Bayesian median autoregression for robust time series forecasting

We develop a Bayesian median autoregressive (BayesMAR) model for time series forecasting. The proposed method utilizes time-varying quantile regression at the median, favorably inheriting the robustness of median regression in contrast to the widely used mean-based methods. Motivated by a working Laplace likelihood approach in Bayesian quantile regression, BayesMAR adopts a parametric model bearing the same structure as autoregressive models by altering the Gaussian error to Laplace, leading to a simple, robust, and interpretable modeling strategy for time series forecasting. We estimate model parameters by Markov chain Monte Carlo. Bayesian model averaging is used to account for model uncertainty, including the uncertainty in the autoregressive order, in addition to a Bayesian model selection approach. The proposed methods are illustrated using simulations and real data applications. An application to U.S. macroeconomic data forecasting shows that BayesMAR leads to favorable and often superior predictive performance compared to the selected mean-based alternatives under various loss functions that encompass both point and probabilistic forecasts. The proposed methods are generic and can be used to complement a rich class of methods that build on autoregressive models.     

Forecasting unstable and nonstationary time series

Many time series are asymptotically unstable and intrinsically nonstationary, i.e. satisfy difference equations with roots greater than one (in modulus) and with time-varying parameters. Models developed by Box–Jenkins solve these problems by imposing on data two transformations: differencing (unit-roots) and exponential (Box–Cox). Owing to the Jensen inequality, these techniques are not optimal for forecasting and sometimes may be arbitrary. This paper develops a method for modeling time series with unstable roots and changing parameters. In particular, the effectiveness of recursive estimators in tracking time-varying unstable parameters is shown with applications to data-sets of Box–Jenkins. The method is useful for forecasting time series with trends and cycles whose pattern changes over time.     

基于Sieve Bootstrap方法的季节时间序列模型检验问题研究

研究目标：完善季节时间序列模型建模理论,解决建模过程烦琐、各类检验方法的结论差异大以及模型误设定问题。研究方法：基于对各季节时间序列模型的数理分析及比较,提出合理的模型检验程序;再运用Sieve Bootstrap方法,给出季节性单位根检验及确定性季节过程检验的统计量的临界值,并比较基于Sieve Bootstrap的检验方法与HEGY检验、BT检验的异同。研究发现：本文提出的检验程序能有效识别模型,检验统计量有限样本性质优良;实证分析表明,本文提出的检验程序及方法能更有效地识别中国宏观经济数据中的季节性。研究创新：将Sieve Bootstrap方法应用于季节时间序列的平稳性检验及趋势性检验中。研究价值：提出季节时间序列模型检验程序及检验方法,促进其在季节性经济数据中的应用。     

Modeling time-varying higher-order conditional moments: A survey

Since the introduction of the Autoregressive Conditional Heteroscedasticity (ARCH) model, the literature on modeling the time-varying second-order conditional moment has become increasingly popular in the last four decades. Its popularity is partly due to its success in capturing volatility in financial time series, which is useful for modeling and predicting risk for financial assets. A natural extension of this is to model time variation in higher-order conditional moments, such as the third and fourth moments, which are related to skewness and kurtosis (tail risk). This leads to an emerging literature on time-varying higher-order conditional moments in the last two decades. This paper outlines recent developments in modeling time-varying higher-order conditional moments in the economics and finance literature. Using the Generalized Autoregressive Conditional Heteroscedasticity (GARCH) framework as a foundation, this paper provides an overview of the two most common approaches for modeling time-varying higher-order conditional moments: autoregressive conditional density (ARCD) and autoregressive conditional moment (ARCM). The discussion covers both the theoretical and empirical aspects of the literature. This includes the identification of the associated skewness–kurtosis domain by using the solutions to the classical moment problems, the structural and statistical properties of the models used to model the higher-order conditional moments and the computational challenges in estimating these models. We also advocate the use of a maximum entropy density (MED) as an alternative method, which circumvents some of the issues prevalent in these common approaches.     

On fitting distributed lag models subject to polynomial restrictions

A methodology is presented for fitting distributed lag models with polynomial restrictions on the lag coefficients. The model incorporates autoregressive residuals. Orthogonal methods are employed so that the procedures are numerically sound. Furthermore, these have the effect of allowing for inference to be made about the three integer parameters in the model: (i) the length of the lag, (ii) the degree of the polynomial, and (iii) the order of the autoregression. The methodology is applied to the extended Almon data. This analysis suggests that estimation of polynomial distributed lags is highly sensitive to autoregressive disturbances. This underlines the importance of modeling the disturbances.     

Semiparametric time series regression modeling with a diverging number of parameters

Variable selection and error structure determination of a partially linear model with time series errors are important issues. In this paper, we investigate the regression coefficient and autoregressive order shrinkage and selection via the smoothly clipped absolute deviation penalty for a partially linear model with a divergent number of covariates and finite order autoregressive time series errors. Both consistency and asymptotic normality of the proposed penalized estimators are derived. The oracle property of the resultant estimators is proved. Simulation studies are carried out to assess the finite‐sample performance of the proposed procedure. A real data analysis is made to illustrate the usefulness of the proposed procedure as well.     

