General characterizations of valid confidence sets and tests in problems which involve locally almost unidentified (LAU) parameters are provided and applied to several econometric models. Two types of inference problems are studied: (1) inference about parameters which are not identifiable on certain subsets of the parameter space, and (2) inference about parameter transformations with singularities (discontinuities). When a LAU parameter or parametric function has an unbounded range, it is shown under general regularity conditions that any valid confidence set with level 1 \Gamma ff for this parameter should be unbounded with probability close to 1 \Gamma ff in the neighborhood of nonidentification subsets and should as well have a non-zero probability of being unbounded under any distribution compatible with the model: no valid confidence set which is bounded with probability one does exist. These properties hold even if "identifying restrictions" are imposed. Similar results also ob...

Posterior distributions in limited information analysis of the simultaneous equations model using the

This paper provides a first order asymptotic theory for generalized method of moments (GMM) estimators when the number of moment conditions is allowed to increase with the sample size and the moment conditions may be weak. Examples in which these asymptotics are relevant include instrumental variable (IV) estimation with many (possibly weak or uninformed) instruments and some panel data models that cover moderate time spans and have correspondingly large numbers of instruments. Under certain regularity conditions, the GMM estimators are shown to converge in probability but not necessarily to the true parameter, and conditions for consistent GMM estimation are given. A general framework for the GMM limit distribution theory is developed based on epiconvergence methods. Some illustrations are provided, including consistent GMM estimation of a panel model with time varying individual effects, consistent limited information maximum likelihood estimation as a continuously updated GMM estimator, and consistent IV structural estimation using large numbers of weak or irrelevant instruments. Some simulations are reported.

We investigate the finite sample properties of the two-pass cross-sectional regression (CSR) method-ology, which is popular for estimating risk premia and testing beta pricing models. We find that the finite sample distributions of the estimated risk premia differ significantly from their asymp-totic distributions. In particular, the risk premia estimates obtained from the second-pass CSR of average returns on estimated betas can be seriously biased even when the number of time series observations is reasonably large. In addition, the standard error of the estimated risk premia based on the asymptotic distribution overstates the actual standard error. We show that popular adjusted estimators in the literature have no finite integral moments and therefore cannot be used to correct the bias. We propose a new bias adjustment of the estimated zero-beta rate and risk premia and we show that the adjusted version has a smaller bias than the unadjusted version. In the empirical asset pricing literature, the popular two-pass cross-sectional regression (CSR) methodology developed by Black, Jensen, and Scholes (1972) and Fama and MacBeth (1973) is often used for estimating of risk premia and testing beta asset pricing models. Although there are many variations on this two-pass methodology, its basis setup always involves two steps. In the first

evaluation of top-order invariant polynomials and moments of ratio of quadratic forms in normal random variables. ” Hillier first became involved as a referee of that earlier paper. His contribution has been confined mainly to suggesting the generating function approach, simplifying some of the proofs, and contributing a few additional results. We are grateful to Plamen Koev, Peter Phillips, Serge Provost, Marko Riedel and two anonymous referees for helpful comments and suggestions. Kan gratefully acknowledges financial support from the National Bank Financial of Canada.

We provide results on properties of the IV estimator in the presence of weak instruments, beginning with the derivation of analytical formulae for the asymptotic bias (ABIAS) and mean squared error (AMSE), within the local-to-zero asymptotic framework of Staiger and Stock (1997). These results add to the results of Staiger and Stock (1997), who have provided an approximate ABIAS measure for the two-stage least squares (2SLS) estimator relative to that of the OLS estimator. In fact, with respect to ABIAS and AMSE, we are able to prove the conjecture put forth by Staiger and Stock (1997) that the limiting distribution of the 2SLS estimator under the local-to-zero assumption is the same as the exact distribution of this estimator under the more restrictive assumptions of xed instruments and Gaussian errors. We also obtain approximations for the ABIAS and AMSE formulae based on an asymptotic scheme; which, loosely speaking, requires the expectation of the first stage F-statistic to converge ...

1This paper was inspired by reading Nielsen (2009). We thank two referees and the CoEditor for helpful comments on the original version. Magdalinos thanks the Cowles Foundation, Yale University, for hospitality in the Fall of 2009 when the paper was …rst

Some exact distribution theory is developed for structural equation models with and without identities. The theory includes LIML, IV and OLS. We relate the new results to earlier studies in the literature, including the pioneering work of Bergstrom (1962). General IV exact distribution formulae for a structural equation model without an identity are shown to apply also to models with an identity by specializing along a certain asymptotic parameter sequence. Some of the new exact results are obtained by means of a uniform asymptotic expansion. An interesting consequence of the new theory is that the uniform asymptotic approximation provides the exact distribution of the OLS estimator in the model considered by Bergstrom (1962). This example appears to be the first instance in the statistical literature of a uniform approximation delivering an exact expression for a probability density.

In a simple model composed of a structural equation and identity, the finite sample distribution of the IV/LIML estimator is always bimodal and this is most apparent when the concentration parameter is small. Weak instrumentation is the energy that feeds the secondary mode and the coefficient in the structural identity provides a point of compression in the density that gives rise to it. The IV limit distribution can be normal, bimodal, or inverse normal depending on the behavior of the concentration parameter and the weakness of the instruments. The limit distribution of the OLS estimator is normal in all cases and has a much faster rate of convergence under very weak instrumentation. The IV estimator is therefore more resistant to the attractive effect of the identity than OLS. Some of these limit results differ from conventional weak instrument asymptotics, including convergence to a constant in very weak instrument cases and limit distributions that are inverse normal.

Two distinguished New Zealanders pioneered some of the foundations of modern econometrics. Alec Aitken, one of the most famous and well-documented mental arithmeticians of all time, contributed the matrix formulation and projection geometry of linear regression, generalized least squares (GLS) estimation, algorithms for Hodrick Prescott (HP) style data smoothing (six decades before their use in economics), and statistical estimation theory leading to the Cramér Rao bound. Rex Bergstrom constructed and estimated by limited information maximum likelihood (LIML) the largest empirical structural model in the early 1950s, opened up the field of exact distribution theory, developed cyclical growth models in economic theory, and spent nearly 40 years of his life developing the theory of continuous time econometric modeling and its empirical application. We provide an overview of their lives, discuss some of their accomplishments, and develop some new econometric theory that connects with their foundational work.