Testing for changes in autocovariances of nonparametric time series models

In the literature on change-point analysis, much attention has been paid to detecting changes in certain marginal characteristics, such as mean, variance, and marginal distribution. For time series data with nonparametric time trend, we study the change-point problem for the autocovariance structure of the unobservable error process. To derive the asymptotic distribution of the cumulative sum test statistic, we develop substantial theory for uniform convergence of weighted partial sums and weighted quadratic forms. Our asymptotic results improve upon existing works in several important aspects. The performance of the test statistic is examined through simulations and an application to interest rates data.     

Central Limit Theorems of Local Polynomial Threshold Estimator for Diffusion Processes with Jumps

Central limit theorems play an important role in the study of statistical inference for stochastic processes. However, when the non‐parametric local polynomial threshold estimator, especially local linear case, is employed to estimate the diffusion coefficients of diffusion processes, the adaptive and predictable structure of the estimator conditionally on the σ ‐field generated by diffusion processes is destroyed, so the classical central limit theorem for martingale difference sequences cannot work. In high‐frequency data, we proved the central limit theorems of local polynomial threshold estimators for the volatility function in diffusion processes with jumps by Jacod's stable convergence theorem. We believe that our proof procedure for local polynomial threshold estimators provides a new method in this field, especially in the local linear case.     

Flexible Threshold Models for Modelling Interest Rate Volatility

This paper focuses on interest rate models with regime switching and extends previous nonlinear threshold models by relaxing the assumption of a fixed number of regimes. Instead we suggest automatic model determination through Bayesian inference via the reversible jump Markov Chain Monte Carlo (MCMC) algorithm. Moreover, we allow the thresholds in the volatility to be driven not only by the interest rate but also by other economic factors. We illustrate our methodology by applying it to interest rates and other economic factors of the American economy.     

Flexible Threshold Models for Modelling Interest Rate Volatility

This paper focuses on interest rate models with regime switching and extends previous nonlinear threshold models by relaxing the assumption of a fixed number of regimes. Instead we suggest automatic model determination through Bayesian inference via the reversible jump Markov Chain Monte Carlo (MCMC) algorithm. Moreover, we allow the thresholds in the volatility to be driven not only by the interest rate but also by other economic factors. We illustrate our methodology by applying it to interest rates and other economic factors of the American economy.     

Threshold Estimation via Group Orthogonal Greedy Algorithm

A threshold autoregressive (TAR) model is an important class of nonlinear time series models that possess many desirable features such as asymmetric limit cycles and amplitude-dependent frequencies. Statistical inference for the TAR model encounters a major difficulty in the estimation of thresholds, however. This article develops an efficient procedure to estimate the thresholds. The procedure first transforms multiple-threshold detection to a regression variable selection problem, and then employs a group orthogonal greedy algorithm to obtain the threshold estimates. Desirable theoretical results are derived to lend support to the proposed methodology. Simulation experiments are conducted to illustrate the empirical performances of the method. Applications to U.S. GNP data are investigated.     

Non Parametric Estimation of Second-Order Diffusion Equation by Using Asymmetric Kernels

In this paper, we study the non parametric estimation of drift coefficient and diffusion coefficient in the second-order diffusion equation by using the asymmetric kernel functions, based on the difference of discrete time observations. The basic idea relies upon replacing the symmetric kernel by asymmetric kernel and provides a new way of obtaining the non parametric estimation for second-order diffusion equation. Under the appropriate assumptions, we prove that the proposed estimators of second-order diffusion equation are consistent and asymptotically follow normal distribution.     

Predictability of Interest Rates and Interest-Rate Portfolios

Due to the near unit-root behavior of interest rates, changes in individual interest-rate series are difficult to forecast. We propose an innovative way of applying dynamic term structure models to predict future changes in interest-rate portfolios. Instead of directly forecasting the movements based on the estimated factor dynamics, we use the dynamic term structure model as a decomposition tool and decompose each interest-rate series into two components: a persistent component captured by the dynamic factors, and a strongly mean-reverting component given by the pricing residuals of the model. With this decomposition, we form interest-rate portfolios that are first-order neutral to the persistent dynamic factors, but are exposed to the strongly mean-reverting residuals. We show that the predictability on the changes of these interest-rate portfolios is significant both statistically and economically. We explore the implications of the predictability in future interest-rate modeling.     

