Curved exponential models have the property that the dimension of the minimal sufficient statistic is larger than the number of parameters in the model. Many econometric models share this feature. The first part of the paper shows that, in fact, econometric models with this property are necessarily curved exponential. A method for constructing an explicit set of minimal sufficient statistics, based on partial scores and likelihood ratios, is given. The difference in dimension between parameterand statistic and the curvature of these models have important consequences for inference. It is not the purpose of this paper to contribute significantly to the theory of curved exponential models, other than to show that the theory applies to many econometric models and to highlight some multivariate aspects. Using the methods developed in the first part, we show that demand systems, the single structural equation model, the seemingly unrelated regressions, and autoregressive models are all curved exponential models.

Second-order approximations to the variances of OLS and GLS estimators are compared when the covariance matrix is locally nonscalar. Using a result of Rothenberg, the comparison of OLS and GLS variances is shown to be asymptotically equivalent to a weighted mean square error comparison of the error covariance parameter estimators used in those two procedures. When there is only one covariance parameter, this comparison depends only on the noncentrality parameter of a classical hypothesis test for a scalar covariance matrix.

We examine the higher order asymptotic properties of semiparametric regression estimators that were obtained by the general MINPIN method described in Andrews (1989, Semiparametric Econometric Models: I. Estimation, Discussion paper 908, Cowles Foundation). We derive an order n−1 stochastic expansion and give a theorem justifying order n−1 distributional approximation of the Edgeworth type.

This article proposes frequentist multiple-equation least-squares averaging approaches for multistep forecasting with vector autoregressive (VAR) models. The proposed VAR forecast averaging methods are based on the multivariate Mallows model averaging (MMMA) and multivariate leave-h-out cross-validation averaging (MCVAh) criteria (with h denoting the forecast horizon), which are valid for iterative and direct multistep forecast averaging, respectively. Under the framework of stationary VAR processes of infinite order, we provide theoretical justifications by establishing asymptotic unbiasedness and asymptotic optimality of the proposed forecast averaging approaches. Specifically, MMMA exhibits asymptotic optimality for one-step-ahead forecast averaging, whereas for direct multistep forecast averaging, the asymptotically optimal combination weights are determined separately for each forecast horizon based on the MCVAh procedure. To present our methodology, we investigate the finite-sample behavior of the proposed averaging procedures under model misspecification via simulation experiments.

This paper introduces tests for the null of cointegration in the presence of I(1) and I(2) variables. These tests use residuals from Park's (1992, Econometrica 60,119–143) canonical cointegrating regression (CCR) and the leads-and-lags regression of Saikkonen (1991, Econometric Theory 9,1–21) and Stock and Watson (1993, Econometrica 61, 783–820). Asymptotic theory for CCR in the presence of I(1) and I(2) variables is also introduced. The distributions of the cointegration tests are nonstandard, and hence their percentiles are tabulated by using simulation. Monte Carlo simulation results to study the finite sample performance of the CCR estimates and the cointegration tests are also reported.

In this paper nearly unstable AR(p) processes (in other words, models with characteristic roots near the unit circle) are studied. Our main aim is to describe the asymptotic behavior of the least-squares estimators of the coefficients. A convergence result is presented for the general complex-valued case. The limit distribution is given by the help of some continuous time AR processes. We apply the results for real-valued nearly unstable AR(p) models. In this case the limit distribution can be identified with the maximum likelihood estimator of the coefficients of the corresponding continuous time AR processes.

This paper gives simple nonuniform bounds on the tail areas of the permutation distribution of the usual Student's t-statistic when the observations are independent with symmetric distributions. As opposed to uniform bounds, nonuniform bounds depend on the observed sample. It is shown that the nonuniform bounds proposed are always tighter than uniform exponential bounds previously suggested. The use of the bounds to perform nonparametric t-tests is discussed and numerical examples are presented. Further, the bounds are extended to t-tests in the context of a simple linear regression.

The paper considers different versions of the Lagrange multiplier (LM) tests for autocorrelation and/or for conditional heteroskedasticity. These versions differ in terms of the residuals, and of the functions of the residuals, used to build the tests. In particular, we compare ordinary least squares versus least absolute deviation (LAD) residuals, and we compare squared residuals versus their absolute value. We show that the LM tests based on LAD residuals are asymptotically distributed as a χ2 and that these tests are robust to nonnormality. The Monte Carlo study provides evidence in favor of the LAD residuals, and of the absolute value of the LAD residuals, to build the LM tests here discussed.

This paper addresses the problem of estimating vector autoregressive models. An approach to handling nonstationary (integrated) time series is briefly discussed, but the main emphasis is upon the estimation of autoregressive approximations to stationary processes. Three alternative estimators are considered–the Yule-Walker, least-squares, and Burg-type estimates–and a complete analysis of their asymptotic properties in the stationary case is given. The results obtained, when placed together with those found elsewhere in the literature, lead to the direct recommendation that the less familiar Burg-type estimator should be used in practice when modeling stationary series. This is particularly so when the underlying objective of the analysis is to investigate the interrelationships between variables of interest via impulse response functions and dynamic multipliers.

