In this paper, new fully nonparametric estimators of the diffusion coefficient of continuous time models are introduced. The estimators are based on Fourier analysis of the state variable trajectory observed and on the estimation of quadratic variation between observations by means of realized volatility. The estimators proposed are shown to be consistent and asymptotically normally distributed. Moreover, the Fourier estimator can be iterated to get a fully nonparametric estimate of the diffusion coefficient in a bivariate model in which one state variable is the volatility of the other. The estimators are shown to be unbiased in small samples using Monte Carlo simulations and are used to estimate univariate and bivariate models for interest rates.

Under general conditions the sample covariance matrix of a vector martingale and its differences converges weakly to the matrix stochastic integral ∫01BdB′, where B is vector Brownian motion. For strictly stationary and ergodic sequences, rather than martingale differences, a similar result obtains. In this case, the limit is ∫01BdB′ + Λ and involves a constant matrix Λ of bias terms whose magnitude depends on the serial correlation properties of the sequence. This note gives a simple proof of the result using martingale approximations.

A local limit theorem is given for the sample mean of a zero energy function of a nonstationary time series involving twin numerical sequences that pass to infinity. The result is applicable in certain nonparametric kernel density estimation and regression problems where the relevant quantities are functions of both sample size and bandwidth. An interesting outcome of the theory in nonparametric regression is that the linear term is eliminated from the asymptotic bias. In consequence and in contrast to the stationary case, the Nadaraya–Watson estimator has the same limit distribution (to the second order including bias) as the local linear nonparametric estimator.

Slowly varying (SV) regressors arise commonly in empirical econometric work, particularly in the form of semilogarithmic regression and log periodogram regression. These regressors are asymptotically collinear. Usual regression formulas for asymptotic standard errors are shown to remain valid, but rates of convergence are affected and the limit distribution of the regression coefficients is shown to be one dimensional. Some asymptotic representations of partial sums of SV functions and central limit theorems with SV weights are given that assist in the development of a regression theory. Multivariate regression and polynomial regression with SV functions are considered and shown to be equivalent, up to standardization, to regression on a polynomial in a logarithmic trend. The theory involves second-, third-, and higher-order forms of slow variation. Some applications to the asymptotic theory of nonlinear trend regression are explored.The author thanks two referees and Pentti Saikkonen for comments and suggestions, Sidney Resnick for references on second-order regular variation, and a Kelly Fellowship and the NSF for partial research support under grants SBR 97-30295 and SES 04-142254. An original draft of the paper was written in June 2000 and circulated under the title “Regression with Slowly Varying Regressors.”

This paper develops an empirical likelihood approach for regular generalized autoregressive conditional heteroskedasticity (GARCH) models and GARCH models with unit roots. For regular GARCH models, it is shown that the log empirical likelihood ratio statistic asymptotically follows a χ2 distribution. For GARCH models with unit roots, two versions of the empirical likelihood methods, the least squares score and the maximum likelihood score functions, are considered. For both cases, the limiting distributions of the log empirical likelihood ratio statistics are established. These two statistics can be used to test for unit roots under the GARCH framework. Finite-sample performances are assessed through simulations for GARCH models with unit roots.This research was supported in part by Hong Kong Research grants Council Grants CUHK4043/02P and HKUST6273/03H. The authors thank two referees and the Co-Editor, Bruce Hansen, for insightful and helpful comments about the relationship between QMLE and MELE, which led to substantial improvement of the presentation. Computational assistance from Jerry Wong and Chun-Yip Yau is also gratefully acknowledged.

This paper considers maximum likelihood estimation for the moving average parameter θ in an MA(1) model when θ is equal to or close to 1. A derivation of the limit distribution of the estimate θLM, defined as the largest of the local maximizers of the likelihood, is given here for the first time. The theory presented covers, in a unified way, cases where the true parameter is strictly inside the unit circle as well as the noninvertible case where it is on the unit circle. The asymptotic distribution of the maximum likelihood estimator subMLE is also described and shown to differ, but only slightly, from that of θLM. Of practical significance is the fact that the asymptotic distribution for either estimate is surprisingly accurate even for small sample sizes and for values of the moving average parameter considerably far from the unit circle.

This article develops an asymptotic theory for estimators of two parameters in the drift function in the fractional Vasicek model when a continuous record of observations is available. The fractional Vasicek model with long-range dependence is assumed to be driven by a fractional Brownian motion with the Hurst parameter greater than or equal to one half. It is shown that, when the Hurst parameter is known, the asymptotic theory for the persistence parameter depends critically on its sign, corresponding asymptotically to the stationary case, the explosive case, and the null recurrent case. In all three cases, the least squares method is considered, and strong consistency and the asymptotic distribution are obtained. When the persistence parameter is positive, the estimation method of Hu and Nualart (2010) is also considered.

