This paper gives new conditions for the functional central limit theorem, and weak convergence of stochastic integrals, for near-epoch-dependent functions of mixing processes. These results have fundamental applications in the theory of unit root testing and cointegrating regressions. The conditions given improve on existing results in the literature in terms of the amount of dependence and heterogeneity permitted, and in particular, these appear to be the first such theorems in which virtually the same assumptions are sufficient for both modes of convergence.

This paper derives a functional central limit theorem for the partial sums of fractionally integrated processes, otherwise known as I(d) processes for |d| < 1/2. Such processes have long memory, and the limit distribution is the so-called fractional Brownian motion, having correlated increments even asymptotically. The underlying shock variables may themselves exhibit quite general weak dependence by being near-epoch-dependent functions of mixing processes. Several weak convergence results for stochastic integrals having fractional integrands and weakly dependent integrators are also obtained. Taken together, these results permit I(p + d) integrands for any integer p ≥ 1.

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.

The classical definitions of GARCH-type processes rely on strong assumptions on the first two conditional moments. The common practice in empirical studies, however, has been to test for GARCH by detecting serial correlations in the squared regression errors. This can be problematic because such autocorrelation structures are compatible with severe misspecifications of the standard GARCH. Numerous examples are provided in the paper. In consequence, standard (quasi-) maximum likelihood procedures can be inconsistent if the conditional first two moments are misspecified. To alleviate these problems of possible misspecification, we consider weak GARCH representations characterized by an ARMA structure for the squared error terms. The weak GARCH representation eliminates the need for correct specification of the first two conditional moments. The parameters of the representation are estimated via two-stage least squares. The estimator is shown to be consistent and asymptotically normal. Forecasting issues are also addressed.

Consider the model y = x′β0 + f*(z) + ε, where ε [d over =] N(0, σ02). We calculate the smallest asymptotic variance that n1/2 consistent regular (n1/2CR) estimators of β0 can have when the only information we possess about f* is that it has a certain shape. We focus on three particular cases: (i) when f* is homogeneous of degree r, (ii) when f* is concave, (iii) when f* is decreasing. Our results show that in the class of all n1/2CR estimators of β0, homogeneity of f* may lead to substantial asymptotic efficiency gains in estimating β0. In contrast, at least asymptotically, concavity and monotonicity of f* do not help in estimating β0 more efficiently, at least for n1/2CR estimators of β0.

Likelihood ratio tests for restrictions on cointegrating vectors are asymptotically χ2 distributed. For some values of the parameters this asymptotic distribution does not give a good approximation to the finite sample distribution. In this paper we derive the Bartlett correction factor for the likelihood ratio test and show by some simulation experiments that it can be a useful tool for making inference.

This paper analyzes the limiting behavior of Dickey–Fuller t-tests when the true generating model is stationary around a broken linear trend. The cases of a break in level and a break in slope are considered separately and found to generate qualitatively different outcomes. In the asymptotic analysis, appropriate normalizations are applied to the break sizes. This leads to theoretical results that generate interesting predictions for sample sizes and break amounts of practical interest. Simulation evidence confirms the value of this approach to an asymptotic theory.