基于非线性ECM模型的贝叶斯门限协整研究

 现有门限协整检验方法由于模型似然函数具有多峰、不连续特征,导致冗余参数识别存在困难,最优化计算相对复杂。本文提出基于非线性误差修正模型的贝叶斯门限协整分析,结合参数的后验条件分布设计MCMC抽样方案,进行贝叶斯门限协整检验;并利用Monte Carlo仿真研究了贝叶斯门限协整检验的有限样本性质,发现贝叶斯门限协整检验方法具有良好的有限样本性质。同时,利用不同期限的美国利率序列进行了实证研究,结果发现1个月与3个月利率之间、3个月与6个月利率之间以及3个月与1年利率之间均存在门限协整关系。研究结果表明：贝叶斯门限协整检验方法解决了冗余参数识别的难题,使计算变得相对简单,并提高了估计的精确度和检验的准确性。     

Estimation of an Ergodic Diffusion from Discrete Observations

We consider a one-dimensional diffusion process X , with ergodic property, with drift b ( x , θ) and diffusion coefficient a ( x , σ) depending on unknown parameters θ and σ. We are interested in the joint estimation of (θ, σ). For that purpose, we dispose of a discretized trajectory, observed at n equidistant times tⁿ_i = ih_n , 1 ≤ i ≤ n . We assume that h_n ← 0 and nh_n ←∞. Under the condition nh_n^p ← 0 for an arbitrary integer p , we exhibit a contrast dependent on p which provides us with an asymptotically normal and efficient estimator of (θ, σ).     

Simple and Explicit Estimating Functions for a Discretely Observed Diffusion Process

We consider a one-dimensional diffusion process X , with ergodic property, with drift b ( x , θ) and diffusion coefficient a ( x , θ) depending on an unknown parameter θ that may be multidimensional. We are interested in the estimation of θ and dispose, for that purpose, of a discretized trajectory, observed at n equidistant times t_i = iΔ , i = 0, ..., n . We study a particular class of estimating functions of the form ∑ f (θ, X _{t i −1}) which, under the assumption that the integral of f with respect to the invariant measure is null, provide us with a consistent and asymptotically normal estimator. We determine the choice of f that yields the estimator with minimum asymptotic variance within the class and indicate how to construct explicit estimating functions based on the generator of the diffusion. Finally the theoretical study is completed with simulations.     

Testing for Serial Independence: Beyond the Portmanteau Approach

Portmanteau tests are typically used to test serial independence even if, by construction, they are generally powerful only in presence of pairwise dependence between lagged variables. In this article, we present a simple statistic defining a new serial independence test, which is able to detect more general forms of dependence. In particular, differently from the Portmanteau tests, the resulting test is powerful also under a dependent process characterized by pairwise independence. A diagram, based on p-values from the proposed test, is introduced to investigate serial dependence. Finally, the effectiveness of the proposal is evaluated in a simulation study and with an application on financial data. Both show that the new test, used in synergy with the existing ones, helps in the identification of the true data-generating process. Supplementary materials for this article are available online.     

Computational Aspects Related to Martingale Estimating Functions for a Discretely Observed Diffusion

Martingale estimating functions for a discretely observed diffusion have turned out to provide estimators with nice asymptotic properties. However, their expression usually involves some conditional expectation that has to be evaluated through Monte Carlo simulations giving rise to an approximated estimator. In this work we study, for ergodic models, the asymptotic properties of the approximated estimator and describe the influence of the number of independent simulated trajectories involved in the Monte Carlo method as well as of the approximation scheme used. Our results are of practical relevance to assess the implementation of martingale estimating functions for discretely observed diffusions.     

Automatic Specification Testing for Vector Autoregressions and Multivariate Nonlinear Time Series Models

