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C8D55 - ASYMPTOTIC AND FINITE-SAMPLE DISTRIBUTIONS OF THE...

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ASYMPTOTIC AND FINITE-SAMPLE DISTRIBUTIONS OF THE IV ESTIMATOR 1 u X Y + + = 2 1 β β ( 29 ( 29 ( 29 ( 29 - - - - = X X Z Z Y Y Z Z b i i i i IV 2 ( 29 ( 29 ( 29 ( 29 2 , 2 2 , 2 2 2 , 2 2 2 IV 2 1 MSD 1 1 1 var IV 2 Z X u Z X i u Z X i u b r X n r X X n n r X X b × = × - = × - = = σ σ σ σ The asymptotic variance of the IV estimator is given by the expression shown. It is the expression for the variance of the OLS estimator, multiplied by the square of the reciprocal of the correlation between X and Z . 2 IV 2 plim β = b
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ASYMPTOTIC AND FINITE-SAMPLE DISTRIBUTIONS OF THE IV ESTIMATOR 2 u X Y + + = 2 1 β β ( 29 ( 29 ( 29 ( 29 - - - - = X X Z Z Y Y Z Z b i i i i IV 2 ( 29 ( 29 ( 29 ( 29 2 , 2 2 , 2 2 2 , 2 2 2 IV 2 1 MSD 1 1 1 var IV 2 Z X u Z X i u Z X i u b r X n r X X n n r X X b × = × - = × - = = σ σ σ σ What does this mean? We have seen that the distribution of the IV estimator degenerates to a spike. So how can it have an asymptotic variance? 2 IV 2 plim β = b
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ASYMPTOTIC AND FINITE-SAMPLE DISTRIBUTIONS OF THE IV ESTIMATOR 3 u X Y + + = 2 1 β β ( 29 ( 29 ( 29 ( 29 - - - - = X X Z Z Y Y Z Z b i i i i IV 2 ( 29 ( 29 ( 29 ( 29 2 , 2 2 , 2 2 2 , 2 2 2 IV 2 1 MSD 1 1 1 var IV 2 Z X u Z X i u Z X i u b r X n r X X n n r X X b × = × - = × - = = σ σ σ σ The contradiction has been caused by compressing several ideas together. We will have to unpick them, taking several small steps. 2 IV 2 plim β = b
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ASYMPTOTIC AND FINITE-SAMPLE DISTRIBUTIONS OF THE IV ESTIMATOR 4 u X Y + + = 2 1 β β ( 29 ( 29 ( 29 ( 29 - - - - = X X Z Z Y Y Z Z b i i i i IV 2 ( 29 ( 29 ( 29 ( 29 2 , 2 2 , 2 2 2 , 2 2 2 IV 2 1 MSD 1 1 1 var IV 2 Z X u Z X i u Z X i u b r X n r X X n n r X X b × = × - = × - = = σ σ σ σ The application of a central limit theorem (CLT) underlies the assertion. To use a CLT, we must first show that a variable has a nondegenerate limiting distribution. The CLT will then show that, under appropriate conditions, this limiting distribution is normal. 2 IV 2 plim β = b
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ASYMPTOTIC AND FINITE-SAMPLE DISTRIBUTIONS OF THE IV ESTIMATOR 5 u X Y + + = 2 1 β β ( 29 ( 29 ( 29 ( 29 - - - - = X X Z Z Y Y Z Z b i i i i IV 2 ( 29 ( 29 ( 29 ( 29 2 , 2 2 , 2 2 2 , 2 2 2 IV 2 1 MSD 1 1 1 var IV 2 Z X u Z X i u Z X i u b r X n r X X n n r X X b × = × - = × - = = σ σ σ σ We cannot apply a CLT to b 2 IV directly, because it does not have a nondegenerate limiting distribution. The expression for the variance may be rewritten as shown. MSD( X ) is the mean square deviation of X . 2 IV 2 plim β = b
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ASYMPTOTIC AND FINITE-SAMPLE DISTRIBUTIONS OF THE IV ESTIMATOR 6 u X Y + + = 2 1 β β ( 29 ( 29 ( 29 ( 29 - - - - = X X Z Z Y Y Z Z b i i i i IV 2 ( 29 ( 29 ( 29 ( 29 2 , 2 2 , 2 2 2 , 2 2 2 IV 2 1 MSD 1 1 1 var IV 2 Z X u Z X i u Z X i u b r X n r X X n n r X X b × = × - = × - = = σ σ σ σ By a law of large numbers, the MSD tends to the population variance of X and so has a well- defined limit.
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