Bootstrap Validity for the Score Test When Instruments May Be Weak

Journal of Econometrics 149 (2009) 52–64 Contents lists available at ScienceDirect Journal of Econometrics journal homepage: www.elsevier.com/locate/jeconom Bootstrap validity for the score test when instruments may be weak Marcelo J. Moreira a,b,∗, Jack R. Porter c, Gustavo A. Suarez d a Columbia University, United States b FGV/EPGE, Brazil c University of Wisconsin, United States d Federal Reserve Board, United States article info a b s t r a c t Article history: It is well-known that size adjustments based on bootstrapping the t-statistic perform poorly when Available online 5 November 2008 instruments are weakly correlated with the endogenous explanatory variable. In this paper, we provide a theoretical proof that guarantees the validity of the bootstrap for the score statistic. This theory does not JEL classification: follow from standard results, since the score statistic is not a smooth function of sample means and some C12 parameters are not consistently estimable when the instruments are uncorrelated with the explanatory C31 variable. Keywords: ' 2008 Elsevier B.V. All rights reserved. Bootstrap t-statistic Score statistic Identification Non-regular case Edgeworth expansion Instrumental variable regression 1. Introduction estimators. Hence, the empirical distribution function of the residuals may differ substantially from their true cumulative Inference in the linear simultaneous equations model with distribution function, which runs counter to the usual argument weak instruments has recently received considerable attention for bootstrap success. Second, the score statistic is not a smooth in the econometrics literature. It is now well understood that function of sample means. In many known non-regular cases1 standard first-order asymptotic theory breaks down when the the usual bootstrap method fails, even in the first-order. Familiar instruments are weakly correlated with the endogenous regressor; cases from the statistics and econometrics literature include cf., Bound et al. (1995), Dufour (1997), Nelson and Startz (1990), estimation on the boundary of the parameter space (Shao, 1994; Staiger and Stock (1997), and Wang and Zivot (1998). It is then Andrews, 2000) and estimating a non-differentiable function of the natural to apply the bootstrap to decrease size distortions of the population mean. Wald statistic (also known as the t-statistic), since the bootstrap is Commonly used fixes for bootstrap failure due to nonregularity valid under some regularity conditions. However, these conditions, are to use the m out of n bootstrap or subsampling. However, which rely on the statistics being smooth functions of sample these methods have two limitations. First, in practice they give moments and the parameters being consistently estimable, break down for the Wald statistic in the weak-instrument case. In fact, quite different results for different choices of the bootstrap sample the bootstrap does not seem to perform well in decreasing the size (or subsample) size m. Second, they do not provide asymptotic distortions of the Wald statistic; cf., Horowitz (2001). refinements in the regular case. For instance, in the non- In this paper, we show that it is valid to bootstrap the score differentiable example above, the function may be differentiable statistic even in the weak-instrument case. Although the score at some values of the population mean and non-differentiable is well-behaved with weak instruments, showing the validity at other values. Then, at the differentiable values, the statistic is of the bootstrap in the unidentified case has several potential typically regular and the usual bootstrap is not only valid but pitfalls. First, the bootstrap replaces parameters with inconsistent 1 ∗ A statistic is said to be regular if, when written as function of sample moments, Corresponding author. Tel.: +1 212 854 3680; fax: +1 212 854 8059. the first derivative of this function evaluated at the population mean exists and is E-mail address: [email protected] (M.J. Moreira). different from zero. 0304-4076/$ – see front matter ' 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jeconom.2008.10.008 M.J. Moreira et al. / Journal of Econometrics 149 (2009) 52–64 53 D 0 −1 0 O 2 DT − U T also provides second-order improvements. Hence, there is a trade- where bβ2SLS .y2NZ y2/ y2NZ y1 and σu 1; bβ2SLS Ω 1; off between robustness (m out of n bootstrap or subsampling) 0 −bβ2SLS U . It is now well understood that the Wald statistic has and refinements (the usual bootstrap). Lastly, subsampling does important size distortions when the