of S,v is given by AE Titjy - 6)(S,v - 8) where AE denotes asymptotic expectation. It is customary to assume that we can substitute plim for AE. This gives the asymptotic covariance matrix as plim r(S,v - 8)(8,v - 8) /1. /1. and using (9A.7) we get this = plim -Z,Z, I . pUmI -Z,uuZ, plimZ,Z, = (T plimiz,Z, (9A.8) (9A.9) since £(uu) = and Z,Z, = ZJZ,. In practice we estimate by 2 = (y - Y,p - X,7)(y - Y,p - X,7) Also, note that Y,Y, = Y;mMY, = Y;MY„ where M = X(XX)-X is an idempotent matrix and YJX = YJX. Also, in practice we estimate /1 - plim -z;z, by y;my, y;x, x,y, x,x,
YiMY, y;x, XY xx, and estimate cr- by & given by (9A.9). In the 2SLS estimation method, we use Y, as regressors rather than instrumental variables; that is, we substitute Y, for Y, on the right-hand side of (9A.4) and estimate the equation by OLS. The equation is = Y,p + x,7 + (u + V,p) = Z,5 + (u + V,P) §2SLs = (Z;Z,)- Z,y = (Z;Z,) Zi(Z,5 + u + V,P) = 5 + (Z;z,)-Z,u since z;v, = 0 Since Z;Z, = ZiZ„ it follows from (9A.7) that SV = S2SLS That is, it does not make any difference whether Y, is used as a regressor or as an instrument. This shows that the 2SLS estimator is also an IV estimator. We can also show that the 2SLS estimator is the best IV estimator. We shall, however, omit the proof here. -For a proof, see Maddala, Econometrics, p. 477. In practice, we consider the variance of 8,v (not VT S,v) and we write var(8,v) = cr
Models of Expectations 10.1 Introduction 10.2 Naive Models of Expectations 10.3 The Adaptive Expectations Model 10.4 Estimation with the Adaptive Expectations Model 10.5 Two Illustrative Examples 10.6 Expectational Variables and Adjustment Lags 10.7 Partial Adjustment with Adaptive Expectations 10.8 Alternative Distributed Lags: Polynomial Lags 10.9 Rational Lags 10.10 Rational Expectations 10.11 Tests for Rationality 10.12 Estimation of a Demand and Supply Model Under Rational Expectations 10.13 The Serial Correlation Problem in Rational Expectations Models Summary Exercises 10.1 Introduction Expectations play a crucial role in almost every economic activity. Production depends on expected sales, investment depends on expected profits, long term interest rates depend on expected short-term rates, expected inflation rates,
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