This paper uses the concept of dual likelihood to develop some higher order asymptotic theory for the empirical likelihood ratio test for parameters defined implicitly by a set of estimating equations. The resulting theory is likelihood based in the sense that it relies on methods developed for ordinary parametric likelihood models to obtain valid Edgeworth expansions for the maximum dual likelihood estimator and for the dual/empirical likelihood ratio statistic. In particular, the theory relies on certain Bartlett-type identities that can be used to produce a simple proof of the existence of a Bartlett correction for the dual/empirical likelihood ratio. The paper also shows that a bootstrap version of the dual/empirical likelihood ratio achieves the same higher order accuracy as the Bartlett-corrected dual/empirical likelihood ratio.This paper is based on Chapter 2 of my Ph.D. dissertation at the University of Southampton. Partial financial support under E.S.R.C. grant R00429634019 is gratefully acknowledged. I thank my supervisor, Grant Hillier, for many stimulating conversations and Peter Phillips, Andrew Chesher, and Jan Podivisnky for some useful suggestions. In addition, I am very grateful to the co-editor Donald Andrews and two referees for many valuable comments that have improved noticeably the original draft. All remaining errors are my own responsibility.

Semiparametric generalized additive models are a powerful tool in quantitative econometrics. With response Y, covariates X,T, the considered model is E(Y |X;T) = G{XTβ + α + m1(T1) + ··· + md(Td)}. Here, G is a known link, α and β are unknown parameters, and m1,…,md are unknown (smooth) functions of possibly higher dimensional covariates T1,…,Td. Estimates of m1,…,md, α, and β are presented, and asymptotic distributions are given for both the nonparametric and the parametric part. The main focus of the paper is application of bootstrap methods. It is shown how bootstrap can be used for bias correction, hypothesis testing (e.g., component-wise analysis), and the construction of uniform confidence bands. Further, bootstrap tests for model specification and parametrization are given, in particular for testing additivity and link function specification. The practical performance of the methods is illustrated in a simulation study.This research was supported by the Deutsche Forschungsgemeinschaft, Sonderforschungsbereich 373 “Quantifikation und Simulation ökonomischer Prozesse,” Humboldt-Universität zu Berlin, DFG project MA 1026/6-2, the Spanish “Dirección General de Enseñanza Superior,” no. BEC2001-1270, and the grant “Nonparametric methods in finance and insurance” from the Danish Social Science Research Council. We thank Marlene Müller, Oliver Linton, and two anonymous referees for helpful discussion.

This paper studies the smooth transition regression model where regressors are I(1) and errors are I(0). The regressors and errors are assumed to be dependent both serially and contemporaneously. Using the triangular array asymptotics, the nonlinear least squares estimator is shown to be consistent, and its asymptotic distribution is derived. It is found that the asymptotic distribution involves a bias under the regressor-error dependence, which implies that the nonlinear least squares estimator is inefficient and unsuitable for use in hypothesis testing. Thus, this paper proposes a Gauss–Newton type estimator that uses the nonlinear least squares estimator as an initial estimator and is based on regressions augmented by leads and lags. Using leads and lags enables the Gauss–Newton estimator to eliminate the bias and have a mixture normal distribution in the limit, which makes it more efficient than the nonlinear least squares estimator and suitable for use in hypothesis testing. Simulation results indicate that the results obtained from the triangular array asymptotics provide reasonable approximations for the finite-sample properties of the estimators and t-tests when sample sizes are moderately large. The cointegrating smooth transition regression model is applied to the Korean and Indonesian data from the Asian currency crisis of 1997. The estimation results partially support the interest Laffer curve hypothesis. But overall the effects of interest rate on spot exchange rate are shown to be quite negligible in both nations.This paper was partly written while the first author was visiting the Institute of Statistics and Econometrics at Humboldt University, Berlin. This author acknowledges financial support from the Alexander von Humboldt Foundation under a Humboldt Research Award and from the Yrjö Jahnsson Foundation. The second author wrote this paper while visiting the Cowles Foundation for Research in Economics, Yale University. This author thanks the faculty and staff of the Cowles Foundation, especially Don Andrews, John Geanakoplos, David Pearce, Peter Phillips, and Nora Wiedenbach, for their support and hospitality. The second author was financially supported for the research in this paper by Kookmin University. The authors thank Don Andrews, Helmut Lütkepohl, Peter Phillips, Bruce Hansen, and two referees for their valuable comments on this paper. Part of the data studied in this paper was provided by Chi-Young Song, whom we thank.

This paper proves that the fully modified vector autoregression (FM-VAR) estimator has second-order bias effects when some roots are local to unity. These bias effects are shown to result in potentially severe size distortions in FM-VAR testing when the hypothesis involves near unit root variables. In addition, the paper reveals that with the FM-VAR method near unit roots become estimated as exact unit roots with convergence speed faster than the order of the sample size. Also this result implies problems for FM-VAR testing, as such “hyperconsistent” estimates give rise to degenerate limit distributions under the null hypothesis.I am grateful to Pentti Saikkonen, Jim Stock, Markku Lanne, Jukka Nyblom, and three referees for very helpful comments on earlier drafts. This paper is a part of the research program of the Research Unit on Economic Structures and Growth (RUESG) at the Department of Economics at the University of Helsinki. Financial support from the ASLA Fulbright, the Yrjö Jahnsson Foundation, and the Finnish Cultural Foundation is gratefully acknowledged. The usual disclaimer applies.

The rank-based unit root tests proposed by Hasan and Koenker (1997, Econometrica 65, 133–161) have power equal to size for normal innovations. Unit root tests based on M-estimators exhibit the same behavior. The problem occurs because the test statistics are transformed to obtain computationally convenient critical values. I describe a convenient way to compute critical values without transforming the test statistics. The resulting tests are almost as powerful as least squares–based tests for normal errors and much more powerful for thicker tailed distributions.I thank Thomas Rothenberg for many useful comments. This paper is based on my dissertation, which he supervised. I also thank Jack Porter, Jim Powell, Richard Stanton, and Jim Stock for good advice. Comments of Bruce Hansen, the editor, and two anonymous referees improved the exposition.

We present a goodness-of-fit test for time series models based on the discrete spectral average estimator. Unlike current tests of goodness of fit, the asymptotic distribution of our test statistic allows the null hypothesis to be either a short- or long-range dependence model. Our test is in the frequency domain, is easy to compute, and does not require the calculation of residuals from the fitted model. This is especially advantageous when the fitted model is not a finite-order autoregressive model. The test statistic is a frequency domain analogue of the test by Hong (1996, Econometrica 64, 837–864), which is a generalization of the Box and Pierce (1970, Journal of the American Statistical Association 65, 1509–1526) test statistic. A simulation study shows that our test has power comparable to that of Hong's test and superior to that of another frequency domain test by Milhoj (1981, Biometrika 68, 177–187).

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.

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.

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.

See solution below.

The 12th meeting of the New Zealand Econometrics Study Group (NZESG) was held in Wellington at the Reserve Bank of New Zealand on Friday afternoon, 17 October 2003, and at Old Government Buildings, Victoria University, on Saturday, 18 October 2003. It was well attended by approximately 50 participants (7 from overseas) with a full program of 20 speakers, including an opening address by Dr Alan Bollard, Governor, Reserve Bank of New Zealand. Full details of the program, including abstracts and photographs of the event, are available at the conference web site http://www.vuw.ac.nz/nzesg/ and at the NZESG home web site http://yoda.eco.auckland.ac.nz/nzesg/.