We propose an optimal test procedure for testing the marginal density functions of a class of nonlinear diffusion processes. The proposed test is not only an optimal one but also avoids undersmoothing. An adaptive test is constructed, and its asymptotic properties are investigated. To show the asymptotic properties, we establish some general results for moment inequalities and asymptotic distributions for strictly stationary processes under the α-mixing condition. These results are applicable to some other estimation and testing of strictly stationary processes with the α-mixing condition. An example of implementation is given to demonstrate that the proposed model specification procedure is applicable to economic and financial model specification and can be implemented in practice. To ensure the applicability and implementation, we propose a computer-intensive simulation scheme for the choice of a suitable bandwidth involved in the kernel estimation and also a simulated critical value for the proposed adaptive test. Our finite sample studies support both the proposed theory and the simulation procedure.The authors thank the co-editor and three anonymous referees for their constructive comments and suggestions. The first author also thanks Song Xi Chen for some constructive suggestions, in particular the suggestion on using the local linear form instead of the Nadaraya–Watson kernel form in equation (2.6), and Yongmiao Hong for sending a working paper. The authors acknowledge comments from seminar participants at the International Chinese Statistical Association Meeting in Hong Kong in July 2001, the Western Australian Branch Meeting of the Statistical Society of Australia in September 2001, the University of Western Australia, and Monash University. Thanks also go to the Australian Research Council for its financial support.

In this paper I present an alternative derivation of the asymptotic distribution of Kremers, Ericsson, and Dolado's (1992, Oxford Bulletin of Economics and Statistics 54, 325–348) conditional error correction model (ECM)–based t-test for cointegration with a single prespecified cointegrating vector. This alternative distribution, which is identical to the distribution of Hansen's (1995, Econometric Theory 11, 1148–1171) covariate augmented t-test for a unit root, is valid for weakly exogenous regressors and depends on a consistently estimable nuisance parameter that takes on values in the unit interval. I show analytically, using asymptotic power functions based on near-cointegrated alternatives, that the ECM t-test with a prespecified cointegrating vector can have much higher power than single equation tests for cointegration based on estimating the cointegrating vector. I also characterize situations in which the ECM t-test computed with a misspecified cointegrating vector will have high power.

Kernel smoothing techniques free the traditional parametric estimators of volatility from the constraints related to their specific models. In this paper the nonparametric local exponential estimator is applied to estimate conditional volatility functions, ensuring its nonnegativity. Its asymptotic properties are established and compared with those for the local linear estimator. It theoretically enables us to determine when the exponential is expected to be superior to the linear estimator. A very strong and novel result is achieved: the exponential estimator is asymptotically fully adaptive to unknown conditional mean functions. Also, our simulation study shows superior performance of the exponential estimator.

This paper proposes a new nonparametric test for conditional parametric distribution functions based on the first-order linear expansion of the Kullback–Leibler information function and the kernel estimation of the underlying distributions. The test statistic is shown to be asymptotically distributed standard normal under the null hypothesis that the parametric distribution is correctly specified, whereas asymptotically rejecting the null with probability one if the parametric distribution is misspecified. The test is also shown to have power against any local alternatives approaching the null at rates slower than the parametric rate n−1/2. The finite sample performance of the test is evaluated via a Monte Carlo simulation.

This paper studies the properties of naive block bootstrap tests that are scaled by zero frequency spectral density estimators (long-run variance estimators). The naive bootstrap is a bootstrap where the formula used in the bootstrap world to compute standard errors is the same as the formula used on the original data. Simulation evidence shows that the naive bootstrap can be much more accurate than the standard normal approximation. The larger the HAC bandwidth, the greater the improvement. This improvement holds for a large number of popular kernels, including the Bartlett kernel, and it holds when the independent and identically distributed (i.i.d.) bootstrap is used and yet the data are serially correlated. Using recently developed fixed-b asymptotics for HAC robust tests, we provide theoretical results that can explain the finite sample patterns. We show that the block bootstrap, including the special case of the i.i.d. bootstrap, has the same limiting distribution as the fixed-b asymptotic distribution. For the special case of a location model, we provide theoretical results that suggest the naive bootstrap can be more accurate than the standard normal approximation depending on the choice of the bandwidth and the number of finite moments in the data. Our theoretical results lay the foundation for a bootstrap asymptotic theory that is an alternative to the traditional approach based on Edgeworth expansions.

The main aim of the paper is to reevaluate the methodological contributions of Tinbergen and Haavelmo in the context of the current discussions on econometric modeling and propose a reformulation of the Haavelmo methodology. The paper argues that the textbook methodology constitutes a less flexible version of Tinbergen's approach and apart from the probabilistic language, it has little in common with the methodology in Haavelmo's 1944 monograph, commonly acknowledged as having founded modern econometrics. The methodology in this monograph includes several important elements which have either been discarded or never fully integrated within the textbook approach. A re-synthesis of these elements gives rise to an alternative methodological framework. This framework can be used to meet most of the objections to the textbook methodology and provides a framework in the context of which the recent methodological controversies can be evaluated.

