Tests for structural breaks in memory parameters of long-memory heterogeneous autoregressive models

Tests for structural breaks in the coefficients of the long-memory heterogeneous autoregressive (HAR) models are developed. The tests are based on the partial sum process of the normalized efficient score vector. The tests have the nice property of identifying the parameters of the daily, weekly, and monthly regressors in which breaks occur. Limiting null distributions of the proposed tests are proven to be derived from standard Brownian bridges. A finite sample Monte-Carlo experiment shows reasonable size and power properties of the proposed tests. The proposed method is illustrated by a real data analysis.     

Vector error correction heterogeneous autoregressive forecast model of realized volatility and implied volatility

A vector error correction model is proposed for forecasting realized volatility which takes advantage of the cointegration relation between realized volatility and implied volatility. The model is constructed by adding a cointegration error term to a vector-and-unit-root version of the heterogeneous autoregressive (HAR) model of Corsi (2009 Corsi, F. 2009. A simple approximate long-memory model of realized volatility. Journal of Financial Econometrics 7 (2):174–96. [Google Scholar]). The proposed model is easier to implement, extend, and interpret than fractional cointegration models. A Monte Carlo simulation and real data analysis reveal advantages of the proposed model over other existing models of Corsi (2009 Corsi, F. 2009. A simple approximate long-memory model of realized volatility. Journal of Financial Econometrics 7 (2):174–96. [Google Scholar]), Busch Christensen and Nielsen (2011 Busch, T., B. J. Christensen, and M. Nielsen. 2011. The role of implied volatility in forecasting future realized volatility and jumps in foreign exchange, stock, and bond markets. Journal of Econometrics 160 (1):48–57. [Google Scholar]), Cho and Shin (2016 Cho, S. J. and D. W. Shin. 2016. An integrated heteroscedastic autoregressive model for forecasting long-memory volatilities. Journal of the Korean Statistical Society, 45:371–380. [Google Scholar]), and Bollerslev Patton, and Quaedvlieg (2016 Bollerslev, T., A. J. Patton, and R. Quaedvlieg. 2016. Exploiting the errors:A simple approach for improved volatility forecasting. Journal of Econometrics 192:1-18. [Google Scholar]).     

An empirical Bayesian forecast in the threshold stochastic volatility models

In the area of finance, the stochastic volatility (SV) model is a useful tool for modelling stock market returns. However, there is evidence that asymmetric behaviour of stock returns exists. A threshold SV (THSV) model is provided to capture this behaviour. In this study, we introduce a robust model created through empirical Bayesian analysis to deal with the uncertainty between the SV and THSV models. A Markov chain Monte Carlo algorithm is applied to empirically select the hyperparameters of the prior distribution. Furthermore, the value at risk from the resulting predictive distribution is also given. Simulation studies show that the proposed empirical Bayes model not only clarifies the acceptability of prediction but also reduces the risk of model uncertainty.     

A nonparametric test of conditional autoregressive heteroscedasticity for threshold autoregressive models

Threshold autoregressive models are widely used in time‐series applications. When building or using such a model, it is important to know whether conditional heteroscedasticity exists. The authors propose a nonparametric test of this hypothesis. They develop the large‐sample theory of a test of nonlinear conditional heteroscedasticity adapted to nonlinear autoregressive models and study its finite‐sample properties through simulations. They also provide percentage points for carrying out this test, which is found to have very good power overall.     

On periodic autoregressive stochastic volatility models: structure and estimation

To capture both the volatility evolution and the periodicity feature in the autocorrelation structure exhibited by many nonlinear time series, a Periodic AutoRegressive Stochastic Volatility (PAR-SV ) model is proposed. Some probabilistic properties, namely the strict and second-order periodic stationarity, are provided. Furthermore, conditions for the existence of higher-order moments are established. The autocovariance structure of the squares and higher order powers of the PAR-SV process is studied. Its dynamic properties are shown to be consistent with financial time series empirical findings. Ways in which the model may be estimated are discussed. Finally, a simulation study of the performance of the proposed estimation methods is provided and the PAR-SV is applied to model the spot rates of the euro and US dollar both against the Algerian dinar. The empirical analysis shows that the proposed PAR-SV model can be considered as a viable alternative to the periodic generalized autoregressive conditionally heteroscedastic (PGARCH) model.     

Evaluation of realized volatility predictions from models with leptokurtically and asymmetrically distributed forecast errors

Accurate volatility forecasting is a key determinant for portfolio management, risk management and economic policy. The paper provides evidence that the sum of squared standardized forecast errors is a reliable measure for model evaluation when the predicted variable is the intra-day realized volatility. The forecasting evaluation is valid for standardized forecast errors with leptokurtic distribution as well as with leptokurtic and asymmetric distributions. Additionally, the widely applied forecasting evaluation function, the predicted mean-squared error, fails to select the adequate model in the case of models with residuals that are leptokurtically and asymmetrically distributed. Hence, the realized volatility forecasting evaluation should be based on the standardized forecast errors instead of their unstandardized version.     

