Robust estimation of conditional variance of time series using density power divergences

Suppose Z _t is the square of a time series Y _t whose conditional mean is zero. We do not specify a model for Y _t , but assume that there exists a p ×1 parameter vector Φ such that the conditional distribution of Z _t | Z _{t ?1} is the same as that of , where Z _{t ?1}=(Z _{t ?1},…,Z _{t ?p} )^T for some lag p ?1. Consequently, the conditional variance of Y _t is some function of . To estimate Φ , we propose a robust estimation methodology based on density power divergences (DPD) indexed by a tuning parameter α ∈[0,1], which yields a continuum of estimators, , where α controls the trade‐off between robustness and efficiency of the DPD estimators. For each α , is shown to be strongly consistent. We develop data‐dependent criteria for the selection of optimal α and lag p in practice. We illustrate the usefulness of our DPD methodology via simulation studies for ARCH‐type models, where the errors are drawn from a gross‐error contamination model and the conditional variance is a linear and/or nonlinear function of . Furthermore, we analyze the Chicago Board Options Exchange Dow Jones volatility index data and show that our DPD approach yields viable models for the conditional variance, which are as good as, or superior to, ARCH/GARCH models and two other divergence‐based models in terms of in‐sample and out‐of‐sample forecasts.     

Robust exponential smoothing

The paper is devoted to robust modifications of exponential smoothing for time series with outliers or long-tailed distributions. Classical exponential smoothing applied to such time series is sensitive to the presence of outliers or long-tailed distributions and may give inadequate smoothing and forecasting results. First, simple and double exponential smoothing in the L₁ norm (i.e. based on the least absolute deviations) are discussed in detail. Then, general exponential smoothing is made robust, replacing the least squares approach by M-estimation in such a way that the recursive character of the final formulas is preserved. The paper gives simple algorithmic procedures which preserve advantageous features of classical exponential smoothing and, in addition, which are less sensitive to outliers. Robust versions are compared numerically with classical ones.     

Quantile Double AR Time Series Models for Financial Returns

We develop a novel quantile double autoregressive model for modelling financial time series. This is done by specifying a generalized lambda distribution to the quantile function of the location‐scale double autoregressive model developed by Ling (2004, 2007). Parameter estimation uses Markov chain Monte Carlo Bayesian methods. A simulation technique is introduced for forecasting the conditional distribution of financial returns m periods ahead, and hence any for predictive quantities of interest. The application to forecasting value‐at‐risk at different time horizons and coverage probabilities for Dow Jones Industrial Average shows that our method works very well in practice. Copyright © 2013 John Wiley & Sons, Ltd.     

Forecasting high‐frequency financial data with the ARFIMA–ARCH model

Financial data series are often described as exhibiting two non‐standard time series features. First, variance often changes over time, with alternating phases of high and low volatility. Such behaviour is well captured by ARCH models. Second, long memory may cause a slower decay of the autocorrelation function than would be implied by ARMA models. Fractionally integrated models have been offered as explanations. Recently, the ARFIMA–ARCH model class has been suggested as a way of coping with both phenomena simultaneously. For estimation we implement the bias correction of Cox and Reid ( 1987 ). For daily data on the Swiss 1‐month Euromarket interest rate during the period 1986–1989, the ARFIMA–ARCH (5,d,2/4) model with non‐integer d is selected by AIC. Model‐based out‐of‐sample forecasts for the mean are better than predictions based on conditionally homoscedastic white noise only for longer horizons (τ > 40). Regarding volatility forecasts, however, the selected ARFIMA–ARCH models dominate. Copyright © 2001 John Wiley & Sons, Ltd.     

On the Predictive Content of Autoregression Residuals: A Semiparametric,Copula‐Based Approach to Time Series Prediction

This paper proposes an adjustment of linear autoregressive conditional mean forecasts that exploits the predictive content of uncorrelated model residuals. The adjustment is motivated by non‐Gaussian characteristics of model residuals, and implemented in a semiparametric fashion by means of conditional moments of simulated bivariate distributions. A pseudo ex ante forecasting comparison is conducted for a set of 494 macroeconomic time series recently collected by Dees et al. (Journal of Applied Econometrics 2007; 22: 1–38). In total, 10,374 time series realizations are contrasted against competing short‐, medium‐ and longer‐term purely autoregressive and adjusted predictors. With regard to all forecast horizons, the adjusted predictions consistently outperform conditionally Gaussian forecasts according to cross‐sectional mean group evaluation of absolute forecast errors and directional accuracy. Copyright © 2012 John Wiley & Sons, Ltd.     

