The presence of permanent volatility shifts in key macroeconomic and financial variables in developed economies appears to be relatively common. Conventional unit root tests are unreliable in the presence of such behavior, having nonpivotal asymptotic null distributions. In this paper we propose a bootstrap approach to unit root testing that is valid in the presence of a wide class of permanent variance changes that includes single and multiple (abrupt and smooth transition) volatility change processes as special cases. We make use of the so-called wild bootstrap principle, which preserves the heteroskedasticity present in the original shocks. Our proposed method does not require the practitioner to specify any parametric model for the volatility process. Numerical evidence suggests that the bootstrap tests perform well in finite samples against a range of nonstationary volatility processes.We thank two anonymous referees, Paulo Rodrigues, Peter Phillips, and seminar participants at the URCT conference held in Faro, Portugal, September 29 to October 1, 2005, for helpful comments on previous versions of this paper.

Based on an idea of Granger (1986, Oxford Bulletin of Economics and Statistics 48, 213–228), we analyze a new vector autoregressive model defined from the fractional lag operator 1 − (1 − L)d. We first derive conditions in terms of the coefficients for the model to generate processes that are fractional of order zero. We then show that if there is a unit root, the model generates a fractional process Xt of order d, d > 0, for which there are vectors β so that β‼Xt is fractional of order d − b, 0 < b ≤ d. We find a representation of the solution that demonstrates the fractional properties. Finally we suggest a model that allows for a polynomial fractional vector, that is, the process Xt is fractional of order d, β‼Xt is fractional of order d − b, and a linear combination of β‼Xt and ΔbXt is fractional of order d − 2b. The representations and conditions are analogous to the well-known conditions for I(0), I(1), and I(2) variables.

This paper develops a generalized autoregressive conditional correlation (GARCC) model when the standardized residuals follow a random coefficient vector autoregressive process. As a multivariate generalization of the Tsay (1987, Journal of the American Statistical Association 82, 590–604) random coefficient autoregressive (RCA) model, the GARCC model provides a motivation for the conditional correlations to be time varying. GARCC is also more general than the Engle (2002, Journal of Business & Economic Statistics 20, 339–350) dynamic conditional correlation (DCC) and the Tse and Tsui (2002, Journal of Business & Economic Statistics 20, 351–362) varying conditional correlation (VCC) models and does not impose unduly restrictive conditions on the parameters of the DCC model. The structural properties of the GARCC model, specifically, the analytical forms of the regularity conditions, are derived, and the asymptotic theory is established. The Baba, Engle, Kraft, and Kroner (BEKK) model of Engle and Kroner (1995, Econometric Theory 11, 122–150) is demonstrated to be a special case of a multivariate RCA process. A likelihood ratio test is proposed for several special cases of GARCC. The empirical usefulness of GARCC and the practicality of the likelihood ratio test are demonstrated for the daily returns of the Standard and Poor's 500, Nikkei, and Hang Seng indexes.

We consider the properties of the highest posterior probability model in a linear regression setting. Under a spike and slab hierarchy we find that although highest posterior model selection is total risk consistent, it possesses hidden undesirable properties. One such property is a marked underfitting in finite samples, a phenomenon well noted for Bayesian information criterion (BIC) related procedures but not often associated with highest posterior model selection. Another concern is the substantial effect the prior has on model selection. We employ a rescaling of the hierarchy and show that the resulting rescaled spike and slab models mitigate the effects of underfitting because of a perfect cancellation of a BIC-like penalty term. Furthermore, by drawing upon an equivalence between the highest posterior model and the median model, we find that the effect of the prior is less influential on model selection, as long as the underlying true model is sparse. Nonsparse settings are, however, problematic. Using the posterior mean for variable selection instead of posterior inclusion probabilities avoids these issues.

We propose an easy to use derivative-based two-step estimation procedure for semiparametric index models, where the number of indexes is not known a priori. In the first step various functionals involving the derivatives of the unknown function are estimated using nonparametric kernel estimators, in particular the average outer product of the gradient (AOPG). By testing the rank of the AOPG we determine the required number of indexes. Subsequently, we estimate the index parameters in a method of moments framework, with moment conditions constructed using the estimated average derivative functionals. The estimator readily extends to multiple equation models and is shown to be root-N-consistent and asymptotically normal.

We consider the limiting behavior of a vector autoregressive model of order one (VAR(1)) with independent and identically distributed (i.i.d.) innovations vector with dependent components in the domain of attraction of a multivariate stable law with possibly different indices of stability. It is shown that in some cases the ordinary least squares (OLS) estimates are inconsistent. This inconsistency basically originates from the fact that each coordinate of the partial sum processes of dependent i.i.d. vectors of innovations in the domain of attraction of stable laws needs a different normalizer to converge to a limiting process. It is also revealed that certain M-estimates, with some regularity conditions, as an appropriate alternative, not only resolve inconsistency of the OLS estimates but also give higher consistency rates in all cases.