For autoregressive processes, we propose new estimators whose pivotal statistics have the standard normal limiting distribution for all ranges of the autoregressive parameters. The proposed estimators are approximately median unbiased. For seasonal time series, the new estimators give us unit root tests that have limiting normal distribution regardless of period of the seasonality. Using the estimators, confidence intervals of the autoregressive parameters are constructed. A Monte-Carlo simulation for first-order autoregressions shows that the proposed tests for unit roots are locally more powerful than the tests based on the ordinary least squares estimators. It also shows that the proposed confidence intervals have shorter average lengths than those of Andrews (1993, Econometrica 61, 139–165) based on the ordinary least squares estimators when the autoregressive coefficient is close to one.

In this paper the testing and estimation problems are discussed in the case of serial correlation. Various models are particular cases of the general framework considered: the nonlinear simultaneous equations models, the probit models, the tobit models, the disequilibrium models, the frontier models, etc. In this context, it is shown that the score test can be written explicitly and that the statistic obtained is a generalization of that of Durbin and Watson; moreover, the maximum likelihood estimation procedure is shown to be robust with respect to serial correlation.

Designed to have power against all alternatives, omnibus consistent tests are the primary econometric tools for testing the correct specification of parametric conditional means when there is no information about the possible alternative. The main purpose of this paper is to show that, contrary to what is generally believed, omnibus specification tests only have substantial local power against alternatives in a finite-dimensional space (usually unknown to the researcher). We call such a space the principal space. We characterize and estimate the principal space for Cramér–von Mises tests. These results are some of the by-products of a detailed theoretical power analysis carried out in the paper. This investigation focuses on the class of the so-called integrated consistent tests under possibly heteroskedastic time series. A Monte Carlo experiment examines the finite-sample properties of tests and estimators of preferred alternatives. Finally, an application of our theory to test the martingale difference hypothesis of some exchange rates provides new information on the rejection of omnibus tests and illustrates our findings.

This paper considers testing for normality for correlated data. The proposed test procedure employs the skewness-kurtosis test statistic, but studentized by standard error estimators that are consistent under serial dependence of the observations. The standard error estimators are sample versions of the asymptotic quantities that do not incorporate any downweighting, and, hence, no smoothing parameter is needed. Therefore, the main feature of our proposed test is its simplicity, because it does not require the selection of any user-chosen parameter such as a smoothing number or the order of an approximating model.We are very grateful to Don Andrews and two referees for useful comments and suggestions. We are especially thankful to a referee who provided a FORTRAN code. Lobato acknowledges financial support from Asociación Mexicana de Cultura and from Consejo Nacional de Ciencia y Tecnologìa (CONACYT) under project grant 41893-S. Velasco acknowledges financial support from Spanish Dirección General de Enseñanza Superior, BEC 2001-1270.

Several economic and financial time series are bounded by an upper and lower finite limit (e.g., interest rates). It is not possible to say that these time series are random walks because random walks are limitless with probability one (as time goes to infinity). Yet, some of these time series behave just like random walks. In this paper we propose a new approach that takes into account these ideas. We propose a discrete-time and a continuous-time process (diffusion process) that generate bounded random walks. These paths are almost indistinguishable from random walks, although they are stochastically bounded by an upper and lower finite limit. We derive for both cases the ergodic conditions, and for the diffusion process we present a closed expression for the stationary distribution. This approach suggests that many time series with random walk behavior can in fact be stationarity processes.

The classical nonstationary autoregressive models are both linear and Markov. They include unit root and cointegration models. A possible nonlinear extension is to relax the linearity and at the same time keep general properties such as nonstationarity and the Markov property. A null recurrent Markov chain is nonstationary, and β-null recurrence is of vital importance for statistical inference in nonstationary Markov models, such as, e.g., in nonparametric estimation in nonlinear cointegration within the Markov models. The standard random walk is an example of a null recurrent Markov chain.

In this paper we suggest that the concept of null recurrence is an appropriate nonlinear generalization of the linear unit root concept and as such it may be a starting point for a nonlinear cointegration concept within the Markov framework. In fact, we establish the link between null recurrent processes and autoregressive unit root models. It turns out that null recurrence is closely related to the location of the roots of the characteristic polynomial of the state space matrix and the associated eigenvectors. Roughly speaking the process is β-null recurrent if one root is on the unit circle, null recurrent if two distinct roots are on the unit circle, whereas the others are inside the unit circle. It is transient if there are more than two roots on the unit circle. These results are closely connected to the random walk being null recurrent in one and two dimensions but transient in three dimensions. We also give an example of a process that by appropriate adjustments can be made β-null recurrent for any β ∈ (0, 1) and can also be made null recurrent without being β-null recurrent.

In this paper we propose a new approximate factor model for large cross-section and time dimensions. Factor loadings are assumed to be smooth functions of time, which allows considering the model as locally stationary while permitting empirically observed time-varying second moments. Factor loadings are estimated by the eigenvectors of a nonparametrically estimated covariance matrix. As is well known in the stationary case, this principal components estimator is consistent in approximate factor models if the eigenvalues of the noise covariance matrix are bounded. To show that this carries over to our locally stationary factor model is the main objective of our paper. Under simultaneous asymptotics (cross-section and time dimension go to infinity simultaneously), we give conditions for consistency of our estimators. A simulation study illustrates the performance of these estimators.