This paper establishes stochastic equicontinuity for classes of mixingales. Attention is restricted to Lipschitz-continuous parametric functions. Unlike some other empirical process theory for dependent data, our results do not require bounded functions, stationary processes, or restrictive dependence conditions. Applications are given to martingale difference arrays, strong mixing arrays, and near-epoch dependent arrays.

In this paper we extend some of Phillips's [4] results to nonlinear unobserved components models and develop a posterior odds ratio test of the unit root hypothesis based on flat and Jeffreys priors. In contrast to the analysis presented by Schotman and van Dijk [9], we utilize a nondegenerate structural representation of the components model that allows us to determine well-behaved Jeffreys priors, posterior densities under flat priors and Jeffreys priors, and posterior odds ratios for the unit root hypothesis without a proper prior for the level parameter. The analysis highlights the importance of the treatment of initial values for inference concerning stationarity and unit roots.

We calculate the bias of the profile score for the regression coefficients in a multistratum autoregressive model with stratum-specific intercepts. The bias is free of incidental parameters. Centering the profile score delivers an unbiased estimating equation and, upon integration, an adjusted profile likelihood. A variety of other approaches to constructing modified profile likelihoods are shown to yield equivalent results. However, the global maximizer of the adjusted likelihood lies at infinity for any sample size, and the adjusted profile score has multiple zeros. Consistent parameter estimates are obtained as local maximizers inside or on an ellipsoid centered at the maximum likelihood estimator.

This paper introduces a test for the comparison of multiple misspecified conditional interval models, for the case of dependent observations. Model accuracy is measured using a distributional analog of mean square error, in which the approximation error associated with a given model, say, model i, for a given interval, is measured by the expected squared difference between the conditional confidence interval under model i and the “true” one.

When comparing more than two models, a “benchmark” model is specified, and the test is constructed along the lines of the “reality check” of White (2000, Econometrica 68, 1097–1126). Valid asymptotic critical values are obtained via a version of the block bootstrap that properly captures the effect of parameter estimation error. The results of a small Monte Carlo experiment indicate that the test does not have unreasonable finite sample properties, given small samples of 60 and 120 observations, although the results do suggest that larger samples should likely be used in empirical applications of the test.The authors express their gratitude to Don Andrews and an anonymous referee for providing numerous useful suggestions, all of which we feel have been instrumental in improving earlier drafts of this paper. The authors also thank Russell Davidson, Clive Granger, Lutz Kilian, Christelle Viaroux, and seminar participants at the 2002 UK Econometrics Group meeting in Bristol, the 2002 European Econometric Society meetings, the 2002 University of Pennsylvania NSF-NBER time series conference, the 2002 EC2 Conference in Bologna, Cornell University, the State University of New York at Stony Brook, and the University of California at Davis for many helpful comments and suggestions on previous versions of this paper.

Monotonicity is a key qualitative prediction of a wide array of economic models derived via robust comparative statics. It is therefore important to design effective and practical econometric methods for testing this prediction in empirical analysis. This article develops a general nonparametric framework for testing monotonicity of a regression function. Using this framework, a broad class of new tests is introduced, which gives an empirical researcher a lot of flexibility to incorporate ex ante information she might have. The article also develops new methods for simulating critical values, which are based on the combination of a bootstrap procedure and new selection algorithms. These methods yield tests that have correct asymptotic size and are asymptotically nonconservative. It is also shown how to obtain an adaptive and rate optimal test that has the best attainable rate of uniform consistency against models whose regression function has Lipschitz-continuous first-order derivatives and that automatically adapts to the unknown smoothness of the regression function. Simulations show that the power of the new tests in many cases significantly exceeds that of some prior tests, e.g., that of Ghosal, Sen, and Van der Vaart (2000).

