Template-type: ReDIF-Paper 1.0
Author-Name: Nicolas Van de Sijpe
Author-Workplace-Name: Dept. of Economics, University of Sheffield,
Author-Email: n.vandesijpe@sheffield.ac.uk.
Author-Name: Frank Windmeijer
Author-Workplace-Name: Dept. of Statistics and Nuffield College, University of Oxford
Author-Email: frank.windmeijer@stats.ox.ac.uk  
Title: On the Power of the Conditional Likelihood Ratio and Related Tests for Weak-Instrument Robust Inference
Abstract: Power curves of the Conditional Likelihood Ratio ($CLR$) and related tests for testing $H_{0}\beta=\beta_{0}$
in linear models with a single endogenous variable, $y=x\beta+u$, estimated using potentially weak instrumental variables have been
presented for two different designs. One design keeps the variance matrix of the structural and first-stage errors, $\Sigma$, constant,
the other instead keeps the variance matrix of the reduced-form and first-stage errors, $\Omega$, constant. The values of $\Sigma$
govern the endogeneity features of the model. The fixed-$\Omega$ design changes these endogeneity features with changing values of
$\beta$ in a way that makes it less suitable for an analysis of the behaviour of the tests in low to moderate endogeneity settings, or
when $\beta$ and the correlation of the structural and first-stage errors, $\rho_{uv}$, have the same sign. At larger values of $\left|\beta\right|$,
the fixed-$\Omega$ design implicitly selects values for $\Sigma$ where the power of the $CLR$ test is high. We show that the Likelihood
Ratio statistic is identical to the $t_{0}(\widehat{\beta}_{L})^{2}$ statistic as proposed by Mills, Moreira and Vilela (2014), where
$\widehat{\beta}_{L}$ is the LIML estimator. In fixed-$\Sigma$ design Monte Carlo simulations, we find that LIML- and Fuller-based conditional
Wald tests and the Fuller-based conditional $t_{0}^{2}$ test are more powerful than the $CLR$ test when the degree of endogeneity
is low to moderate. The conditional Wald tests are further the most powerful of these tests when $\beta$ and $\rho_{uv}$ have the same
sign. We show that in the fixed-$\Omega$ design, setting $\beta_{0}=0$ and the diagonal elements of $\Omega$ equal to $1$ is not without
loss of generality, unlike in the fixed-$\Sigma$ design.
JEL codes: C12, C26
Length: 39 pages
Creation-Date: 2021-07-21
Number: 2020-W09
File-URL: http://www.nuffield.ox.ac.uk/economics/Papers/2020/2020W09aCLR260721.pdf
File-Format: application/pdf
Handle: RePEc:nuf:econwp:2009