Consider a linear regression model y1 = x1β + u1, where the u1'S afe weakly dependent random variables, the x1's are known design nonrandom variables, and β is an unknown parameter. We define an M-estimator βn of) β corresponding to a smooth score function. Then, the second-order Edgeworth expansion for βn is derived. Here we do not assume the normality of (u1), and (u1) includes the usual ARMA processes. Second, we give the second-order Edgeworth expansion for a transformation T(βn) of βn. Then, a sufficient condition for T to extinguish the second-order terms is given. The results are applicable to many statistical problems.

This paper sketches the history of how Bayesian inference was adopted and utilized in econometrics during its first 20 years. It focuses on the causes of the Bayesian movement, the ways in which Bayesian inference was applied, the problems that the application was intended to solve, and the results achieved. It shows that Bayesian research has largely followed mainstream econometric development as far as the major econometric ideas and methods are concerned and that Bayesian reformulation of mainstream econometrics has nevertheless helped in deepening econometricians' understanding of many modeling problems by presenting them from a different angle.

It is well-known that conditional heteroskedasticity thickens the tails of the unconditional distribution of an error term relative to its conditional distribution. To what extent do imperfect forecasts of the conditional variance undo this tail thickening? This note considers the effect of changing the quality of the information embodied in a forecast of a conditional variance. Adding noise of a certain form thickens the tails of the normalized errors, but decreasing the amount of information used in the forecast may or may not thicken the tails. We also explore the relation between tail thickness and various notions of “optimal” volatility forecasts. The relationship is surprisingly complicated.

The exponentially tilted family of densities is used to discuss the relationship between the encompassing principle and the M-test or conditional moment testing principle. It is shown that the two principles are capable of generating the same test statistics and in this sense equivalent. However, there are differences in motivation and emphasis underlying the principles that are important in econometric modeling. In addition, parsimonious encompassing interpretations for test statistics, including those of Cox (1961, in Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability I, pp. 105-123; 1962, Journal of the Royal Statistical Society, Series B 24, 406-424) and Atkinson (1970, Journal of the Royal Statistical Society, Series B 32, 323-344) are provided.

We derive the limit distribution of the number of crossings of a level by a random walk with continuously distributed increments, using a Brownian motion local time approximation. This complements the well-known result for the random walk on the integers. Use of the frequency of level crossings to test for a unit root is examined.

This paper establishes stochastic equicontinuity for classes of mixingales. Attention is restricted to Lipschitz-continuous parametric functions. Unlike some other empirical process theory for dependent data, our results do not require bounded functions, stationary processes, or restrictive dependence conditions. Applications are given to martingale difference arrays, strong mixing arrays, and near-epoch dependent arrays.

Said and Dickey (1984, Biometrika 71, 599–608) and Phillips and Perron (1988, Biometrika 75, 335–346) have derived unit root tests that have asymptotic distributions free of nuisance parameters under very general maintained models. Under models as general as those assumed by these authors, the size of the unit root test procedures will converge to one, not the size under the asymptotic distribution. Solving this problem requires restricting attention to a model that is small, in a topological sense, relative to the original. Sufficient conditions for solving the asymptotic size problem yield some suggestions for improving finite-sample size performance of standard tests.

In this paper we derive the asymptotic distribution of the least absolute deviations (LAD) estimator of the autoregressive parameter under the unit root hypothesis, when the errors are assumed to have finite variances, and present LAD-based unit root tests, which, under heavy-tailed errors, are expected to be more powerful than tests based on least squares. The limiting distribution of the LAD estimator is that of a functional of a bivariate Brownian motion, similar to those encountered in cointegrating regressions. By appropriately correcting for serial correlation and other distributional parameters, the test statistics introduced here are found to have either conditional or unconditional normal limiting distributions. The results of the paper complement similar ones obtained by Knight (1991, Canadian Journal of Statistics 17, 261-278) for infinite variance errors. A simulation study is conducted to investigate the finite sample properties of our tests.

Recent advances in the application of game theory to the study of auctions have spawned a growing empirical literature involving both experimental and field data. In this paper, we focus on four different mechanisms (the Dutch, English, first-price sealed-bid, and Vickrey auctions) within one of the most commonly used theoretical models (the independent private values paradigm) to investigate issues of identification, estimation, and testing in parametric structural econometric models of auctions.

This paper establishes conditions for the nonparametric identifiability of the mixed proportional hazards model with time-varying coefficients. Unlike the mixed proportional hazards model, a regressor with two distinct values is not sufficient to identify this model. An unbounded regressor, however, is sufficient for identification.

We derive the exact discrete model and the Gaussian likelihood function of a first-order system of linear stochastic differential equations driven by an observable vector of stochastic trends and a vector of stationary innovations.

The usual standard errors for the regression coefficients in a seemingly unrelated regression model have a substantial downward bias. Bootstrapping the standard errors does not seem to improve inferences. In this paper, Monte Carlo evidence is reported which indicates that bootstrapping can result in substantially better inferences when applied to t-ratios rather than to standard errors.

Recently, Arcones and Giné (1992, pp. 13–47, in R. LePage & L. Billard [eds.], Exploring the Limits of Bootstrap, New York: Wiley) established that the bootstrap distribution of the M-estimator converges weakly to the limit distribution of the estimator in probability. In contrast, Brown and Newey (1992, Bootstrapping for GMM, Seminar note) discovered that the bootstrap distribution of the GMM overidentification test statistic does not converge weakly to the x2 distribution. In this paper, it is shown that the bootstrap distribution of the GMM estimator converges weakly to the limit distribution of the estimator in probability. Asymptotic coverage probabilities of the confidence intervals based on the bootstrap percentile method are thus equal to their nominal coverage probability.

In “Modeling Stock Prices without Knowing How to Induce Stationarity” (1994, Econometric Theory 10, 701–719), we used posterior-odds calculations to evaluate restrictions imposed by a present-value model of stock prices across the equations of a VAR representation of stock prices and dividends. The results we reported are tainted by the omission of two factors: the Jacobians induced by the mapping of our priors over VAR parameters β into the restricted sample spaces relevant under hypotheses H2-H4 (hence, tainting our calculations of p(Hi|y,X) in (22) for i = 2–4), and an integrating constant needed in calculating the unrestricted probability p(Hi|y,X) in (22). Table 1 reports our revised calculations, which differ substantively from those reported previously.

A class of univariate fractional ARIMA models with a continuous time parameter is developed for the purpose of modeling long-memory time series. The spectral density of discretely observed data is derived for both point observations (stock variables) and integral observations (flow variables). A frequency domain maximum likelihood method is proposed for estimating the longmemory parameter and is shown to be consistent and asymptotically normally distributed, and some issues associated with the computation of the spectral density are explored.