We present a method of estimating density-related functionals, without prior knowledge of the density's functional form. The approach revolves around the specification of an explicit formula for a new class of distributions that encompasses many of the known cases in statistics, including the normal, gamma, inverse gamma, and mixtures thereof. The functionals are based on a couple of hypergeometric functions. Their parameters can be estimated, and the estimates then reveal both the functional form of the density and the parameters that determine centering, scaling, etc. The function to be estimated always leads to a valid density, by design, namely, one that is nonnegative everywhere and integrates to 1. Unlike fully nonparametric methods, our approach can be applied to small datasets. To illustrate our methodology, we apply it to finding risk-neutral densities associated with different types of financial options. We show how our approach fits the data uniformly very well. We also find that our estimated densities' functional forms vary over the dataset, so that existing parametric methods will not do uniformly well.We thank Hans-Jürg Büttler, Aleš Černý, Tony Culyer, Les Godfrey, David Hendry, Sam Kotz, Steve Lawford, Peter Phillips, Bas Werker, and three anonymous referees for their comments. We also thank for their feedback the participants at the seminars and conferences where this paper has been invited, in particular the 1998 CEPR Finance Network Workshop, the 1998 METU conference, the 1998 FORC (Warwick) conference “Options: Recent Advances,” Money Macro & Finance Group, the Swiss National Bank, Imperial College, Tilburg University, Université Libre de Bruxelles, the University of Oxford, Southampton University, and UMIST. The first author acknowledges support from the ESRC (UK) grant R000239538. The second author acknowledges help from the HEC Foundation and the European Community TMR grant “Financial Market Efficiency and Economic Efficiency.” This paper was written when the second author was affiliated with HEC-Paris.

In this paper we propose a new nonparametric test for conditional heteroskedasticity based on a measure of nonparametric goodness-of-fit (R2) that is obtained from the local polynomial regression of the residuals from a parametric regression on some covariates. We show that after being appropriately standardized, the nonparametric R2 is asymptotically normally distributed under the null hypothesis and a sequence of Pitman local alternatives. We also prove the consistency of the test and propose a bootstrap method to obtain the bootstrap p-values. We conduct a small set of simulations and compare our test with some popular parametric and nonparametric tests in the literature.

This paper derives some exact power properties of tests for spatial autocorrelation in the context of a linear regression model. In particular, we characterize the circumstances in which the power vanishes as the autocorrelation increases, thus extending the work of Krämer (2005). More generally, the analysis in the paper sheds new light on how the power of tests for spatial autocorrelation is affected by the matrix of regressors and by the spatial structure. We mainly focus on the problem of residual spatial autocorrelation, in which case it is appropriate to restrict attention to the class of invariant tests, but we also consider the case when the autocorrelation is due to the presence of a spatially lagged dependent variable among the regressors. A numerical study aimed at assessing the practical relevance of the theoretical results is included.

Asymptotic local power analysis has become an important and increasingly used technique in econometrics. This paper reviews the history of local power analysis and delineates the contribution of J.Neyman, E.J.G. Pitman, and G. Noether.

One of the most cited studies in recent years within the field of nonstationary panel data analysis is that of Bai and Ng (2004), in which the authors propose PANIC, a new framework for analyzing the nonstationarity of panels with idiosyncratic and common components. The problem is that the asymptotic validity of PANIC as a platform for constructing pooled panel unit root tests based on averaging is not fully proven. This paper provides the required results, whose usefulness is verified through simulations.

The exact mean integrated squared error (MISE) of the nonparametric kernel density estimator is derived for the asymptotically optimal smooth polynomial kernels of Müller (1984, Annals of Statistics 12, 766–774) and the trapezoid kernel of Politis and Romano (1999, Journal of Multivariate Analysis 68, 1–25) and is used to contrast their finite-sample efficiency with the higher order Gaussian kernels of Wand and Schucany (1990Canadian Journal of Statistics 18, 197–204). We find that these three kernels have similar finite-sample efficiency. Of greater importance is the choice of kernel order, as we find that kernel order can have a major impact on finite-sample MISE, even in small samples, but the optimal kernel order depends on the unknown density function. We propose selecting the kernel order by the criterion of minimax regret, where the regret (the loss relative to the infeasible optimum) is maximized over the class of two-component mixture-normal density functions. This minimax regret rule produces a kernel that is a function of sample size only and uniformly bounds the regret below 12% over this density class.

The paper also provides new analytic results for the smooth polynomial kernels, including their characteristic function.This research was supported in part by the National Science Foundation. I thank Oliver Linton and a referee for helpful comments and suggestions that improved the paper.

It is well known that allowing the coefficients to be time-varying in a predictive model with possibly nonstationary regressors can help to deal with instability in predictability associated with linear predictive models. In this paper, an L2-type test statistic is proposed to test the stability of the coefficient vector, and the asymptotic distributions of the proposed test statistic are developed under both null and alternative hypotheses. A Monte Carlo experiment is conducted to evaluate the finite sample performance of the proposed test statistic and an empirical example is examined to demonstrate the practical application of the proposed testing method.

