Optimal Two-Sided Tests for Instrumental Variables Regression with Heteroskedastic and Autocorrelated Errors

Abstract

1 This paper expands upon and supersedes the corresponding sections of our working paper "Contributions to the Theory of Optimal Tests." We thank Benjamin Mills and Gustavo de Castro for outstanding research assistance, and are particularly indebted to Gustavo for suggesting and implementing the designs for power comparisons reported here. We thank Leandro Gorno, Patrik Guggenberger, Alexei Onatski, and Lucas Vilela for helpful comments; Jose Diogo Barbosa, Felipe Flores, and Leonardo Salim for suggestions on earlier drafts of this paper; and Isaiah Andrews and Jose Olea for productive discussions and sharing numerical codes. We gratefully acknowledge the research support of CNPq, FAPERJ, and NSF (via grant SES-0819761). arXiv:1505.06644v1 [math.ST] 25 May 2015Abstract This paper considers two-sided tests for the parameter of an endogenous variable in an instrumental variable (IV) model with heteroskedastic and autocorrelated errors. We develop the finite-sample theory of weighted-average power (WAP) tests with normal errors and a known long-run variance. We introduce two weights which are invariant to orthogonal transformations of the instruments; e.g., changing the order in which the instruments appear. While tests using the MM1 weight can be severely biased, optimal tests based on the MM2 weight are naturally two-sided when errors are homoskedastic.We propose two boundary conditions that yield two-sided tests whether errors are homoskedastic or not. The locally unbiased (LU) condition is related to the power around the null hypothesis and is a weaker requirement than unbiasedness. The strongly unbiased (SU) condition is more restrictive than LU, but the associated WAP tests are easier to implement. Several tests are SU in finite samples or asymptotically, including tests robust to weak IV (such as the Anderson-Rubin, score, conditional quasi-likelihood ratio, and I. Andrews' (2015) PI-CLC tests) and two-sided tests which are optimal when the sample size is large and instruments are strong.We refer to the WAP-SU tests based on our weights as MM1-SU and MM2-SU tests. Dropping the restrictive assumptions of normality and known variance, the theory is shown to remain valid at the cost of asymptotic approximations. The MM2-SU test is optimal under the strong IV asymptotics, and outperforms other existing tests under the weak IV asymptotics.

Conditional Inference With a Functional Nuisance Parameter

“…Given equations (S5) and (S6), however, standard arguments imply that the most powerful test against alternative μ rejects when (μ Σ(θ 0 θ 0 ) −1 g T (θ 0 )) 2 is large, and thus coincides with the test used to construct PE-2, which in the linear IV model also coincides with the power envelope for SU tests derived in Moreira and Moreira (2015).…”

“…for almost every d. Conditions (S5) and (S6) together are sufficient for the SU condition of Moreira and Moreira (2015).…”

“…The power envelope PE-2 likewise bounds the power of the conditional linear combination (CLC) tests studied by Andrews (2016), and (in the linear IV model) of the strongly unbiased (SU) tests studied by Moreira and Moreira (2015). This follows from the fact that all of these tests satisfy a version of the SU condition of Moreira and Moreira (2015).…”

“…As noted in the supplement to Andrews (2016), a sufficient condition for condition (S6) is that φ(g T (θ 0 ) D T ) = φ(−g T (θ 0 ) D T ), which holds for all the tests discussed above save for the SU tests of Moreira and Moreira (2015), which instead impose restriction (S6) by construction. Given equations (S5) and (S6), however, standard arguments imply that the most powerful test against alternative μ rejects when (μ Σ(θ 0 θ 0 ) −1 g T (θ 0 )) 2 is large, and thus coincides with the test used to construct PE-2, which in the linear IV model also coincides with the power envelope for SU tests derived in Moreira and Moreira (2015).…”

“…Yogo uses data from 11 different countries and considers multiple specifications, taking either the risk-free interest rate or an equity return to be the endogenous regressor. As noted by Moreira and Moreira (2015) the instruments are more strongly correlated with the risk-free rate than with the equity return, and there is a correspondingly more severe weak instruments problem when we use equity returns as the endogenous regressor. Since we are interested in behavior for a range of different identification scenarios we follow Andrews (2016) and calibrate our simulations to specifications that use the risk-free rate as the endogenous regressor.…”