For a certain class of martingales, convergence to a mixture of normal distributions is established under convergence in distribution for the conditional variance. This is less restrictive in comparison with the classical martingale limit theorem, where one generally requires convergence in probability. The extension partially removes a barrier in the applications of the classical martingale limit theorem to nonparametric estimation and inference with nonstationarity and enhances the effectiveness of the classical martingale limit theorem as one of the main tools to investigate asymptotics in statistics, econometrics, and other fields. The main result is applied to investigate limit behavior of the conventional kernel estimator in a nonlinear cointegrating regression model, which improves existing works in the literature.

We consider pure-jump transaction-level models for asset prices in continuous time, driven by point processes. In a bivariate model that admits cointegration, we allow for time deformations to account for such effects as intraday seasonal patterns in volatility and nontrading periods that may be different for the two assets. We also allow for asymmetries (leverage effects). We obtain the asymptotic distribution of the log-price process. For the weak fractional cointegration case, we obtain the asymptotic distribution of the ordinary least squares estimator of the cointegrating parameter based on data sampled from an equally spaced discretization of calendar time, and we justify a feasible method of hypothesis testing for the cointegrating parameter based on the corresponding t-statistic. In the strong fractional cointegration case, we obtain the limiting distribution of a continuously averaged tapered estimator as well as other estimators of the cointegrating parameter, and we find that the rate of convergence can be affected by properties of intertrade durations. In particular, the persistence of durations (hence of volatility) can affect the degree of cointegration. We also obtain the rate of convergence of several estimators of the cointegrating parameter in the standard cointegration case. Finally, we consider the properties of the ordinary least squares estimator of the regression parameter in a spurious regression, i.e., in the absence of cointegration.

When estimating integrated volatilities based on high-frequency data, simplifying assumptions are usually imposed on the relationship between the observation times and the price process. In this paper, we establish a central limit theorem for the realized volatility in a general endogenous time setting. We also establish a central limit theorem for the tricity under the hypothesis that there is no endogeneity, based on which we propose a test and document that this endogeneity is present in financial data.

This paper considers Bayesian nonparametric estimation of conditional densities by countable mixtures of location-scale densities with covariate dependent mixing probabilities. The mixing probabilities are modeled in two ways. First, we consider finite covariate dependent mixture models, in which the mixing probabilities are proportional to a product of a constant and a kernel and a prior on the number of mixture components is specified. Second, we consider kernel stick-breaking processes for modeling the mixing probabilities. We show that the posterior in these two models is weakly and strongly consistent for a large class of data-generating processes. A simulation study conducted in the paper demonstrates that the models can perform well in small samples.

This paper considers a binary response model with a partially linear latent equation, where ϕ is an unknown function and β is a finite-dimensional parameter of interest. Using the principle of smoothed maximum score estimation (Horowitz, 1992; Econometrica 60(3), 505–531), a consistent and asymptotically normal (C.A.N.)estimator for β is proposed under the restriction that the median of the error conditional on the covariates is equal to 0. Furthermore, the rate of convergence in probability is close to the parametric rate, if certain functions admit enough derivatives. This method neither restricts the form of heteroskedasticity in the error term nor suffers from the curse of dimensionality whenever ϕ is multivariate. Some Monte Carlo experiments suggest that this estimator performs well compared with conventional estimators.

This paper develops a general procedure to check the bootstrap validity in M-estimation. We apply the procedure in discontinuous threshold regression to show the inconsistency of the nonparametric bootstrap for inference on the threshold point. Especially, the conditional weak limit of the nonparametric bootstrap is shown not to exist. By comparing with two other boundaries in the literature, we show the fact that the threshold point is a boundary of the covariate that makes its bootstrap inference so different. The remedies to the bootstrap failure in the literature are summarized, and the nonparametric posterior interval is suggested by some simulation studies.