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weakinstrumentslides - Introduction to GMM with Weak...

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Introduction to GMM with Weak Instruments Eric Zivot December 9, 2009
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Monte Carlo Experiment to Illustrate Problem with 2SLS with Weak Instru- ments Reference: Zivot, Startz and Nelson (1998). Valid Con fi dence Intervals and Inference in the Presence of Weak Instruments, International Economic Review . y i = z i δ + ε i , i = 1 , . . . , n z i = x 0 i (1 × k ) π ( k × 1) + v i à ε i v i ! iid N Ãà 0 0 ! , à 1 ρ ρ 1 !! x i iid N (0 , I k )
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Remarks 1. ρ captures the degree of endogeneity; ρ 0 low endogeneity; ρ 1 high endogeneity 2. π captures the strength of the instruments; π 0 weak instruments.
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Review of 2SLS estimation of δ Errors are conditionally homoskedastic so e cient GMM is 2SLS ˆ δ 2 SLS = ( z 0 P X z ) 1 z 0 P X y = δ + ( z 0 P X z ) 1 z 0 P X ε For fi xed π 6 = 0 n ³ ˆ δ 2 SLS δ ´ d N (0 , σ 2 ε ( π 0 Σ xx π ) 1 ) ˆ δ 2 SLS N ( δ, n 1 σ 2 ε ( π 0 Σ xx π ) 1 ) With σ ε = 1 and Σ xx = E [ x i x 0 i ] = I k we have avar( ˆ δ 2 SLS ) = ( π 0 π ) 1 Clearly, if π = 0 then avar( ˆ δ 2 SLS ) is not de fi ned.
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Monte Carlo Design Parameters δ = 1 ρ = 0 . 99 (very high endogeneity) k = 1 , 4; n = 100 Instrument cases 1. Irrelevant instruments: k = 1 : π = 0 k = 4 : π = (0 , 0 , 0 , 0) 0 2. Weak instruments: k = 1 : π = 0 . 1 k = 4 : π = (0 . 1 , 0 , 0 , 0) 0
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3. Strong instruments: k = 1 : π = 1 k = 4 : π = (1 , 0 , 0 , 0) 0
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Monte Carlo Experiment Goal: evaluate inference on δ using 2SLS 1. Simulate data from model 10,000 times 2. Compute ˆ δ 2 SLS and t-stat for testing δ = 1 t = ˆ δ 2 SLS 1 c SE( ˆ δ 2 SLS ) 3. Compute asymptotic 95% con fi dence interval for δ ˆ δ 2 SLS ± 1 . 96 · c SE( ˆ δ 2 SLS )
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4. Determine if | t | > 1 . 96 (equivalently, δ 95% CI) Empirical size of 5% t-test # times | t | > 1 . 96
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Monte Carlo Results Empirical Size of 5% t-test Instrument Quality k=1 k=4 Strong .055 .084 Weak .193 .855 Irrelevant .632 .987 Conclusion: Traditional asymptotic theory is not appropriate for 2SLS in the presence of weak instruments, particularly if endogeneity is high and there are many weak instruments.
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Staiger-Stock Weak Instrument Asymptotics Reference: Staiger and Stock (1997), “Instrumental Variables Regression with Weak Instruments, Econometrica .
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