This article introduces an automatic test for the correct specification of a vector autoregression (VAR) model. The proposed test statistic is a Portmanteau statistic with an automatic selection of the order of the residual serial correlation tested. The test presents several attractive characteristics: simplicity, robustness, and high power in finite samples. The test is simple to implement since the researcher does not need to specify the order of the autocorrelation tested and the proposed critical values are simple to approximate, without resorting to bootstrap procedures. In addition, the test is robust to the presence of conditional heteroscedasticity of unknown form and accounts for estimation uncertainty without requiring the computation of large-dimensional inverses of near-to-singularity covariance matrices. The basic methodology is extended to general nonlinear multivariate time series models. Simulations show that the proposed test presents higher power than the existing ones for models commonly employed in empirical macroeconomics and empirical finance. Finally, the test is applied to the classical bivariate VAR model for GNP (gross national product) and unemployment of Blanchard and Quah (1989 Blanchard, O. and Quah, D. 1989. “The Dynamic Effects of Aggregate Demand and Supply Disturbances,”. The American Economic Review, 79: 655–673. [Web of Science ®] , [Google Scholar]) and Evans (1989 Evans, G. W. 1989. “Output and Unemployment Dynamics in the United States: 1950–1985,”. Journal of Applied Econometrics, 4: 213–238. [Crossref], [Web of Science ®] , [Google Scholar]). Online supplementary material includes proofs and additional details.     

A Simulation-Based Specification Test for Diffusion Processes

This article makes two contributions. First, we outline a simple simulation-based framework for constructing conditional distributions for multifactor and multidimensional diffusion processes, for the case where the functional form of the conditional density is unknown. The distributions can be used, for example, to form predictive confidence intervals for time period t + τ, given information up to period t. Second, we use the simulation-based approach to construct a test for the correct specification of a diffusion process. The suggested test is in the spirit of the conditional Kolmogorov test of Andrews. However, in the present context the null conditional distribution is unknown and is replaced by its simulated counterpart. The limiting distribution of the test statistic is not nuisance parameter-free. In light of this, asymptotically valid critical values are obtained via appropriate use of the block bootstrap. The suggested test has power against a larger class of alternatives than tests that are constructed using marginal distributions/densities. The findings of a small Monte Carlo experiment underscore the good finite sample properties of the proposed test, and an empirical illustration underscores the ease with which the proposed simulation and testing methodology can be applied.     

Estimating and Testing Nonlinear Local Dependence Between Two Time Series

ABSTRACT

The most common measure of dependence between two time series is the cross-correlation function. This measure gives a complete characterization of dependence for two linear and jointly Gaussian time series, but it often fails for nonlinear and non-Gaussian time series models, such as the ARCH-type models used in finance. The cross-correlation function is a global measure of dependence. In this article, we apply to bivariate time series the nonlinear local measure of dependence called local Gaussian correlation. It generally works well also for nonlinear models, and it can distinguish between positive and negative local dependence. We construct confidence intervals for the local Gaussian correlation and develop a test based on this measure of dependence. Asymptotic properties are derived for the parameter estimates, for the test functional and for a block bootstrap procedure. For both simulated and financial index data, we construct confidence intervals and we compare the proposed test with one based on the ordinary correlation and with one based on the Brownian distance correlation. Financial indexes are examined over a long time period and their local joint behavior, including tail behavior, is analyzed prior to, during and after the financial crisis. Supplementary material for this article is available online.     

Testing Periodicity in Short Series and Application to Gene Expression Data

Testing for periodicity in microarray time series encounters the challenges of short series length, missing values and presence of non-Fourier frequencies. In this article, a test method for such series has been proposed. The method is completely simulation based and finds p-values for test of periodicity through fitting Pearson Type VI distribution. The simulation results compare and reveal the excellence of this method over Fisher's g test for varying series length, frequencies, and error variance. This approach is applied to Caulobacter crescentus cell cycle data in order to demonstrate the practical performance.     

Bayesian sequential inference for nonlinear multivariate diffusions

In this paper, we adapt recently developed simulation-based sequential algorithms to the problem concerning the Bayesian analysis of discretely observed diffusion processes. The estimation framework involves the introduction of m−1 latent data points between every pair of observations. Sequential MCMC methods are then used to sample the posterior distribution of the latent data and the model parameters on-line. The method is applied to the estimation of parameters in a simple stochastic volatility model (SV) of the U.S. short-term interest rate. We also provide a simulation study to validate our method, using synthetic data generated by the SV model with parameters calibrated to match weekly observations of the U.S. short-term interest rate.     

Statistical inference for some Volterra population processes in a random environment

We study the problem of maximum-likelihood estimation in some random-environment population models, defined through nonlinear ltǒ stochastic differential equations. It is shown that a criticality parameter can be estimated consistently. The asymptotic behavior of the estimators is analyzed, and a goodness-of-fit test is proposed.