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 Curves of the Conditional Likelihood Ratio and Related Tests for Instrumental Variables Regression with Weak Instruments
Abstract: We show that the Likelihood Ratio (LR) statistic for testing the value of the coefficient $\beta$ in a linear instrumental variables model with a single endogenous variable
is identical to the $t_0(\hat{\beta}_L)^2$ statistic as proposed by Mills, Moreira, and Vilela (2014),
where $\hat{\beta}_L$ is the LIML estimator. This implies the equivalence of their conditional
versions that are robust to weak instruments. From this result, properties of the
power of the Conditional LR (CLR) test can be understood; in particular the asymmetric nature of the power curve as a function of the true value of $\beta$ when testing
H0: $\beta =\beta_0$ for fixed $\beta_0$, when the instruments are weak and the variance matrix
of the structural and first-stage errors is held constant. Power curves of the CLR
and related tests have often been presented for a design where instead the variance
matrix of the reduced-form and first-stage errors has been held constant. This latter
design changes the endogeneity features at each value of $\beta$ and results in a power
curve that is close to the points with maximum power in the design with fixed variance of the structural and first-stage errors. As the results for the design with fixed
variance of the structural and first-stage errors are informative for the behaviour of
the test-based confidence intervals, it seems more natural to consider this design.
We find that LIML- and Fuller-based conditional Wald and conditional $t_0({\hat{\beta}_{Full}})^2$
tests, which are not unbiased tests, are more powerful than the CLR test when the
degree of endogeneity is low to moderate.
JEL codes: C12, C26
Length: 22 pages
Creation-Date: 2020-08-04
Number: 2020-W09
File-URL: https://www.nuffield.ox.ac.uk/economics/Papers/2020/2020W09_CLR030820.pdf
File-Format: application/pdf
Handle: RePEc:nuf:econwp:2009