Adaptive maximum likelihood estimators of unit roots in autoregressive processes with possibly non-Gaussian innovations are considered. Unit root tests based on the adaptive estimators are constructed. Limiting distributions of the test statistics are derived, which are linear combinations of two functionals of Brownian motions. A Monte Carlo simulation reveals that the proposed tests have improved powers over the classical Dickey–Fuller tests when the distribution of the innovation is not close to normal. We also compare the proposed tests with those of Lucas (1995, Econometric Theory 11, 331–346) based on M-estimators.

We construct properly scaled functions of Rp-valued partial sums of demeaned data and derive bounds via the functional law of the iterated logarithm for strong mixing processes. If we obtain a value below or equal to the bound we decide in favor of I(0); otherwise we decide in favor of I(1). This provides a consistent rule for classifying time series as being I(1) or I(0). The nice feature of the procedure lies in the almost sure nature of the bound, guaranteeing a lim sup–type result. We finally provide conditions for the strong consistency of estimators of the variance in the dependent and heterogeneous case.

Following the approach proposed by Gregoir and Laroque (1993, Econometric Theory 9, 329–342), we consider a class of multivariate processes that, when differenced enough, yields covariance stationary processes whose determinant of the matrix series associated with their Wold representation has various unit roots with various orders of multiplicity we restrict to be integers. A representation theorem is provided that involves different polynomial error correction terms at each frequency associated with each unit root. An identification criterion for each set of error correction terms is proposed.

Tail approximation of the asymptotic distribution of the log likelihood ratio test for cointegration in a vector autoregressive process is studied. In dimension 2, an approximation of weighted χ2 type is derived by applying multivariate saddlepoint approximation techniques to a Fourier inversion integral.

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.

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.

This paper extends the statistical results obtained by Gregoir and Laroque (1994, Journal of Econometrics 63, 183–214). It develops statistical tools to analyze multivariate time series that can be represented under an autoregressive equation of finite order with various polynomial error correction terms at various frequencies with possibly a non-null deterministic part as introduced by Gregoir (1999, Econometric Theory 15, 435–468). We propose an estimation procedure that proceeds through repeated applications of principal component analysis and a specification test for the omission of a polynomial relation of cointegration at each frequency.

This paper investigates the asymptotic properties of the maximum marginal likelihood estimator for a regression model with a stochastic trend component when the signal-to-noise ratio is near zero. In particular, the local level model in Harvey (1989, Forecasting, Structural Time Series Models and the Kalman Filter, Cambridge: Cambridge University Press) and its variants where a time trend or an intercept is included are considered. A local-to-zero parameterization is adopted. Two sets of asymptotic properties are presented for the local maximizer: consistency and the limiting distribution. The estimator is found to be super-consistent. The limit distribution is derived and found to possess a long tail and a mass point at zero. It yields a good approximation for samples of moderate size. Simulation also documents that the empirical distribution converges less rapidly to the limit distribution as number of regression parameters increases. The results could be viewed as a transition step toward establishing new likelihood ratio–type or Wald-type tests for the stationarity null.

We consider the asymptotic null distribution of the empirical autocorrelation function when the innovations of a moving average process belong to the normal domain of attraction of a Cauchy law. A series expansion for the density of the limiting null distribution is developed, and some critical values of the tests are computed numerically.

This paper deals with the estimation of unequally spaced panel data regression models with AR(1) remainder disturbances. A feasible generalized least squares (GLS) procedure is proposed as a weighted least squares that can handle a wide range of unequally spaced panel data patterns. This procedure is simple to compute and provides natural estimates of the serial correlation and variance components parameters. The paper also provides a locally best invariant test for zero first-order serial correlation against positive or negative serial correlation in case of unequally spaced panel data.

This paper considers cointegrating regressions with time varying coefficients. The coefficients are modeled as smooth functions evolving over time. It is shown that they can be estimated nonparametrically, using suitably modified series estimators. Presented is the efficient method of estimation, which relies on simple prefiltering of the data and preestimation of the model. The test for the adequacy of model specification is also developed. Our model and statistical methods are applied to analyze the U.S. automobile demand function.

