Statistical models of unobserved heterogeneity are typically formalized as mixtures of simple parametric models and interest naturally focuses on testing for homogeneity versus general mixture alternatives. Many tests of this type can be interpreted as C(α) tests, as in Neyman (1959), and shown to be locally asymptotically optimal. These C(α) tests will be contrasted with a new approach to likelihood ratio testing for general mixture models. The latter tests are based on estimation of general nonparametric mixing distribution with the Kiefer and Wolfowitz (1956) maximum likelihood estimator. Recent developments in convex optimization have dramatically improved upon earlier EM methods for computation of these estimators, and recent results on the large sample behavior of likelihood ratios involving such estimators yield a tractable form of asymptotic inference. Improvement in computation efficiency also facilitates the use of a bootstrap method to determine critical values that are shown to work better than the asymptotic critical values in finite samples. Consistency of the bootstrap procedure is also formally established. We compare performance of the two approaches identifying circumstances in which each is preferred.

A popular validation procedure for Dynamic Stochastic General Equilibrium (DSGE) models consists in comparing the structural shocks and impulse-response functions obtained by estimation-calibration of the DSGE with those obtained in an Structural Vector Autoregressions (SVAR) identified by means of some of the DSGE restrictions. I show that this practice can be seriously misleading when the variables used in the SVAR contain measurement errors. If this is the case, for generic values of the parameters of the DSGE, the shocks estimated in the SVAR are not “made of” the corresponding structural shocks plus measurement error. Rather, each of the SVAR shocks is contaminated by noncorresponding structural shocks. We argue that High-Dimensional Dynamic Factor Models are free from this drawback and are the natural model to use in validation procedures for DSGEs.

In this paper, we introduce a family of contrasts for parametric inference in ARCH models the volatility of which exhibits some degeneracy. We focus specifically on ARCH processes with a linear volatility (called LARCH processes), for which the Gaussian quasi-likelihood estimator may be inconsistent. Our approach generalizes that of Beran and Schützner (2009) and gives an interesting alternative to the WLSE used by Francq and Zakoïan (2010) for an autoregressive process with LARCH errors. The family of contrasts is indexed by a single parameter that controls the smoothness of an approximated quasi-likelihood function. Under mild conditions, the resulting estimators are shown to be strongly consistent and asymptotically normal. The optimal asymptotic variance is also given. For LARCH processes, an atypical result is obtained: under assumptions, we show that the limiting distribution of the estimators can be arbitrarily close to a Gaussian distribution supported on a line. Extensions to multivariate processes are also discussed.

In most cases where estimation with averaged data is performed, interest lies on the parameters of a model at the individual level, but grouped data are used because disaggregate data are not observed. In this note we study the conditions under which it is possible to consistently estimate the parameters of the individual data model using averaged data, giving particular attention to the case of endogenous selection into groups.We are grateful to an anonymous referee for helpful comments and suggestions. We also thank Pedro Duarte Neves, Les Godfrey, Paulo Parente, Maximiano Pinheiro, and Pedro Portugal for helpful discussions. The usual disclaimer applies. Machado is consultant for the Research Department of Banco de Portugal. Santos Silva is grateful for the hospitality, working conditions, and financial support provided by Banco de Portugal, which made this work possible. The authors also gratefully acknowledge partial financial support from Fundação para a Ciência e Tecnologia, program POCTI, partially funded by FEDER.

This paper proposes a test of the rank of the submatrix of β, where β is a cointegrating matrix. In addition, the submatrix of β⊥, an orthogonal complement to β, is investigated. We construct the test statistic by using the eigenvalues of the quadratic form of the submatrix. We show that the test statistic has a limiting chi-square distribution when data are nontrending, whereas for trending data we have to consider a conservative test or other testing procedure that requires the pretest of the structure of the matrix. Finite sample simulations show that, although the simulation settings are limited, the proposed test works well for nontrending data, whereas we have to carefully use the test for trending data because it may become too conservative in some cases.I owe special thanks to two anonymous referees, the co-editor, Pierre Perron, and Taku Yamamoto. All errors are my responsibility. This research was supported by the Ministry of Education, Culture, Sports, Science and Technology under grants-in-aid 13730023 and 14203003.

We extend the framework of the fully modified, ordinary least squares (OLS) estimator introduced by Phillips and Hansen (1990) to the case of seasonally cointegrated processes at a given frequency. First we extend a weak convergence result of sample covariance matrices (Phillips, 1988) to the case of seasonal unit roots. Using a complex number framework, we then show that we can take into account the constraints that exist in a situation of seasonal cointegration as illustrated in Gregoir (1999a) and derive estimates of the cointegration vectors that allow for asymptotic normal inference. This allows us to propose a test whose null hypothesis is the existence of seasonal cointegration. A Monte Carlo exercise investigates the finite sample properties of this test procedure. The paper closes with the analysis of situations in which there exist more than one frequency at which seasonal cointegration can be observed.

