We introduce a novel parametrization of the correlation matrix. The reparametrization facilitates modeling of correlation and covariance matrices by an unrestricted vector, where positive definiteness is an innate property. This parametrization can be viewed as a generalization of Fisher's Z ‐transformation to higher dimensions and has a wide range of potential applications. An algorithm for reconstructing the unique n × n correlation matrix from any vector in <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:msup> <a:mrow> <a:mi mathvariant="double-struck">R</a:mi> </a:mrow> <a:mrow> <a:mi>n</a:mi> <a:mo stretchy="false">(</a:mo> <a:mi>n</a:mi> <a:mo>−</a:mo> <a:mn>1</a:mn> <a:mo stretchy="false">)</a:mo> <a:mo stretchy="false">/</a:mo> <a:mn>2</a:mn> </a:mrow> </a:msup> </a:math> is provided, and we derive its numerical complexity.

This paper develops inference methods for the iterated overidentified Generalized Method of Moments (GMM) estimator. We provide conditions for the existence of the iterated estimator and an asymptotic distribution theory, which allows for mild misspecification. Moment misspecification causes bias in conventional GMM variance estimators, which can lead to severely oversized hypothesis tests. We show how to consistently estimate the correct asymptotic variance matrix. Our simulation results show that our methods are properly sized under both correct specification and mild to moderate misspecification. We illustrate the method with an application to the model of Acemoglu, Johnson, Robinson, and Yared (2008).

This paper develops an asymptotic theory of inference for an unrestricted two-regime threshold autoregressive (TAR) model with an autoregressive unit root. We find that the asymptotic null distribution of Wald tests for a threshold are nonstandard and different from the stationary case, and suggest basing inference on a bootstrap approximation. We also study the asymptotic null distributions of tests for an autoregressive unit root, and find that they are nonstandard and dependent on the presence of a threshold effect. We propose both asymptotic and bootstrap-based tests. These tests and distribution theory allow for the joint consideration of nonlinearity (thresholds) and nonstationary (unit roots). Our limit theory is based on a new set of tools that combine unit root asymptotics with empirical process methods. We work with a particular two-parameter empirical process that converges weakly to a two-parameter Brownian motion. Our limit distributions involve stochastic integrals with respect to this two-parameter process. This theory is entirely new and may find applications in other contexts. We illustrate the methods with an application to the U.S. monthly unemployment rate. We find strong evidence of a threshold effect. The point estimates suggest that the threshold effect is in the short-run dynamics, rather than in the dominate root. While the conventional ADF test for a unit root is insignificant, our TAR unit root tests are arguably significant. The evidence is quite strong that the unemployment rate is not a unit root process, and there is considerable evidence that the series is a stationary TAR process.

There now exists a host of results concerning the revenue performance of various auction methods. This note delves deeper into auction markets by examining the effects on sellers' revenue of certain noncash means of payment. The basic result-that bidding mediums which include some contingent pricing feature generally yield the seller more revenue than do cash bids-is intriguing by itself and also points out the limitations of received theory. Since what follows builds on the independent-preferences framework, it is useful to first note the assumptions of that model and the major results pertaining to it. To model an auction in the independent-preferences tradition, one assumes that bidders have reservation values, Vi, that are known only privately and that can be depicted as being drawn independently from some distribution F( V). The most important result for this model is the revenue equivalence theorem: as given in John Riley and William Samuelson (1981), any auction involving risk-neutral bidders for which the following four conditions hold: (a) a buyer can make any bid above some minimum reserve price, (b) the buyer making the highest bid is awarded the object, (c) the auction rules are anonymous, and (d) there is a common equilibrium bidding strategy in which each buyer makes a bid bi, which is a strictly increasing function of his reservation value Vi, yields an expected revenue of

Given the state of affairs in auction theory -there is at least one model, for instance, to support any position one would care to take concerning the revenue of sealed-bid vs. open auctions-it should not come as a surprise that a fair amount of empirical work in auctions is underway. This paper reports the results of some recently completed research. I first discuss papers in which the predictions being tested derive directly from the pure theory of auctions, and then papers in which the predictions arise out of an application of auction theory to a related institution.

This essay examines the problem of inference within a rational expectations model from two perspectives: that of an econometrician and that of the economic agents within the model. The assumption of rational expectations has been and remains an important component to quantitative research. It endows economic decision makers with knowledge of the probability law implied by the economic model. As such, it is an equilibrium concept. Imposing rational expectations removed from consideration the need for separately specifying beliefs or subjective components of uncertainty. Thus, it simplified model specification and implied an array of testable implications that are different from those considered previously. It reframed policy analysis by questioning the effectiveness of policy levers that induce outcomes that differ systematically from individual beliefs.

We obtain a canonical representation for block matrices. The representation facilitates simple computation of the determinant, the matrix inverse, and other powers of a block matrix, as well as the matrix logarithm and the matrix exponential. These results are particularly useful for block covariance and block correlation matrices, where evaluation of the Gaussian log-likelihood and estimation are greatly simplified. We illustrate this with an empirical application using a large panel of daily asset returns. Moreover, the representation paves new ways to model and regularize large covariance/correlation matrices, test block structures in matrices, and estimate regressions with many variables.