Bias and MSE of The IV Estimator Under Weak
Identi¯cation¤
John Chao Norman R.Swanson简单快乐的短句
University of Maryland Texas A&M University
November2000
无助感
Abstract
We provide results on properties of the IV estimator in the prence of weak instruments,beginning with the derivation of analytical formulae for the asymptotic bias(ABIAS)and mean squared error(AMSE),within the local-to-zero asymptotic framework of Staiger and Stock(1997).The results add to the results of Staiger and Stock(1997),who have provided an approximate ABIAS measure for the two-stage least squares(2SLS)estimator relative to that of the OLS estimator.In fact,with respect to ABIAS and AMSE,we are able to prove the conjecture put forth by Staiger and Stock(1997)that the limiting distribution of the2SLS estimator under the local-to-zero assumption is the same as the exact distribution of this estimator under the more restrictive assumptions of¯xed instruments and Gaussian
errors.We also obtain approximations for the ABIAS and AMSE formulae bad on an asymptotic scheme;which,looly speaking,requires the expectation of the ¯rst stage F-statistic to converge to a¯nite(possibly small)positive limit as the number of instruments approaches in¯nity. The approximations so obtained are shown,via regression analysis,to yield excellent approximations for ABIAS and AMSE functions in general.Additionally,we show that the lead term of our bias expansion,when appropriately standardized by the ABIAS of the OLS estimator,is exactly the relative bias measure given in Staiger and Stock(1997)in the ca with one endogenous explanatory variable,and the lead term of the MSE expansion is the square of the lead term of the bias expansion. JEL classi¯cation:C12,C22.
好听的动物名字Keywords:con°uent hypergeometric function,Laplace approximation,local-to-zero asymptotics, weak instruments.
¤John Chao:Department of Economics,University of Maryland,College Park,MD,USA20742, chao@econ.umd.edu.Norman R.Swanson:Department of Economics,Texas A&M University,College Station,TX, USA77843,nswanson@econ.tamu.edu.The authors wish to thank Paul Bekker,Graham Elliot,Roberto Mariano, Carmela Quintos,Jim Stock,Tim Vogelsang,Eric Zivot,and minar participants at the University of California, San Diego,Cornell University,the University of Pennsylvan
山东省工商局ia,the University of Rochester,and the2000meetings of the World Congress of the Econometric Society for helpful comments.In addition,Swanson thanks the Private Enterpri Rearch Center and the Bush Program in the Economics of Public Policy at Texas A&M University for ¯nancial support.
1Introduction
There has been much renewed interest recently in instrumental variables regression with instruments that are only weakly correlated with the endogenous explanatory variables.Important theoretical contributions to this expanding literature include such papers as Nelson and Startz(1990a),Dufour (1996),Staiger and Stock(1997),and Zivot and Wang(1998)1.Much of this literature focus on the impact that using weak instruments has on interval estimation and hypothesis testing. In contrast,fewer theoretical results have been obtained characterizing the properties of point estimators under weak identi¯cation.This is despite the fact that applied rearchers who¯rst documented the weak instrument problem in empirical work were clearly very interested in its conquences for point estimation,as can be en from the papers by Nelson and Startz(1990b), Bound,Jaeger,and Baker(1995),and Angrist and Krueger(1995).In particular,the authors are concerned with the fact that the u of weak instruments will result in the IV estimator be
ing much more verely biad in the direction of the probability limit of the OLS estimator-a concern which in large part has accounted for the recent interest in this problem.
