Statistical Analysis Of Mixture Vector Autoregressive Models

In this paper, we reconsider the mixture vector autoregressive model, which was proposed in the literature for modelling non‐linear time series. We complete and extend the stationarity conditions, derive a matrix formula in closed form for the autocovariance function of the process and prove a result on stable vector autoregressive moving‐average representations of mixture vector autoregressive models. For these results, we apply techniques related to a Markovian representation of vector autoregressive moving‐average processes. Furthermore, we analyse maximum likelihood estimation of model parameters by using the expectation–maximization algorithm and propose a new iterative algorithm for getting the maximum likelihood estimates. Finally, we study the model selection problem and testing procedures. Several examples, simulation experiments and an empirical application based on monthly financial returns illustrate the proposed procedures.     

动态面板阈模型的一种序贯两步估计

动态面板阈模型可以刻画经济变量动态调整过程的非对称性,在实证分析中有广泛的运用,但阈值参数的引入同时增加了参数估计的困难,理论上尚有许多问题没有解决。针对此类模型,本文提出了一种简单而实用的序贯两步估计方法,首先利用格点搜索获得阈值参数的一致估计,基于该参数对数据结构进行合理划分并引入不同类型的矩条件,然后利用广义矩方法获得自回归参数的估计。理论研究与模拟结果表明,序贯两步估计具有良好的大样本性质和有限样本表现;与现有文献的方法相比,序贯两步估计能够有效避免不同类型参数估计偏差的相互影响,减小估计量的偏差与均方根误差。     

Memory properties and aggregation of spatial autoregressive models

A random field displays long (resp. short) memory when its covariance function is absolutely non-summable (resp. summable), or alternatively when its spectral density (spectrum) is unbounded (resp. bounded) at some frequencies. Drawing on the spectrum approach, this paper characterizes both short and long memory features in the spatial autoregressive model. The data generating process is presented as a sequence of spatial autoregressive micro-relationships. The study elaborates the exact conditions under which short and long memories emerge for micro-relationships and for the aggregated field as well. To study the spectrum of the aggregated field, we develop a new general concept referred to as the ‘root order of a function’. This concept might be usefully applied in studying the convergence of some special integrals. We illustrate our findings with simulation experiments and an empirical application based on Gross Domestic Product data for 100 countries spanning over 1960–2004.     

Asymptotics of the weighted least squares estimation for AR(1) processes with applications to confidence intervals

For the first-order autoregressive model, we establish the asymptotic theory of the weighted least squares estimations whether the underlying autoregressive process is stationary, unit root, near integrated or even explosive under a weaker moment condition of innovations. The asymptotic limit of this estimator is always normal. It is shown that the empirical log-likelihood ratio at the true parameter converges to the standard chi-square distribution. An empirical likelihood confidence interval is proposed for interval estimations of the autoregressive coefficient. The results improve the corresponding ones of Chan et al. (Econ Theory 28:705–717, 2012). Some simulations are conducted to illustrate the proposed method.     

Production Frontiers and Panel Data

Value at risk (VaR) is the standard measure of market risk used by financial institutions. Interpreting the VaR as the quantile of future portfolio values conditional on current information, the conditional autoregressive value at risk (CAViaR) model specifies the evolution of the quantile over time using an autoregressive process and estimates the parameters with regression quantiles. Utilizing the criterion that each period the probability of exceeding the VaR must be independent of all the past information, we introduce a new test of model adequacy, the dynamic quantile test. Applications to real data provide empirical support to this methodology.     

A wavelet-based approach for modelling exchange rates

This paper proposes a new approach, based on the recent developments of the wavelet theory, to model the dynamic of the exchange rate. First, we consider the maximum overlap discrete wavelet transform (MODWT) to decompose the level exchange rates into several scales. Second, we focus on modelling the conditional mean of the detrended series as well as their volatilities. In particular, we consider the generalized fractional, one-factor, Gegenbauer process (GARMA) to model the conditional mean and the fractionally integrated generalized autoregressive conditional heteroskedasticity process (FIGARCH) to model the conditional variance. Moreover, we estimate the GARMA-FIGARCH model using the wavelet-based maximum likelihood estimator (Whitcher in Technometrics 46:225–238, 2004). To illustrate the usefulness of our methodology, we carry out an empirical application using the daily Tunisian exchange rates relative to the American Dollar, the Euro and the Japanese Yen. The empirical results show the relevance of the selected modelling approach which contributes to a better forecasting performance of the exchange rate series.     

