This paper presents results on the issue of weak convergence of a TVP-GQARCH-M(1,1) process. These suggest that the weak limit of this endogenous volatility model is an exogenous (stochastic) volatility continuous time process. Under the appropriate assumptions, we derive the invariant distributions at which the process converges in various cases. They reveal an approximate distributional relation between the GARCH or the asymmetric GQARCH variance processes and the continuous time Ornstein–Uhlenbeck models with respect to appropriate nonnegative Levy processes.I am grateful to Ritsa Panagiotou, Phyllis Alexander, Antonis Demos, Bruce Hansen, Nicholas Magginas, Nour Meddahi, Enrique Sentana, Paolo Zaffaroni, two anonymous referees, and the seminar participants at the Department of International and European Economic Studies for their valuable comments.

In many important textbooks the formal statement of the spectral representation theorem is followed by a process version, usually informal, stating that any stationary stochastic process {ξ(t), t ∈ T} is the limit in quadratic mean of a sequence of processes {S(n,t), t ∈ T}, each consisting of a finite sum of harmonic oscillations with stochastic weights. The natural issues, whether the approximation error ξ(t) − S(n,t) is stationary or whether at least it converges to zero uniformly in t, have not been explicitly addressed in the literature. The paper shows that in all relevant cases, for T unbounded the process convergence is not uniform in t (so that ξ(t) − S(n,t) is not stationary). Equivalently, when T is unbounded the number of harmonic oscillations necessary to approximate ξ(t) with a preassigned accuracy depends on t. The conclusion is that the process version of the spectral representation theorem should explicitly mention that in general the approximation of ξ(t) by a finite sum of harmonic oscillations, given the accuracy, is valid for t belonging to a bounded subset of the real axis (of the set of integers in the discrete-parameter case).The author is grateful for very useful suggestions to Francesco Battaglia, Gianluca Cubadda, Domenico Marinucci, Enzo Orsingher, Dag Tjøstheim, and Umberto Triacca and also to an anonymous referee and the Econometric Theory co-editor Benedikt M. Pötscher.

Typically in time series econometrics, for many statistics, a rescaled integrated process is replaced with Brownian motion to find the limit distribution. For averages of functions of a rescaled integrated process, Park and Phillips have shown that this remains true for functions with poles, as long as a sample-size-dependent region around the poles is excluded from consideration, the function is locally integrable, and some other regularity conditions hold. In this addendum, I show that under some regularity conditions on the function under consideration, there is no need for such a sample-size-dependent region around the pole in Park and Phillips' theorem, as long as the function under consideration is locally integrable.The author thanks four anonymous referees and Professor Phillips for their comments and suggestions and Benedikt Pötscher for suggestions and for pointing out an error in a previous version of this paper.

This paper derives transformations for multivariate statistics that eliminate asymptotic skewness, extending the results of Niki and Konishi (1986, Annals of the Institute of Statistical Mathematics 38, 371–383). Within the context of valid Edgeworth expansions for such statistics we first derive the set of equations that such a transformation must satisfy and second propose a local solution that is sufficient up to the desired order. Application of these results yields two useful corollaries. First, it is possible to eliminate the first correction term in an Edgeworth expansion, thereby accelerating convergence to the leading term normal approximation. Second, bootstrapping the transformed statistic can yield the same rate of convergence of the double, or prepivoted, bootstrap of Beran (1988, Journal of the American Statistical Association 83, 687–697), applied to the original statistic, implying a significant computational saving.

The analytic results are illustrated by application to the family of exponential models, in which the transformation is seen to depend only upon the properties of the likelihood. The numerical properties are examined within a class of nonlinear regression models (logit, probit, Poisson, and exponential regressions), where the adequacy of the limiting normal and of the bootstrap (utilizing the k-step procedure of Andrews, 2002, Econometrica 70, 119–162) as distributional approximations is assessed.This paper is derived from my Ph.D. thesis, “Higher-Order Asymptotics for Econometric Estimators and Tests,” for which thanks for patient and helpful supervision go to Grant Hillier. Comments by Karim Abadir, Francesco Bravo, Giovanni Forchini, Soren Johansen, Paul Marriott, Mark Salmon, and Steve Satchell, participants at the conference “Differential Geometric Methods in Econometrics,” held at EUI, Florence, and by two anonymous referees proved most helpful. In particular, I thank Peter Phillips for showing interest in the paper, helping with improving the exposition, and providing me with copies of two unpublished research notes. Financial support in the form of a Leverhulme Special Research Fellowship in Economics and Mathematics is gratefully acknowledged.

