Most of the asymptotic results for Markov regime-switching models with possible unit roots are based on specifications implying that the number of regime switches grows to infinity as the sample size increases. Conversely, in this note we derive some new asymptotic results for the case of Markov regime switches that are infrequent in the sense that their number is bounded in probability, even asymptotically. This is achieved by (inversely) relating the probability of regime switching to the sample size. The proposed asymptotic theory is applied to a well-known stochastic unit root model, where the dynamics of the observed variable switches between a unit root regime and a stationary regime.

We consider the problem of combining forecasts from two different levels (called “macro” and “micro”), where we have access to the forecasts and their precisions but not to the full data set. We develop a theoretical framework and provide Monte Carlo evidence in the cases of both perfect and imperfect aggregation. Our proposed procedure is simple and robust. We also extend the procedure to time series and propose a forecast model for the European zero rates, combining quarterly and monthly observations. We show that forecast accuracy is improved at both levels.We are grateful to the editors of this special issue and to two referees for their constructive comments, which greatly helped improve the paper. Earlier versions of this paper were presented at the NAKE workshop in Amsterdam, at Tilburg University, and at ESEM 2006 in Vienna. We thank the participants for their useful comments. In addition, we thank Steffan Berridge, John Einmahl, and Hans Schumacher for their constructive comments.

Saikkonen (1991, Econometric Theory 7, 1–21) developed an asymptotic optimality theory for the estimation of cointegrated regressions. He proposed the dynamic ordinary least squares (OLS) estimator obtained by augmenting the static cointegrating regression with leads and lags of the first differences of the I(1) regressors. However, the assumptions imposed preclude the use of information criteria such as the Akaike information criterion (AIC) and Bayesian information criterion (BIC) to select the number of leads and lags. We show that his results remain valid under weaker conditions that permit the use of such data dependent rules. Simulations show that, relative to sequential general to specific testing procedures, the use of such information criteria can indeed produce estimates with smaller mean squared errors and confidence intervals with better coverage rates.

We consider the parameter restrictions that need to be imposed to ensure that the conditional variance process of a GARCH(p,q) model remains nonnegative. Previously, Nelson and Cao (1992, Journal of Business ’ Economic Statistics 10, 229–235) provided a set of necessary and sufficient conditions for the aforementioned nonnegativity property for GARCH(p,q) models with p ≤ 2 and derived a sufficient condition for the general case of GARCH(p,q) models with p ≥ 3. In this paper, we show that the sufficient condition of Nelson and Cao (1992) for p ≥ 3 actually is also a necessary condition. In addition, we point out the linkage between the absolute monotonicity of the generalized autoregressive conditional heteroskedastic (GARCH) generating function and the nonnegativity of the GARCH kernel, and we use it to provide examples of sufficient conditions for this nonnegativity property to hold.

This paper compares models of fractional processes and associated weak convergence results based on moving average representations in the time domain with spectral representations. Both approaches have been applied in the literature on fractional processes. We point out that the conventional forms of these models are not equivalent, as is commonly assumed, even under a Gaussianity assumption. We show that it is necessary to distinguish between “two-sided” processes depending on both leads and lags from one-sided or “causal” processes, because in the case of fractional processes these models yield different limiting properties. We derive new representations of fractional Brownian motion and show how different results are obtained for, in particular, the distribution of stochastic integrals in the multivariate context. Our results have implications for valid statistical inference in fractional integration and cointegration models.We thank F. Hashimzade and two anonymous referees for their valuable comments.

In our paper published in vol. 22, no. 1, of Econometric Theory (2006, pp. 69–97) the proof of (22) in Lemma 3.1 is incorrect as given. The problem in the proof occurs after the first display on p. 93 where we attempted to deduce (22) from the already established relation (21). The argument given there is incorrect, because an existence-quantor was mistaken for an all-quantor. We show how the problem can be repaired. In the course of this, we also note a useful variant of another result in Lemma 3.1.

The paper obtains conditions that ensure stationarity of linear long-run equilibrium relations and differenced observations in vector autoregressive error correction models with nonlinear short-run dynamics. The considered models include various threshold error correction models and their smooth transition counterparts. These models assume that the form of the short-run dynamics depends on values of observable transition functions that determine the regime in which the considered process evolves. In related models studied in the paper the transition functions are unobservable. These models are obtained by making the transition functions of threshold error correction models dependent on an unobservable random term. Previous stationarity conditions obtained for these kinds of regime switching error correction models are extended by using recent developments on nonlinear autoregressive models based on the theory of Markov chains and the concept of joint spectral radius of a set of square matrices. In addition to stationarity, existence of second-order moments and beta mixing is also established. The results of the paper enhance the understanding of the considered nonlinear error correction models and pave the way for the development of their asymptotic estimation and testing theory.Financial support from the Research Unit of Economic Structures and Growth (RUESG) in the University of Helsinki and the Yrjö Jahnsson Foundation is gratefully acknowledged. The author thanks Anders Rahbek for stimulating discussions on the topic of this paper and Helmut Lütkepohl and an anonymous referee for useful comments.

We consider the Breitung (2002, Journal of Econometrics 108, 343–363) statistic ξn, which provides a nonparametric test of the I(1) hypothesis. If ξ denotes the limit in distribution of ξn as n → ∞, we prove (Theorem 1) that 0 ≤ ξ ≤ 1/π2, a result that holds under any assumption on the underlying random variables. The result is a special case of a more general result (Theorem 3), which we prove using the so-called cotangent method associated with Cauchy's residue theorem.

In this note, based on the generalized method of moments (GMM) interpretation of the usual ordinary least squares (OLS) and feasible generalized least squares (FGLS) estimators of seemingly unrelated regressions (SUR) models, we show that the OLS estimator is asymptotically as efficient as the FGLS estimator if and only if the cross-equation orthogonality condition is redundant given the within-equation orthogonality condition. Using the condition for redundancy of moment conditions of Breusch, Qian, Schmidt, and Wyhowski (1999, Journal of Econometrics 99, 89–111), we then derive the necessary and sufficient condition for the equal asymptotic efficiency of the OLS and FGLS estimators of SUR models. We also provide several useful sufficient conditions for the equal asymptotic efficiency of OLS and FGLS estimators that can be interpreted as various mixings of the two famous sufficient conditions of Zellner (1962, Journal of the American Statistical Association 57, 348–368).

We are pleased to announce award of the A.R. Bergstrom Prize in Econometrics for 2007 to Melanie Morten, Ph.D. student, Yale University, for her paper, “Healthy and Wealthy? Examining the Causality between Income and Life Expectancy.”