On bootstrap validity for specification tests with weak instruments

Abstract

This paper investigates the asymptotic validity of the bootstrap for Durbin-WuHausman (DWH) specification tests when instrumental variables (IVs) may be arbitrary weak. It is shown that under strong identification, the bootstrap offers a better approximation than the usual asymptotic χ 2 distributions. However, the bootstrap provides only a first-order approximation when instruments are weak. These resultsshow unlike the Wald-statistic based on a k-class type estimator (Moreira et al., 2009), the bootstrap is valid even for the Wald-type of DWH statistics in the presence of weak instruments.

Uniform Inference after Pretesting for Exogeneity

“…It is therefore unclear whether OLS is not seriously biased in these data. In particular, as shown by Guggenberger (2010a) and Doko Tchatoka andDufour (2018, 2020), the Hausman test is not able to reject the null hypothesis of exogeneity in situations where there is only a small degree of endogeneity: for sequences of correlations between the structural and reduced form errors that are local to zero of order n − 1 2 (i.e., local endogeneity), where n is the sample size, the Hausman pretest statistic has a noncentral chi-squared limiting distribution, and its noncentrality parameter is small when IVs strength is not high. Therefore, the pretest has low power and as a result, OLS based inference is selected in the second stage with high probability.…”

“…In his simulations based upon the published regressions (Table XIV), the rejection frequencies can be as low as 0.237 and 0.386 for 1% and 5% significance levels, respectively, for asymptotic Hausman tests, and even as low as 0.105 and 0.208, respectively, for bootstrap Hausman tests. However, Young (2019)'s finding from the AEA data that OLS estimates seem to be not very different from 2SLS estimates may be attributed to the fact that the used IVs are not strong (2SLS is biased towards OLS under weak IVs), and Hausman-type tests also have low power in this case (e.g., see Doko Tchatoka andDufour (2018, 2020)). It is therefore unclear whether OLS is not seriously biased in these data.…”

Testing endogeneity with high dimensional covariates

“…They state that once the distribution of characteristics of federal circuit court judges in a given circuit-year is controlled for, "the realized characteristics of the randomly assigned threejudge panel should be unrelated to other factors besides judicial decisions that may be related to economic outcomes" (page 2405). More broadly, in empirical practice, adding covariates to make IVs more plausibly valid is commonplace; see Card (1999), Cawley et al (2013), and Kosec (2014) for examples as well as review papers in epidemiology and causal inference by Hernán and Robins (2006) and Baiocchi et al (2014). behavior of the DWH test under the null depends on the variance estimate (Doko Tchatoka, 2015;Nakamura and Nakamura, 1981;Staiger and Stock, 1997). Other works study the behavior of the DWH test under different strengths of instruments and/or weak instrument asymptotics (Hahn et al, 2011;Staiger and Stock, 1997) and under a two-stage testing scheme (Guggenberger, 2010).…”

“…behavior of the DWH test under the null depends on the variance estimate (Doko Tchatoka, 2015;Nakamura and Nakamura, 1981;Staiger and Stock, 1997). Other works study the behavior of the DWH test under different strengths of instruments and/or weak instrument asymptotics (Hahn et al, 2011;Staiger and Stock, 1997) and under a two-stage testing scheme (Guggenberger, 2010).…”

Size-Corrected Wild Bootstrap Tests after Pretesting for Exogeneity with Heteroskedastic or Clustered Data

“…However, Young (2022)'s finding from the AEA data that OLS estimates seem to be not very different from 2SLS estimates may be attributed to the fact that the used IVs are not strong so that 2SLS may be biased towards OLS, and Hausman-type tests also have low power in this case [e.g., see Doko Tchatoka andDufour (2018, 2020)]. In particular, as shown by Guggenberger (2010a), the Hausman test is not able to reject the null hypothesis of exogeneity in situations where there is only a small degree of endogeneity, i.e., local endogeneity.…”