We compare the powers of five tests of the coefficient on a single endogenous
regressor in instrumental variables regression. Following Moreira (2003), all tests are
implemented using critical values that depend on a statistic which is sufficient under the
null hypothesis for the (unknown) concentration parameter, so these conditional tests are
asymptotically valid under weak instrument asymptotics. Four of the tests are based on
k-class Wald statistics (two-stage least squares, LIML, Fuller’s (1977), and bias-adjusted
TSLS); the fifth is Moreira’s (2003) conditional likelihood ratio (CLR) test. The
heretofore unstudied conditional Wald tests are found to perform poorly, compared to the
CLR test: in many cases, the conditional Wald tests have almost no power against a wide
range of alternatives. Our analysis is facilitated by a new algorithm, presented here, for
the computation of the asymptotic conditional p-value of the CLR test.

This paper establishes the asymptotic distributions of the likelihood ratio (LR),
Anderson-Rubin (AR), and Lagrange multiplier (LM) test statistics under “many
weak IV asymptotics.” These asymptotics are relevant when the number of IVs is
large and the coefficients on the IVs are relatively small. The asymptotic results hold
under the null and under suitable alternatives. Hence, power comparisons can be
made.
Provided k3 /n
→ 0 as n → ∞, where n is the sample size and k is the number of
instruments, these tests have correct asymptotic size. This holds no matter how weak
the instruments are. Hence, the tests are robust to the strength of the instruments.
The power results show that the conditional LR test is more powerful asymptotically
than the AR and LM tests under many weak IV asymptotics.

This paper considers tests of the parameter on endogenous variables in an instru-
mental variables regression model. The focus is on determining tests that have some
optimal power properties. We start by considering a model with normally distrib-
uted errors and known error covariance matrix. We consider tests that are similar and
satisfy a natural rotational invariance condition. We determine a two-sided power
envelope for invariant similar tests. This allows us to assess and compare the power
properties of tests such as the conditional likelihood ratio (CLR), Lagrange multi-
plier, and Anderson-Rubin tests. We find that the CLR test is quite close to being
uniformly most powerful invariant among a class of two-sided tests.
The finite sample results of the paper are extended to the case of unknown error
covariance matrix and possibly non-normal errors via weak instrument asymptotics.
Strong instrument asymptotic results also are provided because we seek tests that
perform well under both weak and strong instruments.

This paper constructs tests for martingale time variation in regression coefficients in
the regression model yt = xt′βt + ut, where βt is k×1, and Σβ is the covariance matrix of
Δβt. Under the null there is no time variation, so Ho: Σβ = 0; under the alternative there is
time variation in r linear combinations of the coefficients, so Ha: rank(Σβ ) = r, where r
may be less than k. The Gaussian point optimal invariant test for this reduced rank
testing problem is derived, and the test’s asymptotic behavior is studied under local
alternatives. The paper also considers the analogous testing problem in the multivariate
local level model Zt = μt + at, where Zt is a k×1 vector, μt is a level process that is constant
under the null but is subject to reduced rank martingale variation under the alternative,
and at is an I(0) process. The test is used to investigate possible common trend variation
in the growth rate of per-capita GDP in France, Germany and Italy.

This paper considers VAR models incorporating many time series that interact through a
few dynamic factors. Several econometric issues are addressed including estimation of
the number of dynamic factors and tests for the factor restrictions imposed on the VAR.
Structural VAR identification based on timing restrictions, long run restrictions, and
restrictions on factor loadings are discussed and practical computational methods
suggested. Empirical analysis using U.S. data suggest several (7) dynamic factors,
rejection of the exact dynamic factor model but support for an approximate factor model,
and sensible results for a SVAR that identifies money policy shocks using timing
restrictions.

This paper reviews recent developments in methods for dealing with weak instru-
ments (IVs) in IV regression models. The focus is more on tests (and confidence
intervals derived from tests) than estimators.
The paper also presents new testing results under “many weak IV asymptotics,”
which are relevant when the number of IVs is large and the coefficients on the IVs
are relatively small. Asymptotic power envelopes for invariant tests are established.
Power comparisons of the conditional likelihood ratio (CLR), Anderson-Rubin, and
Lagrange multiplier tests are made. Numerical results show that the CLR test is on
the asymptotic power envelope. This holds no matter what the relative magnitude
of the IV strength to the number of IVs.

This paper provides a simple shrinkage representation that describes the
operational characteristics of various forecasting methods that are applicable when there
are a large number of orthogonal predictors (such as principal components). These
methods include pretest methods, Bayesian model averaging, empirical Bayes, and
bagging. We then compare these and other many-predictor forecasting methods in the
context of macroeconomic forecasting (real activity and inflation) using 131 monthly
predictors with monthly U.S. economic time series data, 1959:1 - 2003:12. The
theoretical shrinkage representations serve to inform our empirical comparison of these
forecasting methods.