Control charts for multivariate spatial autoregressive models

This paper deals with spatial detection of changes in model parameters of spatial autoregressive processes. The respective sequential testing problems are formulated. Moreover, we introduce characteristic quantities to monitor means or covariances of multivariate spatial autoregressive processes. Additionally, we also take into account the simultaneous surveillance of the mean vector and the covariance matrix. The aim is to apply control charts, important tools of sequential analysis, to these quantities. The considered control procedures are based on either cumulative sums or exponential smoothing. Further, we illustrate the methodology of statistical process control studying the spectrum of additive colors in a satellite photograph. Via simulation studies, the proposed control procedures are calibrated for a predefined average run length. In addition, we compare the performance of the control procedures considering the out-of-control situation. Eventually, the control charts are applied, and the signals of the different schemes are visualized. The final results are critically discussed.     

Estimation of the parameters of symmetric stable ARMA and ARMA–GARCH models

In this article, we first propose the modified Hannan–Rissanen Method for estimating the parameters of autoregressive moving average (ARMA) process with symmetric stable noise and symmetric stable generalized autoregressive conditional heteroskedastic (GARCH) noise. Next, we propose the modified empirical characteristic function method for the estimation of GARCH parameters with symmetric stable noise. Further, we show the efficiency, accuracy and simplicity of our methods with Monte-Carlo simulation. Finally, we apply our proposed methods to model the financial data.     

A novel and fast methodology for simultaneous multiple structural break estimation and variable selection for nonstationary time series models

A class of nonstationary time series such as locally stationary time series can be approximately modeled by piecewise stationary autoregressive (PSAR) processes. But the number and locations of the piecewise autoregressive segments, as well as the number of nonzero coefficients in each autoregressive process, are unknown. In this paper, by connecting the multiple structural break detection with a variable selection problem for a linear model with a large number of regression coefficients, a novel and fast methodology utilizing modern penalized model selection is introduced for detecting multiple structural breaks in a PSAR process. It also simultaneously performs variable selection for each autoregressive model and hence the order selection. To further its performance, an algorithm is given, which remains very fast in computation. Numerical results from simulation and a real data example show that the algorithm has excellent empirical performance.     

On control charts for monitoring the variance of a time series

In this paper we derive control charts for the variance of a Gaussian process using the likelihood ratio approach, the generalized likelihood ratio approach, the sequential probability ratio method and a generalized sequential probability ratio procedure, the Shiryaev–Roberts procedure and a generalized modified Shiryaev–Roberts approach. Recursive presentations for the calculation of the control statistics are given for autoregressive processes of order 1. In an extensive simulation study these schemes are compared with existing control charts for the variance. In order to asses the performance of the schemes both the average run length and the average delay are used.     

Analyzing multiple vector autoregressions through matrix-variate normal distribution with two covariance matrices

Abstract

This article proposes a new approach to analyze multiple vector autoregressive (VAR) models that render us a newly constructed matrix autoregressive (MtAR) model based on a matrix-variate normal distribution with two covariance matrices. The MtAR is a generalization of VAR models where the two covariance matrices allow the extension of MtAR to a structural MtAR analysis. The proposed MtAR can also incorporate different lag orders across VAR systems that provide more flexibility to the model. The estimation results from a simulation study and an empirical study on macroeconomic application show favorable performance of our proposed models and method.     

Multivariate autoregressive extreme value process and its application for modeling the time series properties of the extreme daily asset prices

Abstract

In this article we suggest a new multivariate autoregressive process for modeling time-dependent extreme value distributed observations. The idea behind the approach is to transform the original observations to latent variables that are univariate normally distributed. Then the vector autoregressive DCC model is fitted to the multivariate latent process. The distributional properties of the suggested model are extensively studied. The process parameters are estimated by applying a two-stage estimation procedure. We derive a prediction interval for future values of the suggested process. The results are applied in an empirically study by modeling the behavior of extreme daily stock prices.     