In this paper,we focus on the point estimation properties of the IV estimator in the prence of weak instruments.To formalize the notion of having weak instruments,we adopt the local-to-zero asymptotic framework of Staiger and Stock(1997)in the context of a simple simultaneous equations tup with a single structural equation and an arbitrary number of available instruments. An important reason for employing the local-to-zero framework is that simulation studies reported in Staiger and Stock(1997)show this framework to yield a very good approximation for the¯nite sample distribution of the IV estimator when the quality of the available instruments is poor. Staiger and Stock(1997)u this framework primarily to derive the asymptotic distributions of single-equation estimators and of related test statistics and did not conduct an extensive study of the bias and MSE of single-equation estimators,although they do provide an approximate measure for the asymptotic bias of the two-stage least squares(2SLS)estimator relative to that of the 1Related to the weak instrument literature is a literature which examines the implications for statistical inference when the underlying simultaneous equations model is underidenti¯ed,in the n of not satisfying the usual rank condition for identi¯cation.Notable contributions to this literature include Phillips(1989),Choi and
有一种爱叫等待Phillips(1992), and Kitamura(1994).The paper by Phillips(1989)is the earliest theoretical work in econometrics,known to the authors,which systematically investigates the properties of conventional statistical procedures under identi¯cation failure.
OLS estimator.Our paper,on the other hand,focus on a general IV estimator which does not necessarily make u of all available instruments,and our primary objective is the derivation of explicit analytical formulae for the asymptotic bias and MSE under the local-to-zero framework. We¯nd that our derived formulae depend only on the size of a concentration parameter,¹0¹,the number of instruments ud,k21,and the cond moments of the disturbances of the underlying model;and we examine how the bias and MSE vary as a function of¹0¹and k21.
Becau the analytical formulae for bias and MSE involve complicated functions of con°uent hypergeometric functions,we also derive approximations for the formulae bad on an expansion which,looly speaking,requires the¯rst stage F-statistic for testing instrument relevance to con-verge to a¯nite(possibly small)positive limit as the number of instruments approaches in¯nity.In order to asss the ufulness of the approximations,we conduct a ries of numerical experiments in which the accuracy of our approximation is evaluated.We¯nd that our approximations are quite accurate,even for k21as small as3or4.Additionally,the approximations allow us to make various inter
esting obrvations.For example,when the approximation method is applied to the bias,the lead term of the expansion(when appropriately standardized by the asymptotic bias of the OLS estimator)is exactly the relative bias measure given in Staiger and Stock(1997)in the ca where there is only one endogenous regressor.Furthermore,the lead term of the MSE expansion is the square of the lead term of the bias expansion,implying that the variance component of the MSE is of a lower order vis-a-vis the bias component in a scenario where the number of instruments ud is large relative to the value of the population analogue of the¯rst stage F-statistic.In order to tie our¯ndings in with the small sample IV literature,we note also that our formulae for the asymptotic bias and MSE,derived under the local-to-zero framework,correspond to the exact bias and MSE functions of the2SLS estimator,as derived by Richardson and Wu(1971),when a¯xed instrument/Gaussian model is assumed.Hence,with respect to bias and MSE,we are able to prove the conjecture put forth by Staiger and Stock(1997)that the limiting distribution of the2SLS estimator under the local-to-zero assumption is the same as the exact distribution of this estimator under the more restrictive assumptions of¯xed instruments and Gaussian errors.
This rest of the paper is organized as follows.Section2contains preliminaries,including the model,assumptions,and notation to be ud.Section3prents formulae for the asymptotic bias and
MSE of the IV estimator under weak identi¯cation,and discuss properties and implica-tions of the formulae.Section4develops bias and MSE approximations.Section5summarizes
various numerical calculations ud to asss the accuracy of our approximations.Conclusions and summarizing remarks are given in Section6.All proofs and technical details are contained in two appendices.