It is common in econometric applications that several hypothesis tests are carried out simultaneously. The problem then becomes how to decide which hypotheses to reject, accounting for the multitude of tests. The classical approach is to control the familywise error rate (FWE), which is the probability of one or more false rejections. But when the number of hypotheses under consideration is large, control of the FWE can become too demanding. As a result, the number of false hypotheses rejected may be small or even zero. This suggests replacing control of the FWE by a more liberal measure. To this end, we review a number of recent proposals from the statistical literature. We briefly discuss how these procedures apply to the general problem of model selection. A simulation study and two empirical applications illustrate the methods.We thank three anonymous referees for helpful comments that have led to an improved presentation of the paper. The research of the third author has been partially supported by the Spanish Ministry of Science and Technology and FEDER, Grant BMF2003-03324.

The paper provides general matrix formulas for minimum mean squared error signal extraction for a finitely sampled time series whose signal and noise components are nonstationary autoregressive integrated moving average processes. These formulas are quite practical; in addition to being simple to implement on a computer, they make it possible to easily derive important general properties of the signal extraction filters. We also extend these formulas to estimates of future values of the unobserved signal, and we show how this result combines signal extraction and forecasting.

Average exponential F tests for structural change in a Gaussian linear regression model and modifications thereof maximize a weighted average power that incorporates specific weighting functions to make the resulting test statistics simple. Generalizations of these tests involve the numerical evaluation of (potentially) complicated integrals. In this paper, we suggest a uniform Laplace approximation to evaluate weighted average power test statistics for which a simple closed form does not exist. We also show that a modification of the avg-F test is optimal under a very large class of weighting functions and can be written as a ratio of quadratic forms so that both its p-values and critical values are easy to calculate using numerical algorithms.

The paper examines various tests for assessing whether a time series model requires a slope component. We first consider the simple t-test on the mean of first differences and show that it achieves high power against the alternative hypothesis of a stochastic nonstationary slope and also against a purely deterministic slope. The test may be modified, parametrically or nonparametrically, to deal with serial correlation. Using both local limiting power arguments and finite-sample Monte Carlo results, we compare the t-test with the nonparametric tests of Vogelsang (1998, Econometrica 66, 123–148) and with a modified stationarity test. Overall the t-test seems a good choice, particularly if it is implemented by fitting a parametric model to the data. When standardized by the square root of the sample size, the simple t-statistic, with no correction for serial correlation, has a limiting distribution if the slope is stochastic. We investigate whether it is a viable test for the null hypothesis of a stochastic slope and conclude that its value may be limited by an inability to reject a small deterministic slope.The second author thanks the Economic and Social Research Council (ESRC) for support as part of a project on Dynamic Common Factor Models for Regional Time Series, grant L138 25 1008. Support from the Bank of Italy is also gratefully acknowledged. Earlier versions of this paper were presented at the meeting on Frontiers in Time Series held in Olbia, Italy, in June 2005 and at the NSF/NBER Time Series conference in Heidelberg, Germany, in September 2005; we are grateful to several participants for helpful comments. We also thank Peter Phillips, Robert Taylor, Jesus Gonzalo, and a number of other participants at the Unit Root and Co-integration Testing meeting in Faro, Portugal, for helpful comments. We are grateful to Paulo Rodrigues and two referees for their comments.

This paper considers various tests of the unit root hypothesis in panels where the cross-section dependence is due to common dynamic factors. Three situations are studied. First, the common factors and idiosyncratic components may both be nonstationary. In this case test statistics based on generalized least squares (GLS) possess a standard normal limiting distribution, whereas test statistics based on ordinary least squares (OLS) are invalid. Second, if the common component is I(1) and the idiosyncratic component is stationary (the case of cross-unit cointegration), then both the OLS and the GLS statistics fail. Finally, if the idiosyncratic components are I(1) but the common factors are stationary, then the OLS-based test statistics are severely biased, whereas the GLS-based test statistics are asymptotically valid in this situation. A Monte Carlo study is conducted to verify the asymptotic results.The research for this paper was carried out within research project “Unit roots and cointegration in panel data” financed by the German Research Association (DFG). We thank Paulo Rodrigues and two anonymous referees for helpful comments and suggestions.

Aragon, Daouia, and Thomas-Agnan (2005, Econometric Theory 21, 358–389) introduced a new nonparametric frontier estimation. We prove the weak convergence of the empirical conditional quantile function. The distribution of the limit depends on the unknown conditional quantile density function. We provide a method to construct uniform confidence bands without estimating the conditional quantile density.

This paper establishes that the availability of instrumental variables enables the identification and the consistent estimation of nonparametric quantile regression models in the presence of measurement error in the regressors. The proposed estimator takes the form of a nonlinear functional of derivatives of conditional expectations and is shown to provide estimated quantile functions that are uniformly consistent over a compact set.