This article develops new model selection methods for forecasting panel data using a set of least squares (LS) vector autoregressions. Model selection is based on minimizing the estimated quadratic forecast risk among candidate models. We provide conditions under which the selection criterion is asymptotically efficient in the sense of Shibata (1980) as n (cross sections) and T (time series) approach infinity. Relative to extant selection criteria, this criterion places a heavier penalty on model dimensionality in order to account for the effects of parameterized forms of cross sectional heterogeneity (such as fixed effects) on forecast loss. We also extend the analysis to bias-corrected least squares, showing that significant reductions in forecast risk can be achieved.

New methods are developed for identifying, estimating, and performing inference with nonstationary time series that have autoregressive roots near unity. The approach subsumes unit-root (UR), local unit-root (LUR), mildly integrated (MI), and mildly explosive (ME) specifications in the new model formulation. It is shown how a new parameterization involving a localizing rate sequence that characterizes departures from unity can be consistently estimated in all cases. Simple pivotal limit distributions that enable valid inference about the form and degree of nonstationarity apply for MI and ME specifications and new limit theory holds in UR and LUR cases. Normalizing and variance stabilizing properties of the new parameterization are explored. Simulations are reported that reveal some of the advantages of this alternative formulation of nonstationary time series. A housing market application of the methods is conducted that distinguishes the differing forms of house price behavior in Australian state capital cities over the past decade.

Theoretical results of frequentist model averaging mainly focus on asymptotic optimality and asymptotic distribution of the model averaging estimator. However, even for basic least squares model averaging, many theoretical problems have not been well addressed yet. This article discusses asymptotic properties of a class of least squares model averaging methods with nested candidate models that includes the Mallows model averaging (MMA) of Hansen (2007, Econometrica 75, 1175–1189) as a special case. Two scenarios are considered: (i) all candidate models are under-fitted; and (ii) the true model is included in the candidate models. We find that in the first scenario, the least squares model averaging method asymptotically assigns weight one to the largest candidate model and the resulting model averaging estimator is asymptotically normal. In the second scenario with a slightly special weight space, if the penalty factor in the weight selection criterion is diverging with certain order, the model averaging estimator is asymptotically optimal by putting weight one to the true model. However, MMA with fixed model dimensions is not asymptotically optimal since it puts nonnegligible weights to over-fitted models. The theoretical results are clearly summarized with their restrictions, and some critical implications are discussed. Monte Carlo simulations confirm our theoretical results.

Positive definiteness of income effect matrices provides a sufficient condition for the law of demand to hold. Given cross section household expenditure data, empirical evidence for the law of demand can be obtained by estimating such matrices. Härdle, Hildenbrand, and Jerison used the bootstrap method to simulate the distribution of the smallest eigenvalue of random matrices and to test their positive definiteness. Here, theoretical aspects of this bootstrap test of positive definiteness are considered. The asymptotic distribution of the smallest eigenvalue , of the matrix estimate is obtained. This theory applies generally to symmetric, asymptotically normal random matrices. A bootstrap approximation to the distribution of is shown to converge in probability to the asymptotic distribution of . The bootstrap test is illustrated using British family expenditure survey data.

We extend the ${\cal M}$ class of unit root tests introduced by Stock (1999, Cointegration, Causality and Forecasting. A Festschrift in Honour of Clive W.J. Granger. Oxford University Press), Perron and Ng (1996, Review of Economic Studies 63, 435–463) and Ng and Perron (2001, Econometrica 69, 1519–1554) to the seasonal case, thereby developing semi-parametric alternatives to the regression-based augmented seasonal unit root tests of Hylleberg, Engle, Granger, and Yoo (1990, Journal of Econometrics 44, 215–238). The success of this class of unit root tests to deliver good finite sample size control even in the most problematic (near-cancellation) case where the shocks contain a strong negative moving average component is shown to carry over to the seasonal case as is the superior size/power trade-off offered by these tests relative to other available tests.

In this paper, we provide a Doob-style consistency theorem for stationary models. Many applications involving Bayesian inference deal with non independent and identically distributed data, in particular, with stationary data. However, for such models, there is still a theoretical gap to be filled regarding the asymptotic properties of Bayesian procedures. The primary goal to be achieved is establishing consistency of the sequence of posterior distributions. Here we provide an answer to the problem. Bayesian methods have recently gained growing popularity in economic modeling, thus implying the timeliness of the present paper. Indeed, we secure Bayesian procedures against possible inconsistencies. No results of such a generality are known up to now.The authors are grateful for the comments and suggestions of two referees. Antonio Lijoi and Igor Prünster were supported by the Italian Ministry of University and Research, grants 2006134525 and 2006133449, respectively. The research of Stephen G. Walker was funded by an EPSRC Advanced Research Fellowship.

Relevant sample quantities such as the sample autocorrelation function and extremes contain useful information about autoregressive time series with heteroskedastic errors. As these quantities usually depend on the tail index of the underlying heteroskedastic time series, estimating the tail index becomes an important task. Since the tail index of such a model is determined by a moment equation, one can estimate the underlying tail index by solving the sample moment equation with the unknown parameters being replaced by their quasi-maximum likelihood estimates. To construct a confidence interval for the tail index, one needs to estimate the complicated asymptotic variance of the tail index estimator, however. In this paper the asymptotic normality of the tail index estimator is first derived, and a profile empirical likelihood method to construct a confidence interval for the tail index is then proposed. A simulation study shows that the proposed empirical likelihood method works better than the bootstrap method in terms of coverage accuracy, especially when the process is nearly nonstationary.