Before proceeding,we brie°y introduce some notation.We u the symbol\=)"to denote convergence in distribution,while\´"denotes equivalence in distribution.P X=X(X0X)¡1X0is the matrix which projects orthogonally onto the range space of X and M X=I¡P X.In addition, P(Z;X)=P X+M X Z(Z0M X Z)¡1Z0M X and M(Z;X)=M X¡M X Z(Z0M X Z)¡1Z0M X:
2Setup
Consider the simultaneous equations model(SEM):
y1=y2¯+X°+u;(1)
y2=Z¦+X©+v;(2)
where y1and y2are T£1vectors of obrvations on two endogenous variables,X is a T£k1 matrix of obrvations on k1exogenous variables included in the structural equation(1),Z is a T£k2matrix of obrvations on k2exogenous variables excluded from the structural equation, and u and v are T£1vectors of random disturbances2.Let u t and v t denote the t th component of the random vectors u and v,respectively;and let Z0t and X0t denote the t th row of the matrices Z and X,respectively.Additionally,let w t=(u t;v t)0and let Z t=(X0t;Z0t)0;assume that E(w t)=0; E(w t w0t)=§=µ¾uu¾uv¾uv¾vv¶;and EZ t w0t=0for all t;and assume that E(w t w0s)=0for all t=s;where t;s=1;:::;T:Following Staiger and Stock(1997),we formalize the notion of weak
instruments by modeling¦to be a parameter quence that is local to zero.In particular,we make the following assumption.
Assumption1:¦=¦T=C=p T,where C is a¯xed k
2£1vector.
Also,following Staiger and Stock(1997),we assume that the data generating process of the ex-ogenous variables,Z=(X;Z);and of the disturbances,(u;v);is such that the following moment convergence results hold.
2Although for notational simplicity we only study the ca with one endogenous explanatory variable in this paper, we do not e any reason why many of the qualitative conclusions reached here will not continue to hold in more general ttings.
爱无价Assumption 2:The following limits hold jointly:(i)(u 0u=T;u 0v=T;v 0v=T )p海边风景图片
!(¾uu ;¾uv ;¾vv );(ii)Z 0Z /T p !Q;and (iii)(T ¡1=2X 0u;T ¡1=2Z 0u;T ¡1=2X 0v;T ¡1=2Z 0v )=)(ÃXu ;ÃZu ;ÃXv;ÃZv );where
Q =E (Z t Z t )and where ô(Ã0Xu ;Ã0Zu ;Ã0Xv ;Ã0Zv )0is distributed N (0;(§-Q )):We consider IV estimation of the parameter ¯in equation (1)above,where the IV estima-
tor may not make u of all available instruments.De¯ne the IV estimator as:b ¯IV =(y 02
(P H ¡P X )y 2)¡1(y 02(P H ¡P X )y 1);where H =(Z 1;X )is a T £(k 21+k 1)matrix of instruments,and Z 1is a T £k 21submatrix of Z formed by column lection.It will prove convenient to partition Z as Z =(Z 1;Z 2),where Z 2is a T £k 22matrix of obrvations of the excluded exogenous variables not ud as instruments in estimation.Note that when Z 1=Z and H =[Z;X ](i.e.when all available instruments are ud),the IV estimator de¯ned above is equivalent to the 2SLS estimator.Addition-ally,partition ¦T ;T ¡12Z 0u;T ¡1
2Z 0v;ÃZu ;and ÃZv conformably with Z =(Z 1;Z 2)by writing ¦T =(¦01;T ;¦02;T
)0=(C 01=p T ;C 02=p T )0;T ¡12Z 0u =(T ¡12(Z 01u )0;T ¡12(Z 02u )0)0;T ¡12Z 0v =(T ¡12(Z 01v )0;T ¡12(Z 02v )0)0,ÃZu =(Ã0Z 1u ;Ã0Z 2u )0;and ÃZv =(Ã0Z 1v ;Ã0Z 2v )0;where from part (iii)of Assumption
2we have that (T ¡1(Z 01u )0;T ¡1(Z 02u )0;T ¡1(Z 01v )0;T ¡1(Z 02v )0)0)(Ã0Z 1u ;Ã0Z 2u ;Ã0Z 1v ;Ã0Z 2v )0:Fur-thermore,partition Q conformably with Z =(X;Z 1;Z 2)as Q =0@Q XX Q XZ 1Q XZ 2Q Z 1X Q Z 1Z 1Q Z 1Z 2Q Z 2X Q Z 2Z 1Q Z 2Z 21A :(3)Finally,de¯ne
