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Weak Instruments and Identification 2 Slides

# Weak Instruments and Identification 2 Slides - NBER Summer...

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Revised July 22, 2008 4-1 NBER Summer Institute What’s New in Econometrics – Time Series Lecture 4 July 14, 2008 Weak Instruments, Weak Identification, and Many Instruments, Part II

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Revised July 22, 2008 4-2 Outline Lecture 3 1) What is weak identification, and why do we care? 2) Classical IV regression I: Setup and asymptotics 3) Classical IV regression II: Detection of weak instruments 4) Classical IV regression III: hypothesis tests and confidence intervals Lecture 4 5) Classical IV regression IV: Estimation 6) GMM I: Setup and asymptotics 7) GMM II: Detection of weak identification 8) GMM III: Hypothesis tests and confidence intervals 9) GMM IV: Estimation 10) Many instruments
5) Classical IV regression IV: Estimation Estimation is much harder than testing or confidence intervals Uniformly unbiased estimation is impossible (among estimators with support on the real line), uniformly in μ 2 Estimation must be divorced from confidence intervals Partially robust estimators (with smaller bias/better MSE than TSLS): Remember k -class estimators? ˆ () = [ Y ( I k k β M Z ) Y ] –1 [ Y ( I k M Z ) y ] TSLS: k = 1 , LIML: k = ˆ LIML k = smallest root of det( Y Y k Y M Z Y ) = 0 Fuller: k = ˆ LIML k c /( T–k–#included exog. ), c > 0 Revised July 22, 2008 4-3

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Comparisons of k -class estimators Anderson, Kunitomo, and Morimune (1986) – using second order theory Hahn, Hausman, and Kuersteiner (2004) – using MC simulations LIML median unbiased to second order HHK simulations – LIML exhibits very low median bias no moments exist! There can be extreme outliers LIML also can be shown to minimize the AR statistic: ˆ LIML β : min AR( ) = 00 () / / ( ) P k M Tk ββ −− Z Z y Y y Y yY so LIML necessarily falls in the AR confidence set if it is nonempty Revised July 22, 2008 4-4
Revised July 22, 2008 4-5 Comparisons of k -class estimators, ctd. Fuller With c = 1, lowest RMSE to second order among a certain class (Rothenberg (1984)) In simulation studies ( m =1), Fuller performs very well with c = 1 Others (Jacknife TSLS; bias-adjusted TSLS) are dominated by Fuller, LIML Summary and recommendations Under strong instruments, LIML, TSLS, k -class will all be close to each other. under weak instruments, TSLS has greatest bias and large MSE LIML has the advantage of minimizing AR – and thus always falling in the AR (and CLR) confidence set. LIML is a reasonable (good) choice as an alternative to TSLS.

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What about the bootstrap or subsampling ? The bootstrap is often used to improve performance of estimators and tests through bias adjustment and approximating the sampling distribution. A straightforward bootstrap algorithm for TSLS: y t = β Y t + u t Y t = Π′ Z t + v t ˆ i) Estimate , Π by , TSLS ˆ Π ii) Compute the residuals , ˆ t u ˆ t v iii) Draw T “errors” and exogenous variables from { , , Z t }, and construct bootstrap data ˆ t u ˆ t v , using , t Y ± ˆ TSLS ˆ Π t y ± iv) Compute TSLS estimator (and t -statistic, etc.) using bootstrap data v) Repeat, and compute bias-adjustments and quantiles from the boostrap distribution, e.g. bias = bootstrap mean of
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Weak Instruments and Identification 2 Slides - NBER Summer...

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