There are many instances in the statistical literature in which inference is based on a normalized quadratic form in a standard normal vector, normalized by the squared length of that vector. Examples include both test statistics (the Durbin–Watson statistic) and estimators (serial correlation coefficients). Although much studied, no general closed-form expression for the density function of such a statistic is known. This paper gives general formulae for the density in each open interval between the characteristic roots of the matrix involved. Results are given for the case of distinct roots, which need not be assumed positive, and when the roots occur with multiplicities greater than one. Starting from a representation of the density as a surface integral over an (n − 2)-dimensional hyperplane, the density is expressed in terms of top-order zonal polynomials involving difference quotients of the characteristic roots of the matrix in the numerator quadratic form.

A new model of near integration is formulated in which the local to unity parameter is identifiable and consistently estimable with time series data. The properties of the model are investigated, new functional laws for near integrated time series are obtained that lead to mixed diffusion processes, and consistent estimators of the localizing parameter are constructed. The model provides a more complete interface between I(0) and I(1) models than the traditional local to unity model and leads to autoregressive coefficient estimates with rates of convergence that vary continuously between the O(√n) rate of stationary autoregression, the O(n) rate of unit root regression, and the power rate of explosive autoregression. Models with deterministic trends are also considered, least squares trend regression is shown to be efficient, and consistent estimates of the localizing parameter are obtained for this case also. Conventional unit root tests are shown to be consistent against local alternatives in the new class.

This paper presents asymptotic results for the seasonal unit root test proposed by Hylleberg, Engle, Granger and Yoo (1990, Journal of Econometrics 44, 215–238) in a near integration context. The findings are important in that they provide the asymptotic power functions of the Hylleberg et al. statistics when the characteristic roots of a seasonal process are local to unity. These conclusions extend the available asymptotic results for this test and serve as a framework for the potential development of more powerful test procedures.

This paper investigates the consistency of the least squares estimators and derives their limiting distributions in an AR(1) model with a single structural break of unknown timing. Let β1 and β2 be the preshift and postshift AR parameter, respectively. Three cases are considered: (i) |β1| < 1 and |β2| < 1; (ii) |β1| < 1 and β2 = 1; and (iii) β1 = 1 and |β2| < 1. Cases (ii) and (iii) are of particular interest but are rarely discussed in the literature. Surprising results are that, in both cases, regardless of the location of the change-point estimate, the unit root can always be consistently estimated and the residual sum of squares divided by the sample size converges to a discontinuous function of the change point. In case (iii), [circumflex over beta]2 does not converge to β2 whenever the change-point estimate is lower than the true change point. Further, the limiting distribution of the break-point estimator for shrinking break is asymmetric for case (ii), whereas those for cases (i) and (iii) are symmetric. The appropriate shrinking rate is found to be different in all cases.

This paper proposes nonparametric tests of change in the distribution function of a time series. The limiting null distributions of the test statistics depend on a nuisance parameter, and critical values cannot be tabulated a priori. To circumvent this problem, a new simulation-based statistical method is developed. The validity of our simulation procedure is established in terms of size, local power, and test consistency. The finite-sample properties of the proposed tests are evaluated in a set of Monte Carlo experiments, and the distributional stability in financial markets is examined.

We show that the standard consistent test for testing the null of conditional homoskedasticity (against conditional heteroskedasticity) can be generalized to a time-series regression model with weakly dependent data and with generated regressors. The test statistic is shown to have an asymptotic normal distribution under the null hypothesis of conditional homoskedastic error. We also discuss extension of our test to the case of testing the null of a parametrically specified conditional variance. We advocate using a bootstrap method to overcome the issue of slow convergence of this test statistic to its limiting distribution.

Let {Xt} follow a discrete Gaussian vector autoregression with deterministic components. We derive the exact finite-sample joint moment generating function (MGF) of the quadratic forms that form the basis for the sufficient statistic. The formula is then specialized to the limiting MGF of functionals involving multivariate and univariate Ornstein–Uhlenbeck processes, drifts, and time trends. Such processes arise asymptotically from more general non-Gaussian processes and also from the Gaussian {Xt} and have also been used in areas other than time series, such as the “goodness of fit” literature.

The range of correlation coefficient of any bivariate discrete random vector with finite or countably infinite values is derived. We show analytically that the normal-transformed discrete bivariate probability system of Van Ophem (1999, Econometric Theory 15, 228–237) has a flexible correlation coefficient. We establish the strictly monotonic relation of its correlation coefficient with the bivariate normal correlation-coefficient parameter in the system.

We prove in this paper the validity of an Edgeworth expansion to the joint distribution of the sample autocorrelations of a stationary Gaussian long memory process. The method of proof relies on a verification of the suitably modified conditions for the validity of a multivariate Edgeworth expansion of Durbin (1980, Biometrika 67, 311–333). A simulation study proves the expansion to be useful and accurate.