Transcription of Weak IV. 1. What are Weak Instruments?
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Weak IV. 1. What are Weak Instruments?
1. What are Weak instruments ? Time Series Analysis, Fall 2007 Professor Anna MikushevaPaul Schrimpf, scribeNovemeber 13, 2007corrected September 2012 LectureWeak lecture extensively uses lectures given by Jim Stock as a part of mini-course at the NBER What are Weak Instruments? Consider the simplest classical homoskedastic IV model:yt= xt+utxt=Zt +vt,whereytare one-dimensional,xtisn 1,Ztisk 1 andEutZt= 0, one observes data{yt,xt,Zt}.Assume thatn k. In generalutandvtare correlated, and thus,xtis an endogenous (since we assumed thatEutZt= 0), if it also relevant (EZ xhas rankn), then it can serveas instrument and the model is identified. The usual TSLS will be TSLS= (x PZx) 1x PZy,wherePZ=Z(Z Z) 1Z .The problem of weak identification arises when moment conditions are not very informative about theparameter of interest, that is, when the rank of the matrix g0( 0)| = R 0=( 0) =EZ xisn, but atthe same time itis very close to a reduced rank matrix (for example, the smallest eigenvalue ofn nmatrixx Z(Z Z) 1Z xis very closeto zero).
Weak instruments asymptotics. Weak instrument asymptotics is the name for asymptotic embedding modeling correlation as converging to zero at speed p T:It is the same as modeling being constant. So, assume that ˇ= C= p T. Then ( + ^ z. TSLS 0) v) 0. z. u; ( + z. v) 0
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