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quantitative application of statistical and mathematical models

Consistency of 2SLS We now present a demonstration of the consistency of the 2SLS estimate for the structural parameter. The following is a set of regularity conditions. Assumption 12.1 1. The observations (yi ; xi; zi); i = 1; :::; n; are independent and identically distributed. 2. E y 2  < 1: 3. E kxk 2 < 1: 4. E kzk 2 < 1: 5. E (zz0 ) is positive deÖnite. 6. E (zx0 ) has full rank k: 7. E (ze) = 0: Assumptions 12.1.2-4 state that all variables have Önite variances. Assumption 12.1.5 states that the instrument vector has an invertible design matrix, which is identical to the core assumption about regressors in the linear regression model. This excludes linearly redundant instruments. Assumptions 12.1.6 and 12.1.7 are the key identiÖcation conditions for instrumental variables. Assumption 12.1.6 states that the instruments and regressors have a full-rank cross-moment matrix. This is often called the relevance condition. Assumption 12.1.7 states that the instrumental variables and structural error are uncorrelated. Assumptions 12.1.5-7 are identical to DeÖnition 12.1. Theorem 12.1 Under Assumption 12.1, b 2sls p ! as n ! 1: The proof of the theorem is provided below. This theorem shows that the 2SLS estimator is consistent for the structural coe¢ cient under similar moment conditions as the least-squares estimator. The key di§erences are the instrumental variables assumption E (ze) = 0 and the identiÖcation assumption rank (E (zx0 )) = k. The result includes the IV estimator (when ` = k) as a special case. The proof of this consistency result is similar to that for the least-squares estimator. Take the structural equation y = X + e in matrix format and substitute it into the expression for the estimator. We obtain b 2sls =  X0Z Z 0Z 1 Z 0X 1 X0Z Z 0Z 1 Z 0 (X + e) = +  X0Z Z 0Z 1 Z 0X 1 X0Z Z 0Z 1 Z 0e: (12.41) This separates out the stochastic component. Re-writing and applying the WLLN and CMT b 2sls =  1 n X0Z   1 n Z 0Z 1  1 n Z 0X !1   1 n X0Z   1 n Z 0Z 1  CHAPTER 12. INSTRUMENTAL VARIABLES 424 p ! QxzQ1 zz Qzx1 QxzQ1 zzE (ziei) = 0 where Qxz = E xiz 0 i  Qzz = E ziz 0 i  Qzx = E zix 0 i  : The WLLN holds under the i.i.d. Assumption 12.1.1 and the Önite second moment Assumptions 12.1.2-4. The continuous mapping theorem applies if the matrices Qzz and QxzQ1 zz Qzx are invertible, which hold under the identiÖcation Assumptions 12.1.5 and 12.1.6. The Önal equality uses Assumptio