An autoregressive model based on the generalized hyperbolic distribution

We define a nonlinear autoregressive time series model based on the generalized hyperbolic distribution in an attempt to model time series with non-Gaussian features such as skewness and heavy tails. We show that the resulting process has a simple condition for stationarity and it is also ergodic. An empirical example with a forecasting experiment is presented to illustrate the features of the proposed model.     

A class of observation-driven random coefficient INAR(1) processes based on negative binomial thinning

Integer-valued time series models and their applications have attracted a lot of attention over the last years. In this paper, we introduce a class of observation-driven random coefficient integer-valued autoregressive processes based on negative binomial thinning, where the autoregressive parameter depends on the observed values of the previous moment. Basic probability and statistics properties of the process are established. The unknown parameters are estimated by the conditional least squares and empirical likelihood methods. Specially, we consider three aspects of the empirical likelihood method: maximum empirical likelihood estimate, confidence region and EL test. The performance of the two estimation methods is compared through simulation studies. Finally, an application to a real data example is provided.     

Analysis of a jump-diffusion option pricing model with serially correlated jump sizes

This paper extends the classical jump-diffusion option pricing model to incorporate serially correlated jump sizes which have been documented in recent empirical studies. We model the series of jump sizes by an autoregressive process and provide an analysis on the underlying stock return process. Based on this analysis, the European option price and the hedging parameters under the extended model are derived analytically. Through numerical examples, we investigate how the autocorrelation of jump sizes influences stock returns, option prices and hedging parameters, and demonstrate its effects on hedging portfolios and implied volatility smiles. A calibration example based on real market data is provided to show the advantage of incorporating the autocorrelation of jump sizes.     

Empirical Likelihood-based Inference for Stationary-ergodicity of the Generalized Random Coefficient Autoregressive Model

In this article, we consider the application of the empirical likelihood method to the generalized random coefficient autoregressive (GRCA) model. When the order of the model is 1, we derive an empirical likelihood ratio test statistic to test the stationary-ergodicity. Some simulation studies are also conducted to investigate the finite sample performances of the proposed test.     

Bootstrap entropy test for general location-scale time series models with heteroscedasticity

This study considers a goodness-of-fit test for location-scale time series models with heteroscedasticity, including a broad class of generalized autoregressive conditional heteroscedastic-type models. In financial time series analysis, the correct identification of model innovations is crucial for further inferences in diverse applications such as risk management analysis. To implement a goodness-of-fit test, we employ the residual-based entropy test generated from the residual empirical process. Since this test often shows size distortions and is affected by parameter estimation, its bootstrap version is considered. It is shown that the bootstrap entropy test is weakly consistent, and thereby its usage is justified. A simulation study and data analysis are conducted by way of an illustration.     

Specification testing in nonparametric AR‐ARCH models

In this paper, an autoregressive time series model with conditional heteroscedasticity is considered, where both conditional mean and conditional variance function are modeled nonparametrically. Tests for the model assumption of independence of innovations from past time series values are suggested. Tests based on weighted L²‐distances of empirical characteristic functions are considered as well as a Cramér–von Mises‐type test. The asymptotic distributions under the null hypothesis of independence are derived, and the consistency against fixed alternatives is shown. A smooth autoregressive residual bootstrap procedure is suggested, and its performance is shown in a simulation study.     

Some asymptotic properties in INAR(1) processes with Poisson marginals

The first-order integer-valued autoregressive (INAR(1)) process with Poisson marginal distributions is considered. It is shown that the sample autocovariance function of the model is asymptotically normally distributed. We derive asymptotic distribution of Yule-Walker type estimators of parameters. It turns out that our Yule-Walker type estimators are better than the conditional least squares estimators proposed by Klimko and Nelson (1978) and Al-Osh and Alzaid (1987). also, we study the relationship between the model andM/M/∞ queueing system.