| Achoimre: | Power curves of the Conditional Likelihood Ratio (CLR) and related tests for testing
H0:β = β0 in linear models with a single endogenous variable, y = xβ+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, Σ, constant, the other instead keeps the variance matrix of the reduced-form
and first-stage errors, Ω, constant. The values of Σ govern the endogeneity features
of the model. The fixed-Ω design changes these endogeneity features with changing
values of β 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 β and the correlation of
the structural and first-stage errors, ρuv, have the same sign. At larger values of |β|,
the fixed-Ω design implicitly selects values for Σ where the power of the CLR test is
high. We further show that the Likelihood Ratio statistic is identical to the t0(βb
L)
2
statistic as proposed by Mills et al. (2014), where βb
L is the Liml estimator. In fixedΣ design Monte Carlo simulations, we find that Liml- and Fuller-based conditional
Wald tests and the Fuller-based conditional t
2
0
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 β and ρuv have the same sign.
We show that in the fixed-Ω design, setting β0 = 0 and the diagonal elements of Ω
equal to 1 is not without loss of generality, unlike in the fixed-Σ design.
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