Model identification for infinite variance autoregressive processes

We consider model identification for infinite variance autoregressive time series processes. It is shown that a consistent estimate of autoregressive model order can be obtained by minimizing Akaike’s information criterion, and we use all-pass models to identify noncausal autoregressive processes and estimate the order of noncausality (the number of roots of the autoregressive polynomial inside the unit circle in the complex plane). We examine the performance of the order selection procedures for finite samples via simulation, and use the techniques to fit a noncausal autoregressive model to stock market trading volume data.     

Ordinal-response GARCH models for transaction data: A forecasting exercise

We use numerous high-frequency transaction data sets to evaluate the forecasting performances of several dynamic ordinal-response time series models with generalized autoregressive conditional heteroscedasticity (GARCH). The specifications account for three components: leverage effects, in-mean effects and moving average error terms. We estimate the model parameters by developing Markov chain Monte Carlo algorithms. Our empirical analysis shows that the proposed ordinal-response GARCH models achieve better point and density forecasts than standard benchmarks.     

Dynamic spatial autoregressive models with autoregressive and heteroskedastic disturbances

We propose a new class of models specifically tailored for spatiotemporal data analysis. To this end, we generalize the spatial autoregressive model with autoregressive and heteroskedastic disturbances, that is, SARAR(1, 1), by exploiting the recent advancements in score‐driven (SD) models typically used in time series econometrics. In particular, we allow for time‐varying spatial autoregressive coefficients as well as time‐varying regressor coefficients and cross‐sectional standard deviations. We report an extensive Monte Carlo simulation study in order to investigate the finite‐sample properties of the maximum likelihood estimator for the new class of models as well as its flexibility in explaining a misspecified dynamic spatial dependence process. The new proposed class of models is found to be economically preferred by rational investors through an application to portfolio optimization.     

Modeling and forecasting of stock index volatility with APARCH models under ordered restriction

This article examines volatility models for modeling and forecasting the Standard & Poor 500 (S&P 500) daily stock index returns, including the autoregressive moving average, the Taylor and Schwert generalized autoregressive conditional heteroscedasticity (GARCH), the Glosten, Jagannathan and Runkle GARCH and asymmetric power ARCH (APARCH) with the following conditional distributions: normal, Student's t and skewed Student's t‐distributions. In addition, we undertake unit root (augmented Dickey–Fuller and Phillip–Perron) tests, co‐integration test and error correction model. We study the stationary APARCH (p) model with parameters, and the uniform convergence, strong consistency and asymptotic normality are prove under simple ordered restriction. In fitting these models to S&P 500 daily stock index return data over the period 1 January 2002 to 31 December 2012, we found that the APARCH model using a skewed Student's t‐distribution is the most effective and successful for modeling and forecasting the daily stock index returns series. The results of this study would be of great value to policy makers and investors in managing risk in stock markets trading.     

POSTERIOR‐PREDICTIVE EVIDENCE ON US INFLATION USING EXTENDED NEW KEYNESIAN PHILLIPS CURVE MODELS WITH NON‐FILTERED DATA

Changing time series properties of US inflation and economic activity, measured as marginal costs, are modeled within a set of extended New Keynesian Phillips curve (NKPC) models. It is shown that mechanical removal or modeling of simple low‐frequency movements in the data may yield poor predictive results which depend on the model specification used. Basic NKPC models are extended to include structural time series models that describe typical time‐varying patterns in levels and volatilities. Forward‐ and backward‐looking expectation components for inflation are incorporated and their relative importance is evaluated. Survey data on expected inflation are introduced to strengthen the information in the likelihood. Use is made of simulation‐based Bayesian techniques for the empirical analysis. No credible evidence is found on endogeneity and long‐run stability between inflation and marginal costs. Backward‐looking inflation appears stronger than forward‐looking inflation. Levels and volatilities of inflation are estimated more precisely using rich NKPC models. The extended NKPC structures compare favorably with existing basic Bayesian vector autoregressive and stochastic volatility models in terms of fit and prediction. Tails of the complete predictive distributions indicate an increase in the probability of deflation in recent years. Copyright © 2014 John Wiley & Sons, Ltd.     

Graphical interaction models for multivariate time series1

In this paper we extend the concept of graphical models for multivariate data to multivariate time series. We define a partial correlation graph for time series and use the partial spectral coherence between two components given the remaining components to identify the edges of the graph. As an example we consider multivariate autoregressive processes. The method is applied to air pollution data. Received: June 1999     

Local influence analysis for Poisson autoregression with an application to stock transaction data

In statistical diagnostics and sensitivity analysis, the local influence method plays an important role and has certain advantages over other methods in several situations. In this paper, we use this method to study time series of count data when employing a Poisson autoregressive model. We consider case‐weights, scale, data, and additive perturbation schemes to obtain their corresponding vectors and matrices of derivatives for the measures of slope and normal curvatures. Based on the curvature diagnostics, we take a stepwise local influence approach to deal with data with possible masking effects. Finally, our established results are illustrated to be effective by analyzing a stock transactions data set.     