-=µ-11-12-012-22¶=µQ Z 1Z 1¡Q Z 1X Q ¡1XX Q XZ 1Q Z 1Z 2¡Q Z 1X Q ¡1XX Q XZ 2Q Z 2Z 1¡Q Z 2X Q ¡1XX Q XZ 1Q Z 2Z 2¡Q Z 2X Q ¡1XX Q XZ 2
¶(4)and -1¤=(-11;-12):To ensure that the asymptotic bias and MSE of the IV estimator are
well-behaved,we make the following additional assumption.Assumption 3:sup T
E (j U T j 2+±)<1;for some ±>0;where U T =b ¯IV;T ¡¯0;b ¯IV;T denotes the IV estimator of ¯for a sample of size T ,and ¯0is the true value of ¯.Note that Assumption 3is su±cient for the uniform integrability of (b ¯
IV;T ¡¯0)2(e Billingsley (1968),pp.32).Under Assumption 3,lim T !1E (b ¯IV;T ¡¯0)=E (U )and lim T !1E (b ¯IV;T ¡¯0)2=E (U 2),where U is the limiting random variable of the quence f U T g who explicit form is given in Lemma A1in Appendix A.Hence,under Assumption 3,the asymptotic bias and MSE correspond
to the bias and MSE implied by the limiting distribution of b ¯
IV;T .Note also that for the special
ca where(u t;v t)0»i:i:d:N(0;§);k21¸4implies Assumption3,since it is well-known that the IV estimator of¯under Gaussianity has¯nite sample moments which exist up to and including the degree of apparent overidenti¯cation,as given by the order Sawa(1969)). Throughout this paper,we shall assume k21¸4so as to ensure that our results apply in the Gaussian ca.In addition,note that Assumption3rules out the limited information maximum likelihood(LIML)estimator in the Gaussian ca since it is well-known that the¯nite sample distribution of LIML in this ca has Cauchy-like tails so that no positive integer moment exists. (See Mariano and McDonald(1979)and Phillips(1984,1985)for various results documenting the non-existence of moments of the¯nite sample distribution of LIML.)Conquently,we do not consider the
LIML estimator in this paper.Since the primary motivation for adopting a local-to-zero framework in this paper is to try to obtain better approximations for the¯nite sample moments of estimators in the ca of weak identi¯cation,we shall focus attention here only on the IV estimator who¯nite sample distribution is known to have moments which exist up to and including the degree of overidenti¯cation in the benchmark Gaussian ca.
3Asymptotic Bias and MSE:Formulae and Properties
We begin with two theorems which give explicit analytical formulae for the asymptotic bias and MSE of the IV estimator in the ca with weak instruments.The theorems also characterize some of the properties of the bias and MSE functions.煮菜
Theorem3.1(Bias)Given the SEM described by equations(1)and(2),and under Assumptions 1,2,and3,the following results hold for k21¸4:
(a)
µk21¡1;k21;¹0¹¶;(5)
b b¯IV(¹0¹;k21)=¾1=2uu¾¡1=2vv½e¡¹0¹21F1
where b b¯IV(¹0¹;k21)=lim T!1E(b¯IV;T¡¯0)is the asymptotic bias function of the IV estimator which we write as a function of¹0¹=¾¡1vv C0-01¤-¡111-1¤C and k21;and where½=¾uv¾¡12uu¾¡12vv,¡(¢) denotes the gamma function,and1F1(¢;¢;¢)denotes the con°uent hypergeometric function.
(b)For k21¯xed,as¹0¹!1;b b¯IV(¹0¹;k21)!0:
(c)For¹0¹¯xed,as k21!1;b b¯IV(¹0¹;k21)!¾uv=¾vv=¾1=2uu¾¡1=2vv½:
