This paper develops a general approach to robust, regression-based specification tests for (possibly) dynamic econometric models. A useful feature of the proposed tests is that, in addition to estimation under the null hypothesis, computation requires only a matrix linear least-squares regression and then an ordinary least-squares regression similar to those employed in popular nonrobust tests. For the leading cases of conditional mean and/or conditional variance tests, the proposed statistics are robust to departures from distributional assumptions that are not being tested, while maintaining asymptotic efficiency under ideal conditions. Moreover, the statistics can be computed using any √T-consistent estimator, resulting in significant simplifications in some otherwise difficult contexts. Among the examples covered are conditional mean tests for models estimated by weighted nonlinear least squares under misspecification of the conditional variance, tests of jointly parameterized conditional means and variances estimated by quasi-maximum likelihood under nonnormality, and some robust specification tests for a dynamic linear model estimated by two-stage least squares.

This paper establishes asymptotic properties of quasi-maximum likelihood estimators for spatial dynamic panel data with both time and individual fixed effects when the number of individuals n and the number of time periods T can be large. We propose a data transformation approach to eliminate the time effects. When n / T → 0, the estimators are consistent and asymptotically centered normal; when n is asymptotically proportional to T, they are consistent and asymptotically normal, but the limit distribution is not centered around 0; when n / T → ∞, the estimators are consistent with rate T and have a degenerate limit distribution. We also propose a bias correction for our estimators. When n1/3 / T → 0, the correction will asymptotically eliminate the bias and yield a centered confidence interval. The estimates from the transformation approach can be consistent when n is a fixed finite number.

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.

An asymptotic theory for stochastic processes generated from nonlinear transformations of nonstationary integrated time series is developed. Various nonlinear functions of integrated series such as ARIMA time series are studied, and the asymptotic distributions of sample moments of such functions are obtained and analyzed. The transformations considered in the paper include a variety of functions that are used in practical nonlinear statistical analysis. It is shown that their asymptotic theory is quite different from that of integrated processes and stationary time series. When the transformation function is exponentially explosive, for instance, the convergence rate of sample functions is path dependent. In particular, the convergence rate depends not only on the size of the sample but also on the realized sample path. Some brief applications of these asymptotics are given to illustrate the effects of nonlinearly transformed integrated processes on regression. The methods developed in the paper are useful in a project of greater scope concerned with the development of a general theory of nonlinear regression for nonstationary time series.

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.

Although econometricians have been using Bollerslev's (1986, Journal of Econometrics 31, 307–327) GARCH(r, s) model for over a decade, the higher order moment structure of the model remains unresolved. The sufficient condition for the existence of the higher order moments of the GARCH(r, s) model was given by Ling (1999a, Journal of Applied Probability 36, 688–705). This paper shows that Ling's condition is also necessary. As an extension, the necessary and sufficient moment conditions are established for Ding, Granger, and Engle's (1993, Journal of Empirical Finance, 1, 83–106) asymmetric power GARCH(r, s) model.

We consider the estimation and identification of the functional structures of nonlinear econometric systems of the ARCH type. We employ nonparametric kernel estimates for the nonlinear functions characterizing the systems, and we establish strong consistency along with sharp rates of convergence under mild regularity conditions. We also prove the asymptotic normality of the estimates.

Quantile regression methods are suggested for a class of ARCH models. Because conditional quantiles are readily interpretable in semiparametric ARCH models and are inherendy easier to estimate robustly than population moments, they offer some advantages over more familiar methods based on Gaussian likelihoods. Related inference methods, including the construction of prediction intervals, are also briefly discussed.

We define the Bernstein copula and study its statistical properties in terms of both distributions and densities. We also develop a theory of approximation for multivariate distributions in terms of Bernstein copulas. Rates of consistency when the Bernstein copula density is estimated empirically are given. In order of magnitude, this estimator has variance equal to the square root of the variance of common nonparametric estimators, e.g., kernel smoothers, but it is biased as a histogram estimator.We would thank Mark Salmon for interesting us in the copula function and Peter Phillips, an associate editor, and the referees for many valuable comments. All remaining errors are our sole responsibility.

Econometric applications of kernel estimators are proliferating, suggesting the need for convenient variance estimates and conditions for asymptotic normality. This paper develops a general “delta-method” variance estimator for functionals of kernel estimators. Also, regularity conditions for asymptotic normality are given, along with a guide to verify them for particular estimators. The general results are applied to partial means, which are averages of kernel estimators over some of their arguments with other arguments held fixed. Partial means have econometric applications, such as consumer surplus estimation, and are useful for estimation of additive nonparametric models.

This paper considers estimation and inference in panel vector autoregressions where (i) the individual effects are either random or fixed, (ii) the time-series properties of the model variables are unknown a priori and may feature unit roots and cointegrating relations, and (iii) the time dimension of the panel is short and its cross-sectional dimension is large. Generalized method of moments (GMM) and quasi maximum likelihood (QML) estimators are obtained and compared in terms of their asymptotic and finite-sample properties. It is shown that the asymptotic variances of the GMM estimators that are based on levels in addition to first differences of the model variables depend on the variance of the individual effects, whereas by construction the fixed effects QML estimator is not subject to this problem. Monte Carlo evidence is provided showing that the fixed effects QML estimator tends to outperform the various GMM estimators in finite sample under both normal and nonnormal errors. The paper also shows how the fixed effects QML estimator can be successfully used for unit root and cointegration tests in short panels.We are grateful to Karim Abadir, Stephen Bond, Jinyong Hahn, Marc Nerlove, Ingmar Prucha, and, especially, Manuel Arellano, Peter Schmidt, Peter Phillips (the editor), and four anonymous referees for helpful and constructive comments. We have also benefited from useful suggestions by participants at various seminars and conferences.

In frontier analysis, most of the nonparametric approaches (free disposal hull [FDH], data envelopment analysis [DEA]) are based on envelopment ideas, and their statistical theory is now mostly available. However, by construction, they are very sensitive to outliers. Recently, a robust nonparametric estimator has been suggested by Cazals, Florens, and Simar (2002, Journal of Econometrics 1, 1–25). In place of estimating the full frontier, they propose rather to estimate an expected frontier of order m. Similarly, we construct a new nonparametric estimator of the efficient frontier. It is based on conditional quantiles of an appropriate distribution associated with the production process. We show how these quantiles are interesting in efficiency analysis. We provide the statistical theory of the obtained estimators. We illustrate with some simulated examples and a frontier analysis of French post offices, showing the advantage of our estimators compared with the estimators of the expected maximal output frontiers of order m.We thank J.P. Florens for helpful discussions and C. Cazals for providing the post office data set. We also are very grateful to the referees for useful suggestions.

This paper studies a broad class of nonnegative ARCH(∞) models. Sufficient conditions for the existence of a stationary solution are established and an explicit representation of the solution as a Volterra type series is found. Under our assumptions, the covariance function can decay slowly like a power function, falling just short of the long memory structure. A moving average representation in martingale differences is established, and the central limit theorem is proved.

This paper provides L1 and weak laws of large numbers for uniformly integrable L1-mixingales. The L1-mixingale condition is a condition of asymptotic weak temporal dependence that is weaker than most conditions considered in the literature. Processes covered by the laws of large numbers include martingale difference, ø(·), ρ(·), and α(·) mixing, autoregressive moving average, infinite-order moving average, near epoch dependent, L1-near epoch dependent, and mixingale sequences and triangular arrays. The random variables need not possess more than one finite moment and the L1-mixingale numbers need not decay to zero at any particular rate. The proof of the results is remarkably simple and completely self-contained.

This paper proposes a version of the generalized method of moments procedure that handles both the case where the number of moment conditions is finite and the case where there is a continuum of moment conditions. Typically, the moment conditions are indexed by an index parameter that takes its values in an interval. The objective function to minimize is then the norm of the moment conditions in a Hilbert space. The estimator is shown to be consistent and asymptotically normal. The optimal estimator is obtained by minimizing the norm of the moment conditions in the reproducing kernel Hilbert space associated with the covariance. We show an easy way to calculate this estimator. Finally, we study properties of a specification test using overidentifying restrictions. Results of this paper are useful in many instances where a continuum of moment conditions arises. Examples include efficient estimation of continuous time regression models, cross-sectional models that satisfy conditional moment restrictions, and scalar diffusion processes.

This paper presents several generic uniform convergence results that include generic uniform laws of large numbers. These results provide conditions under which pointwise convergence almost surely or in probability can be strengthened to uniform convergence. The results are useful for establishing asymptotic properties of estimators and test statistics.

The results given here have the following attributes, (1) they extendresults of Newey to cover convergence almost surely as well as convergence in probability, (2) they apply to totally bounded parameter spaces (rather than just to compact parameter spaces), (3) they introduce a set of conditions for a generic uniform law of large numbers that has the attribute of giving the weakest conditions available for i.i.d. contexts, but which apply in some dependent nonidentically distributed contexts as well, and (4) they incorporate and extend themain results in the literature in a parsimonious fashion.

The definition of causation, discussed in Granger (1980) and elsewhere, has been widely applied in economics and in other disciplines. For this definition, a series yt is said to cause xt+l if it contains information about the forecastability for xt+l contained nowhere else in some large information set, which includes xt−j, j ≥ 0. However, it would be convenient to think of causality being different in extent or direction at seasonal or low frequencies, say, than at other frequencies. The fact that a stationary series is effectively the (uncountably infinite) sum of uncorrelated components, each of which is associated with a single frequency, or a narrow frequency band, introduces the possibility that the full causal relationship can be decomposed by frequency. This is known as the Wiener decomposition or the spectral decomposition of the series, as discussed by Hannan (1970). For any series generated by , where xt, and are both stationary, with finite variances and a(B) is a backward filter

with B the backward operator, there is a simple, well-known relationship between the spectral decompositions of the two series.

Asymptotic theory is developed for local time density estimation for a general class of functionals of integrated and fractionally integrated time series. The main result provides a convenient basis for developing a limit theory for nonparametric cointegrating regression and nonstationary autoregression. The treatment directly involves local time estimation and the density function of the processes under consideration, providing an alternative approach to the Markov chain and Fourier integral methods that have been used in other recent work on these problems.

Let Yt satisfy the stochastic difference equation for t = 1,2,…, where et are independent and identically distributed random variables with mean zero and variance σ2 and the initial conditions (Y−p+1,…, Y0) are fixed constants. It is assumed that the process is invertible and that the true, but unknown, roots m1,m2,…,mp of satisfy the hypothesis Hd: m1 = … = md = 1 and |mj| < 1 for j = d + 1,…,p. We present a reparameterization of the model for Yt that is convenient for testing the hypothesis Hd. We consider the asymptotic properties of (i) a likelihood ratio type “F-statistic” for testing the hypothesis Hd, (ii) a likelihood ratio type t-statistic for testing the hypothesis Hd against the alternative Hd−1. Using these asymptotic results, we obtain two sequential testing procedures that are asymptotically consistent.