When considering the general Gauss–Markov model Y = Xβ + e, where E(e) = 0, Cov(e) = V, one often needs to know whether the range inclusion range(X) ⊆ range(V) holds.

The central limit theorem for student's distribution—solution.

Solution to Normal's deconvolution and the independence of sample mean and variance problem.

We study some methods of combining procedures for forecasting a continuous random variable. Statistical risk bounds under the square error loss are obtained under distributional assumptions on the future given the current outside information and the past observations. The risk bounds show that the combined forecast automatically achieves the best performance among the candidate procedures up to a constant factor and an additive penalty term. In terms of the rate of convergence, the combined forecast performs as well as if the best candidate forecasting procedure were known in advance.

Empirical studies suggest that combining procedures can sometimes improve forecasting accuracy over the original procedures. Risk bounds are derived to theoretically quantify the potential gain and price of linearly combining forecasts for improvement. The result supports the empirical finding that it is not automatically a good idea to combine forecasts. Indiscriminate combining can degrade performance dramatically as a result of the large variability in estimating the best combining weights. An automated combining method is shown in theory to achieve a balance between the potential gain and the complexity penalty (the price of combining), to take advantage (if any) of sparse combining, and to maintain the best performance (in rate) among the candidate forecasting procedures if linear or sparse combining does not help.This research was supported by U.S. National Security Agency Grant MDA9049910060 and U.S. National Science Foundation CAREER Grant DMS0094323. The author sincerely thanks three reviewers and Poti Giannakouros for their very valuable comments, which led to a substantial improvement of the paper.

A concise derivation of the Wallace and Hussain fixed effects transformation.

This note discusses the concept of aliasing and its use in the paper by H.J. Bierens (2001, Econometric Theory 17, 962–983), in the framework of a second-order autoregression with complex unit roots. The condition on the range of the angular frequency φ is extended to φ ∈ (0,2π) − {π}.Financial support from UPV-EHU research project 9/UPV-00038.321-13503/2001, Basque Government project PI9970 and Ministerio de ciencia y tecnología BEC2003-02028 is gratefully acknowledged.

Unbiasedness of the OLS estimator with random regressors—solution.

A Hausman test based on the difference between fixed effects two-stage least squares and error components two-stage least squares.

Let P in be a projector with P+ its Moore–Penrose inverse and PH its conjugate transpose. Werner (2002) provides a list of equivalent conditions for P to be Hermitian.

Solution to the asymptotic distribution of the Dickey–Fuller statistic under nonnegativity constraint problem.

Consistent standard errors for target variance approach to GARCH estimation.

This problem discusses an I(2) model in the VAR(1) case. The I(2) representation theorem of Johansen (1992) (JRT) holds also for VAR(1) processes. The I(2) model for VAR(k) processes has been discussed for k ≥ 2 in Johansen (1996, Ch. 9; 1997). We here discuss a parametrization for the I(2) case of VAR(1) that differs from the VAR(k) model.

Correcting for heteroskedasticity of unspecified form.

Applications are now being sought for the seventh A.R. Bergstrom Prize in Econometrics.

The objective of the Prize is to reward the achievement of excellence in econometrics, as evidenced by a research paper in any area of econometrics. The Prize is open to New Zealand citizens or permanent residents of New Zealand who, on the closing date of applications, have current or recent (i.e., within two years) student status for a higher degree. It is intended that the awardee will utilize the proceeds to assist in financing further study or research in econometrics in New Zealand or overseas.

In Econometric Theory, vol. 20, no. 2 (April 2004), on page 428 the number of the Solution should be changed to Solution 03.2.1.