instruments may be weak. In not provide a general method of controlling size uniformly in particular, under the weak-instrument asymptotics of Staiger and cases where the bootstrap fails (Andrews and Guggenberger, Stock (1997), the limiting distribution of the Wald statistic is not forthcoming). standard normal. An alternative statistic is the score (LM) used by In this paper, we find that weak instruments are not, in Kleibergen (2002) and Moreira (2002): general, the cause of bootstrap failure. Although parameters are p 0 0 not consistently estimable when instruments are weak and the LM D S T = T T ; (3) score statistic is not differentiable, we show that the re-centered D 0 −1=2 0 · 0 −1=2 D 0 −1=2 0 −1 where S .Z Z/ Z Yb0 .b0Ωb0/ and T .Z Z/ Z Y Ω residual bootstrap for the score is valid regardless of instrument · 0 −1 −1=2 a0 .a0Ω a0/ . The (two-sided) score test rejects the null if the strength. In light of the recent negative results on the bootstrap, it is LM2 statistic is larger than the 1 − α quantile of the chi-square- notable that the bootstrap can still work in some non-regular cases. one distribution. This test is similar if the errors are normal with Still, we additionally find that the higher-order improvements known variance Ω, since the LM statistic is pivotal. With unknown provided by the bootstrapped score statistic when instruments are error distribution, the score test is no longer similar. However, un- strong do not extend to the case of weak instruments. like the Wald test, the score test is asymptotically similar under The remainder of this paper is organized as follows. In Section 2, both weak-instrument and standard asymptotics. we present the model and establish some notation. In Section 3, we In practice, the covariance matrix Ω is typically unknown, so summarize some folk theorems showing the size improvements 0 we replace it with the consistent estimator Ωe D Y MZ Y =n: based on the bootstrap for the Wald and score tests under D 0 −1=2 0 · 0 −1=2 standard asymptotics. In Section 4, we present the main results. eS .Z Z/ Z Yb0 .b0Ωeb0/ ; We establish the validity of the bootstrap for the score statistic, and D 0 −1=2 0 −1 · 0 −1 −1=2 eT .Z Z/ Z Y Ωe a0 .a0Ωe a0/ ; show that the bootstrap will not in general provide second-order 0 p 0 improvements in the unidentified case. In Section 5, we present LMf D eS eT = eT eT : Monte Carlo simulations that suggest that the bootstrap methods O 2 Q 2 DT − U T − U0 For the Wald statistic, replace σu by σu 1; bβ2SLS Ωe 1; bβ2SLS may lead to improvements, although in general they do not lead to to obtain We. Below we present results for We and LMf although higher-order adjustments in the weak-instrument case. Section 6 analogous results for the known covariance case are also similarly concludes. In Appendix A, we provide all proofs pertaining to the available. score statistic. In Appendix B, we provide some additional useful results and extensions. 3. Preliminary results 2. The model In this section, we summarize some folk theorems for the strong-instrument case. Some of the results are already known, The structural equation of interest is and those that are new follow from standard results. The results in this section provide a foundation for the weak-instrument results y D y β C u; (1) 1 2 to be presented in Section 4. where y1 and y2 are n × 1 vectors of observations on two For any symmetric matrix A, let vech.A/ denote the column endogenous variables, u is an n×1 unobserved disturbance vector, vector containing the column by column vectorization of the non- and β is an unknown scalar parameter. This equation is assumed redundant elements of A. The test statistics given in the previous to be part of a larger linear simultaneous equations model, section can be written as functions of 0 which implies that y2 is correlated with u. The complete system 0 0 0 0 Rn D vech Y ; Z Y ; Z contains exogenous variables that can be used as instruments for n n n n conducting inference on β. Specifically, it is assumed that the D 0 0 0 0 0 f1 Yn; Zn ;:::; f` Yn; Zn ; reduced form for Y D [y1; y2] can be written as where fi, i D 1; : : : ; `, ` D .k C 2/.k C 3/ =2, are elements of the 0 y D Zπβ C v (2) 0 0 0 0 1 1 matrix Yn; Zn Yn; Zn . Both We and LMf statistics can be written y2 D Zπ C v2; in the form p where Z is an n × k matrix of exogenous variables having full n H Rn − H .µ/ ; (4) × column rank k with probability one (w.p.1) and π is a k 1 vector. where µ D E .Rn/. The n rows of Z are i.i.d., and F is the distribution of each row Let k·k be the Euclidean norm and k·k1 the supremum norm.

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