Econometric modelling in finance and risk management: An overview

This paper gives an overview about the sixteen papers included in this special issue. The papers in this special issue cover a wide range of topics. Such topics include discussing a class of tests for correlation, estimation of realized volatility, modeling time series and continuous-time models with long-range dependence, estimation and specification testing of time series models, estimation in a factor model with high-dimensional problems, finite-sample examination of quasi-maximum likelihood estimation in an autoregressive conditional duration model, and estimation in a dynamic additive quantile model.     

Linear models, smooth transition autoregressions, and neural networks for forecasting macroeconomic time series: A re-examination

In this paper, we examine the forecast accuracy of linear autoregressive, smooth transition autoregressive (STAR), and neural network (NN) time series models for 47 monthly macroeconomic variables of the G7 economies. Unlike previous studies that typically consider multiple but fixed model specifications, we use a single but dynamic specification for each model class. The point forecast results indicate that the STAR model generally outperforms linear autoregressive models. It also improves upon several fixed STAR models, demonstrating that careful specification of nonlinear time series models is of crucial importance. The results for neural network models are mixed in the sense that at long forecast horizons, an NN model obtained using Bayesian regularization produces more accurate forecasts than a corresponding model specified using the specific-to-general approach. Reasons for this outcome are discussed.     

The ACR Model: A Multivariate Dynamic Mixture Autoregression*

This paper proposes and analyses the autoregressive conditional root (ACR) time‐series model. This multivariate dynamic mixture autoregression allows for non‐stationary epochs. It proves to be an appealing alternative to existing nonlinear models, e.g. the threshold autoregressive or Markov switching class of models, which are commonly used to describe nonlinear dynamics as implied by arbitrage in presence of transaction costs. Simple conditions on the parameters of the ACR process and its innovations are shown to imply geometric ergodicity, stationarity and existence of moments. Furthermore, consistency and asymptotic normality of the maximum likelihood estimators are established. An application to real exchange rate data illustrates the analysis.     

Using qualitative approach to forecasting regime switches

Drastic changes (named regime switches) often exist in economic and financial time series causing the forecasting of time series difficult. Hence, we need robust models to detect and forecast the regime switches. Most previous studies apply quantitative methods to forecast time series and regime switches. Contrast to these studies, this study attempts a novel approach to use a qualitative method to forecast regime switches. Fuzzy set/qualitative comparative analysis (fsQCA), based on fuzzy set and logic theory, yields the relationships between antecedent combinations and outcome. Studies support fsQCA analysis is more proper to reflect the real situations. Hence, this study uses fsQCA to analyze the autoregressive relationships of the upward and downward regime switches in the in-sample data. Then, the relationships are used to forecast the regime switches in the out-of-sample data. Taiwan Capitalization Weighted Stock Index is taken as the data for analysis. The empirical results show that fsQCA provides strong predictive validities.     

A new marked point process model for the federal funds rate target: Methodology and forecast evaluation

Forecasts of key interest rates set by central banks are of paramount concern for investors and policy makers. Recently it has been shown that forecasts of the federal funds rate target, the most anticipated indicator of the Federal Reserve Bank's monetary policy stance, can be improved considerably when its evolution is modeled as a marked point process (MPP). This is due to the fact that target changes occur in discrete time with discrete increments, have an autoregressive nature and are usually in the same direction. We propose a model which is able to account for these dynamic features of the data. In particular, we combine Hamilton and Jordà's [2002. A model for the federal funds rate target. Journal of Political Economy 110(5), 1135–1167] autoregressive conditional hazard (ACH) and Russell and Engle's [2005. A discrete-state continuous-time model of financial transactions prices and times: the autoregressive conditional multinomial-autoregressive conditional duration model. Journal of Business and Economic Statistics 23(2), 166 – 180] autoregressive conditional multinomial (ACM) model. The paper also puts forth a methodology to evaluate probability function forecasts of MPP models. By improving goodness of fit and point forecasts of the target, the ACH–ACM qualifies as a sensible modeling framework. Furthermore, our results show that MPP models deliver useful probability function forecasts at short and medium term horizons.     

Recent developments of the autoregressive distributed lag modelling framework

We review the literature on the autoregressive distributed lag (ARDL) model, from its origins in the analysis of autocorrelated trend stationary processes to its subsequent applications in the analysis of cointegrated non-stationary time series. We then survey several recent extensions of the ARDL model, including asymmetric and non-linear generalisations of the ARDL model, the quantile ARDL model, the pooled mean group dynamic panel data model and the spatio-temporal ARDL model.     

Numerical issues in threshold autoregressive modeling of time series

This paper analyses the contribution of various numerical approaches to making the estimation of threshold autoregressive time series more efficient. It relies on the computational advantages of QR factorizations and proposes Givens transformations to update these factors for sequential LS problems. By showing that the residual sum of squares is a continuous rational function over threshold intervals it develops a new fitting method based on rational interpolation and the standard necessary optimality condition. Taking as benchmark a simple grid search, the paper illustrates via Monte Carlo simulations the efficiency gains of the proposed tools.