Bayesian analysis of multivariate threshold autoregressive models with missing data

In some fields, we are forced to work with missing data in multivariate time series. Unfortunately, the data analysis in this context cannot be carried out in the same way as in the case of complete data. To deal with this problem, a Bayesian analysis of multivariate threshold autoregressive models with exogenous inputs and missing data is carried out. In this paper, Markov chain Monte Carlo methods are used to obtain samples from the involved posterior distributions, including threshold values and missing data. In order to identify autoregressive orders, we adapt the Bayesian variable selection method in this class of multivariate process. The number of regimes is estimated using marginal likelihood or product parameter-space strategies.     

A new generalization of skew-T distribution with volatility models

In this paper, we propose a new generalized alpha-skew-T (GAST) distribution for generalized autoregressive conditional heteroskedasticity (GARCH) models in modelling daily Value-at-Risk (VaR). Some mathematical properties of the proposed distribution are derived including density function, moments and stochastic representation. The maximum likelihood estimation method is discussed to estimate parameters via a simulation study. Then, the real data application on S&P-500 index is performed to investigate the performance of GARCH models specified under GAST innovation distribution with respect to normal, Student's-t and Skew-T models in terms of the VaR accuracy. Backtesting methodology is used to compare the out-of-sample performance of the VaR models. The results show that GARCH models with GAST innovation distribution outperforms among others and generates the most conservative VaR forecasts for all confidence levels and for both long and short positions.     

Kalman filter-based modelling and forecasting of stochastic volatility with threshold

We propose a parametric nonlinear time-series model, namely the Autoregressive-Stochastic volatility with threshold (AR-SVT) model with mean equation for forecasting level and volatility. Methodology for estimation of parameters of this model is developed by first obtaining recursive Kalman filter time-update equation and then employing the unrestricted quasi-maximum likelihood method. Furthermore, optimal one-step and two-step-ahead out-of-sample forecasts formulae along with forecast error variances are derived analytically by recursive use of conditional expectation and variance. As an illustration, volatile all-India monthly spices export during the period January 2006 to January 2012 is considered. Entire data analysis is carried out using EViews and matrix laboratory (MATLAB) software packages. The AR-SVT model is fitted and interval forecasts for 10 hold-out data points are obtained. Superiority of this model for describing and forecasting over other competing models for volatility, namely AR-Generalized autoregressive conditional heteroscedastic, AR-Exponential GARCH, AR-Threshold GARCH, and AR-Stochastic volatility models is shown for the data under consideration. Finally, for the AR-SVT model, optimal out-of-sample forecasts along with forecasts of one-step-ahead variances are obtained.     

Detection of jump location curve in spatial linear regression model with two-dimensional threshold

In the context of spatial linear regression, we discuss detection of jump location curve treated as threshold curve which cannot be expressed by independent variables but indirectly determines two specific model forms. The threshold curve in this paper is described by a straight line with two location variables, longitude and latitude, and can be estimated by maximizing the coefficient difference between two one-sided linear regression models. Theoretical results show that the estimator is consistent. Our method performs well by numerical studies.     

Multivariate stochastic volatility with Bayesian dynamic linear models

This paper develops a Bayesian procedure for estimation and forecasting of the volatility of multivariate time series. The foundation of this work is the matrix-variate dynamic linear model, for the volatility of which we adopt a multiplicative stochastic evolution, using Wishart and singular multivariate beta distributions. A diagonal matrix of discount factors is employed in order to discount the variances element by element and therefore allowing a flexible and pragmatic variance modelling approach. Diagnostic tests and sequential model monitoring are discussed in some detail. The proposed estimation theory is applied to a four-dimensional time series, comprising spot prices of aluminium, copper, lead and zinc of the London metal exchange. The empirical findings suggest that the proposed Bayesian procedure can be effectively applied to financial data, overcoming many of the disadvantages of existing volatility models.     

Continuous threshold models with two-way interactions in survival analysis

Proportional hazards model with the biomarker–treatment interaction plays an important role in the survival analysis of the subset treatment effect. A threshold parameter for a continuous biomarker variable defines the subset of patients who can benefit or lose from a certain new treatment. In this article, we focus on a continuous threshold effect using the rectified linear unit and propose a gradient descent method to obtain the maximum likelihood estimation of the regression coefficients and the threshold parameter simultaneously. Under certain regularity conditions, we prove the consistency, asymptotic normality and provide a robust estimate of the covariance matrix when the model is misspecified. To illustrate the finite sample properties of the proposed methods, we simulate data to evaluate the empirical biases, the standard errors and the coverage probabilities for both the correctly specified models and misspecified models. The proposed continuous threshold model is applied to a prostate cancer data with serum prostatic acid phosphatase as a biomarker.     

Self-exciting threshold models for time series of counts with a finite range

In this article, an integer-valued self-exciting threshold model with a finite range based on the binomial INARCH(1) model is proposed. Important stochastic properties are derived, and approaches for parameter estimation are discussed. A real-data example about the regional spread of public drunkenness in Pittsburgh demonstrates the applicability of the new model in comparison to existing models. Feasible modifications of the model are presented, which are designed to handle special features such as zero-inflation.