Semiparametric estimates of long memory seem useful in the analysis of long financial time series because they are consistent under much broader conditions than parametric estimates. However, recent large sample theory for semiparametric estimates forbids conditional heteroskedasticity. We show that a leading semiparametric estimate, the Gaussian or local Whittle one, can be consistent and have the same limiting distribution under conditional heteroskedasticity as under the conditional homoskedasticity assumed by Robinson (1995, Annals of Statistics 23, 1630–61). Indeed, noting that long memory has been observed in the squares of financial time series, we allow, under regularity conditions, for conditional heteroskedasticity of the general form introduced by Robinson (1991, Journal of Econometrics 47, 67–84), which may include long memory behavior for the squares, such as the fractional noise and autoregressive fractionally integrated moving average form, and also standard short memory ARCH and GARCH specifications.

This paper considers classical test statistics, namely, the likelihood ratio, efficient score, and Wald statistics, for econometric models under simulation estimation. The simulated likelihood ratio, simulated efficient score, and simulated Wald test statistics are shown to be asymptotically equivalent. Because the simulated score vector can be asymptotically biased, limiting distributions of these simulated statistics can be asymptotically noncentral χ2 distributed. This paper studies inference issues with various simulated test statistics. Monte Carlo results are also provided to compare and demonstrate finite sample properties of simulated test statistics.

In this paper nearly unstable AR(p) processes (in other words, models with characteristic roots near the unit circle) are studied. Our main aim is to describe the asymptotic behavior of the least-squares estimators of the coefficients. A convergence result is presented for the general complex-valued case. The limit distribution is given by the help of some continuous time AR processes. We apply the results for real-valued nearly unstable AR(p) models. In this case the limit distribution can be identified with the maximum likelihood estimator of the coefficients of the corresponding continuous time AR processes.

This paper studies efficient detrending in cointegrating regression and develops modified tests for cointegration that use efficient detrending procedures. Asymptotics for these tests are derived. Monte Carlo experiments are conducted to evaluate the detrending procedures in finite samples and to compare tests for cointegration based on different detrending procedures. The limit theory allows for increasingly remote initial condition effects as the sample size goes to infinity.

Likelihood ratio (LR) tests for the cointegrating rank of a vector autoregressive process have been developed under different assumptions regarding deterministic terms. For instance, nonzero mean terms and linear trends have been accounted for in some of the tests. In this paper we provide a general framework for deriving the local power properties of these tests. Thereby it is possible to assess the virtue of utilizing varying amounts of prior information by making assumptions regarding the deterministic terms. One interesting result from this analysis is that if no assumptions regarding the specific form of the mean term are made whereas a linear trend is excluded then a test is available that has the same local power as an LR test derived under a zero mean assumption.

It is possible to construct a test of the null of no fractional integration that has nontrivial asymptotic power against a sequence of alternatives specifying that the series is I(d) with d = O(T−1/2), where T is the sample size. In this paper, I show that tests for fractional integration that are based on the partial sum process of the time series have only trivial asymptotic power (i.e., equal to the size) against this sequence of local alternatives. These tests include the rescaled-range test. In this sense, despite its widespread use in empirical work, the rescaled-range test is a poor test for fractional integration.

In this paper, a necessary and sufficient condition for the existence of the unconditional fourth moment of the GARCH(p,q) process is given and also an expression for the moment itself. Furthermore, the autocorrelation function of the centered and squared observations of this process is derived. The statistical theory is further illustrated by a few special cases such as the GARCH(2,2) process and the ARCH(q) process.

We provide a theoretical framework in which to study the accuracy of bootstrap P values, which may be based on a parametric or nonparametric bootstrap. In the parametric case, the accuracy of a bootstrap test will depend on the shape of what we call the critical value function. We show that, in many circumstances, the error in rejection probability of a bootstrap test will be one whole order of magnitude smaller than that of the corresponding asymptotic test. We also propose a simulation method for estimating this error that requires the calculation of only two test statistics per replication.

This paper deals with a scalar I(d) process {yj}, where the integration order d is any real number. Under this setting, we first explore asymptotic properties of various statistics associated with {yj}, assuming that d is known and is greater than or equal to ½. Note that {yj} becomes stationary when d < ½, whose case is not our concern here. It turns out that the case of d = ½ needs a separate treatment from d > ½. We then consider, under the normality assumption, testing and estimation for d, allowing for any value of d. The tests suggested here are asymptotically uniformly most powerful invariant, whereas the maximum likelihood estimator is asymptotically efficient. The asymptotic theory for these results will not assume normality. Unlike in the usual unit root problem based on autoregressive models, standard asymptotic results hold for test statistics and estimators, where d need not be restricted to d ≥ ½. Simulation experiments are conducted to examine the finite sample performance of both the tests and estimators.