Nonparametric kernel estimation of density and conditional mean is widely used, but many of the pointwise and global asymptotic results for the estimators are not available unless the density is continuous and appropriately smooth; in kernel estimation for discrete-continuous cases smoothness is required for the continuous variables. Nonsmooth density and mass points in distributions arise in various situations that are examined in empirical studies; some examples and explanations are discussed in the paper. Generally, any distribution function consists of absolutely continuous, discrete, and singular components, but only a few special cases of nonparametric estimation involving singularity have been examined in the literature, and asymptotic theory under the general setup has not been developed. In this paper the asymptotic process for the kernel estimator is examined by means of the generalized functions and generalized random processes approach; it provides a unified theory because density and its derivatives can be defined as generalized functions for any distribution, including cases with singular components. The limit process for the kernel estimator of density is fully characterized in terms of a generalized Gaussian process. Asymptotic results for the Nadaraya–Watson conditional mean estimator are also provided.

An unknown constant matrix M is observed with additive random error. The basic problem considered is to devise an estimator of M that trades off bias against variance so as to achieve relatively low quadratic risk. This paper develops an adaptive total least squares estimator and an adaptive total shrinkage estimator of M that minimize estimated risk over certain large classes of linear estimators. It is shown that the asymptotic risk of the adaptive total least squares estimator is the smallest attainable among reduced rank total least squares fits to the data matrix. The asymptotic risk of the adaptive total shrinkage estimator is shown to be smaller still. A close link is established between total shrinkage and the Efron–Morris estimator of M. In the asymptotics, the row dimension of M tends to infinity, and the column dimension stays fixed. The risks converge uniformly when the signal-to-noise ratio and the measurement error variance are both bounded. A second problem treated is estimation of M under the assumption that a linear relation holds among its columns. In this formulation of the errors-in-variables linear regression model, rank constrained adaptive total least squares asymptotically dominates the usual total least squares estimator of M, and rank constrained adaptive total shrinkage is better still.This research was supported in part by National Science Foundation Grant DMS 0404547.

This paper studies a class of Markov models that consist of two components. Typically, one of the components is observable and the other is unobservable or “hidden.” Conditions under which geometric ergodicity of the unobservable component is inherited by the joint process formed of the two components are given. This implies existence of initial values such that the joint process is strictly stationary and β-mixing. In addition to this, conditions for the existence of moments are also obtained, and extensions to the case of nonstationary initial values are provided. All these results are applied to a general model that includes as special cases various first-order generalized autoregressive conditional heteroskedasticity (GARCH) and autoregressive conditional duration (ACD) models with possibly complicated nonlinear structures. The results only require mild moment assumptions and in some cases provide necessary and sufficient conditions for geometric ergodicity.

Integration for seasonal time series can take the form of seasonal periodic or nonperiodic integration. When seasonal time series are periodically integrated, we show that any cointegration is either full periodic cointegration or full nonperiodic cointegration, with no possibility of cointegration applying for only some seasons. In contrast, seasonally integrated series can be seasonally, periodically or nonperiodically cointegrated, with the possibility of cointegration applying for a subset of seasons. Cointegration tests are analyzed for periodically integrated series. A residual-based test is examined, and its asymptotic distribution is derived under the null hypothesis of no cointegration. A Monte Carlo analysis shows good performance in terms of size and power. The role of deterministic terms in the cointegrating test regression is also investigated. Further, we show that the asymptotic distribution of the error-correction test for periodic cointegration derived by Boswijk and Franses (1995, Review of Economics and Statistics 77, 436–454) does not apply for periodically integrated processes.The authors gratefully acknowledge the comments of participants at the conference on Unit Root and Cointegration Testing, University of the Algave, September–October 2005, and they particularly thank two anonymous referees and Helmut Lütkepohl (co-editor of this issue of Econometric Theory) for their constructive comments, which have substantially improved the generality of the results in the paper. Tomás del Barrio Castro acknowledges financial support from Ministerio de Educación y Ciencia SEJ2005-07781/ECON.

We suggest using a class of semiparametric dynamic panel data models to capture individual variations in panel data. The model assumes linearity in some continuous/discrete variables that can be exogenous/endogenous and allows for nonlinearity in other weakly exogenous variables. We propose a nonparametric generalized method of moments (NPGMM) procedure to estimate the functional coefficients, and we establish the consistency and asymptotic normality of the resulting estimators.

We consider statistical tests concerning various relationships between two absolute concentration curves (ACCs). In particular, we consider tests for determining if the two ACCs coincide, if one is above another in a specified order, or if they do not intersect without specifying which one is above/below the other one. These problems are of interest in the context of marginal conditional stochastic dominance (MCSD). Constructing statistical tests for the MCSD relies on ideas and also on their modifications developed by Linton, Maasoumi, and Whang (2005, Review of Economic Studies 72, 735–765) in the context of stochastic dominance for distribution functions. Our theoretical considerations are supplemented with a simulation study.

Research on model/procedure selection has focused on selecting a single model globally. In many applications, especially for high-dimensional or complex data, however, the relative performance of the candidate procedures typically depends on the location, and the globally best procedure can often be improved when selection of a model is allowed to depend on location. We consider localized model selection methods and derive their theoretical properties.This research was supported by U.S. National Science Foundation CAREER Grant DMS0094323. We thank three referees and the editors for helpful comments on improving the paper.