This paper derives exact discrete time representations for data generated by a continuous time autoregressive moving average (ARMA) system with mixed stock and flow data. The representations for systems comprised entirely of stocks or of flows are also given. In each case the discrete time representations are shown to be of ARMA form, the orders depending on those of the continuous time system. Three examples and applications are also provided, two of which concern the stationary ARMA(2, 1) model with stock variables (with applications to sunspot data and a short-term interest rate) and one concerning the nonstationary ARMA(2, 1) model with a flow variable (with an application to U.S. nondurable consumers’ expenditure). In all three examples the presence of an MA(1) component in the continuous time system has a dramatic impact on eradicating unaccounted-for serial correlation that is present in the discrete time version of the ARMA(2, 0) specification, even though the form of the discrete time model is ARMA(2, 1) for both models.

Sharp upper and lower uniform bounds are established for a general class of functionals of integrated and fractionally integrated time series. The main result is used to develop optimal uniform convergence for the Nadaraya-Watson estimator and the local linear nonparametric estimator in a nonlinear cointegrating regression model. Unlike the point-wise situation, it is shown that the performance of the local linear nonparametric estimator is superior to that of the Nadaraya-Watson estimator in uniform asymptotics.

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.

In this paper, a consistent model specification test is proposed. Some consistent model specification tests have been discussed in econometrics literature. Those tests are consistent by randomization, display a discontinuity in sample size, or have an asymptotic distribution that depends on the data-generating process and on the model, whereas our test does not have one of those disadvantages. Our test can be viewed upon as a conditional moment test as proposed by Newey but instead of a fixed number of conditional moments, an asymptotically infinite number of moment conditions is employed. The use of an asymptotically infinite number of conditional moments will make it possible to obtain a consistent test. Computation of the test statistic is particularly simple, since in finite samples our statistic is equivalent to a chi-square conditional moment test of a finite number of conditional moments.

I review the part played by John Denis Sargan in the formation of the “LSE approach” to dynamic econometric modeling. Despite his unassuming demeanor and his location at LSE—which had earlier dismissed a substantive role for econometric evidence—Sargan nevertheless radically altered the econometric approach of a generation, establishing a powerful approach to empirical modeling of economic time series. His main contributions to econometric methodology, and the subsequent research, are discussed as a complement to the other papers in this memorial volume.Financial support from the U.K. Economic and Social Research Council under grant L138251009 is gratefully acknowledged. I am indebted to Alok Bhargava, Julia Campos, Meghnad Desai, Neil Ericsson, Toni Espasa, Grayham Mizon, Peter Phillips, Timo Teräsvirta, and Ken Wallis for helpful comments and to Peter Phillips for the invitation to contribute this paper to Econometric Theory.

An I(0) process is commonly defined as a process that satisfies a functional central limit theorem, i.e., whose scaled partial sums converge weakly to a Wiener process, and an I(1) process as a process whose first differences are I(0). This paper establishes that with this definition, it is impossible to consistently discriminate between I(0) and I(1) processes. At the same time, on a more constructive note, there exist consistent unit root tests and also nontrivial inconsistent stationarity tests with correct asymptotic size.

Asymptotic normality of the Bayesian posterior is a well-known result for stationary dynamic models or nondynamic models. This paper extends the analysis to a time series model with a possible nonstationary process. We spell out conditions under which asymptotic normality of the posterior is obtained even if the true data-generation process is a nonstationary process.

Deciding the order of differencing is an important part in the specification of an autoregressive integrated moving average (ARIMA) mode. In most, though not all, cases this means deciding whether to use the original observations or their first differences. Common test procedures used in this context are some variants of autoregressive unit root tests. In these tests, one tests the null hypothesis that the order of differencing is one against the alternative that it is zero. The null hypothesis thus states that the original series is nonstationary and integrated of order one, whereas the alternative assumes that it is stationary. In this paper the situation is reversed so that our null hypothesis states that the original series is stationary, whereas the alternative states that it is integrated of order one. In our approach the use of a differenced series thus means overdifferencing and, consequently, a model with a moving average unit root. Testing for this moving average unit root is the topic of this paper. As discussed by Saikkonen and Luukkonen [26] and Tanaka [31], test procedures obtained for this null hypothesis can also be used to test the null hypothesis that a multivariate time series is cointegrated with a given theoretical cointegrating vector. Since the null hypothesis of cointegration is often of interest and cannot be naturally tested by autoregressive unit root tests, this connection provides an important motivation for the test procedures of this paper.