We consider the problem of hypothesis testing in a modified version of the stochastic integration and cointegration framework of Harris, McCabe, and Leybourne (2002, Journal of Econometrics 111, 363–384). This nonlinear setup allows for volatility in excess of that catered for by the standard integration/cointegration paradigm through the introduction of nonstationary heteroskedasticity. We propose a test for stochastic cointegration against the alternative of no cointegration and a secondary test for stationary cointegration against the heteroskedastic alternative. Asymptotic distributions of these tests under their respective null hypotheses are derived, and consistency under their respective alternatives is established. Monte Carlo evidence suggests that the tests will perform well in practice. An empirical application to the term structure of interest rates is also given.We are most grateful to the Associate Editor and two anonymous referees for providing helpful comments on earlier versions of this paper.

We show that the empirical distribution of the roots of the vector autoregression (VAR) of order p fitted to T observations of a general stationary or nonstationary process converges to the uniform distribution over the unit circle on the complex plane, when both T and p tend to infinity so that (ln T)/p → 0 and p3/T → 0. In particular, even if the process is a white noise, nearly all roots of the estimated VAR will converge by absolute value to unity. For fixed p, we derive an asymptotic approximation to the expected empirical distribution of the estimated roots as T → ∞. The approximation is concentrated in a circular region in the complex plane for various data generating processes and sample sizes.

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.

This paper considers spurious regression between integrated processes with stable errors. Our results show that the t-ratios diverge at the rate of √T, which is identical to what Phillips (1986, Journal of Econometrics 33, 311–340) has obtained for the Gaussian case. Therefore, it is the long memory in the dependent variable and regressors, instead of the moment conditions of the error terms, that causes the spurious regression.

The inverse of a noncentral Wishart matrix occurs in a variety of contexts in multivariate statistical work, including instrumental variable (IV) regression, but there has been very little work on its properties. In this paper, we first provide an expression for the expectation of the inverse of a noncentral Wishart matrix, and then go on to do the same for a number of scalar-valued functions of the inverse. The main result is obtained by exploiting simple but powerful group-equivariance properties of the expectation map involved. Subsequent results exploit the consequences of other invariance properties.

This is an analytical study of the effect of level-shift and temporary-change components, when present but neglected, on the trace test for cointegration. The contribution is threefold. First, we discuss in a multivariate framework, and jointly, effects that in the previous literature have been discussed in a univariate setting and in isolation. Second, we consider a rather general specification of shifts and outliers with random size, number, and timing and with flexible dynamics. It nests the classical cases of additive shifts, innovational outliers, and additive outliers. Third, as an instrument for this analysis we develop an asymptotic theory for product moment matrices of linear processes with stochastic level-shift components, generalizing results of Leipus and Viano (2003, Statistics and Probability Letters 61, 177–190).

We propose a statistical procedure to determine the dimension of the nonstationary subspace of cointegrated functional time series taking values in the Hilbert space of square-integrable functions defined on a compact interval. The procedure is based on sequential application of a proposed test for the dimension of the nonstationary subspace. To avoid estimation of the long-run covariance operator, our test is based on a variance ratio-type statistic. We derive the asymptotic null distribution and prove consistency of the test. Monte Carlo simulations show good performance of our test and provide evidence that it outperforms the existing testing procedure. We apply our methodology to three empirical examples: age-specific U.S. employment rates, Australian temperature curves, and Ontario electricity demand.

I am pleased to announce that Guido Kuersteiner and Robert Taylor have joined the Board of Co-Editors of Econometric Theory for five-year terms beginning on 1 January 2007.

We consider weak convergence of sample averages of nonlinearly transformed stochastic triangular arrays satisfying a functional invariance principle. A fundamental paradigm for such processes is constituted by integrated processes. The results obtained are extensions of recent work in the literature to the multivariate and non-Gaussian case. As admissible nonlinear transformation, a new class of functionals (so-called locally p-integrable functions) is introduced that adapts the concept of locally integrable functions in Pötscher (2004, Econometric Theory 20, 1–22) to the multidimensional setting.

We study the problem of identifying a forecaster’s loss function from observations on forecasts, realizations, and the forecaster’s information set. Essentially different loss functions can lead to the same forecasts in all situations, though within the class of all continuous loss functions, this is strongly nongeneric. With the small set of exceptional cases ruled out, generic nonparametric preference recovery is theoretically possible, but identification depends critically on the amount of variation in the conditional distributions of the process being forecast. There exist processes with sufficient variability to guarantee identification, and much of this variation is also necessary for a process to have universal identifying power. We also briefly address the case in which the econometrician does not fully observe the conditional distributions used by the forecaster, and in this context we provide a practically useful set identification result for loss functions used in forecasting binary variables.