Specification versus data fitting: SEM prediction and the Q‐class estimator

We propose a new class of limited information estimators built upon an explicit trade‐off between data fitting and a priori model specification. The estimators offer the researcher a continuum of estimators that range from an extreme emphasis on data fitting and robust reduced‐form estimation to the other extreme of exact model specification and efficient estimation. The approach used to generate the estimators illustrates why ULS often outperforms 2SLS‐PRRF even in the context of a correctly specified model, provides a new interpretation of 2SLS, and integrates Wonnacott and Wonnacott's (1970) least weighted variance estimators with other techniques. We apply the new class of estimators to Klein's Model I and generate forecasts. We find for this example that an emphasis on specification (as opposed to data fitting) produces better out‐of‐sample predictions. Copyright © 1999 John Wiley & Sons, Ltd.     

Estimating and forecasting the long‐memory parameter in the presence of periodicity

We consider one parametric and five semiparametric approaches to estimate D in SARFIMA (0, D, 0)_s processes, that is, when the process is a fractionally integrated ARMA model with seasonality s. We also consider h‐step‐ahead forecasting for these processes. We present the proof of some features of this model and also a study based on a Monte Carlo simulation for different sample sizes and different seasonal periods. We compare the different estimation procedures analyzing the bias, the mean squared error values, and the confidence intervals for the estimators. We also consider three different methods to choose the total number of regressors in the regression analysis for the semiparametric class of estimation procedures. We apply the methodology to the Nile River flow monthly data, and also to a simulated seasonal fractionally integrated time series. Copyright © 2007 John Wiley & Sons, Ltd.     

Volatility forecasting with double Markov switching GARCH models

This paper investigates inference and volatility forecasting using a Markov switching heteroscedastic model with a fat‐tailed error distribution to analyze asymmetric effects on both the conditional mean and conditional volatility of financial time series. The motivation for extending the Markov switching GARCH model, previously developed to capture mean asymmetry, is that the switching variable, assumed to be a first‐order Markov process, is unobserved. The proposed model extends this work to incorporate Markov switching in the mean and variance simultaneously. Parameter estimation and inference are performed in a Bayesian framework via a Markov chain Monte Carlo scheme. We compare competing models using Bayesian forecasting in a comparative value‐at‐risk study. The proposed methods are illustrated using both simulations and eight international stock market return series. The results generally favor the proposed double Markov switching GARCH model with an exogenous variable. Copyright © 2008 John Wiley & Sons, Ltd.     

A Study of Value‐at‐Risk Based on M‐Estimators of the Conditional Heteroscedastic Models

In this paper, we investigate the performance of a class of M‐estimators for both symmetric and asymmetric conditional heteroscedastic models in the prediction of value‐at‐risk. The class of estimators includes the least absolute deviation (LAD), Huber's, Cauchy and B‐estimator, as well as the well‐known quasi maximum likelihood estimator (QMLE). We use a wide range of summary statistics to compare both the in‐sample and out‐of‐sample VaR estimates of three well‐known stock indices. Our empirical study suggests that in general Cauchy, Huber and B‐estimator have better performance in predicting one‐step‐ahead VaR than the commonly used QMLE. Copyright © 2011 John Wiley & Sons, Ltd.     

Forecasting volatility of emerging stock markets: linear versus non‐linear GARCH models

ARCH and GARCH models are substantially used for modelling volatility of time series data. It is proven by many studies that if variables are significantly skewed, linear versions of these models are not sufficient for both explaining the past volatility and forecasting the future volatility. In this paper, we compare the linear(GARCH(1,1)) and non‐linear(EGARCH) versions of GARCH model by using the monthly stock market returns of seven emerging countries from February 1988 to December 1996. We find that for emerging stock markets GARCH(1,1) model performs better than EGARCH model, even if stock market return series display skewed distributions. Copyright © 2000 John Wiley & Sons, Ltd.     

Prediction‐based adaptive compositional model for seasonal time series analysis

In this paper we propose a new class of seasonal time series models, based on a stable seasonal composition assumption. With the objective of forecasting the sum of the next ? observations, the concept of rolling season is adopted and a structure of rolling conditional distributions is formulated. The probabilistic properties, estimation and prediction procedures, and the forecasting performance of the model are studied and demonstrated with simulations and real examples.     

Forecasting commodity prices: GARCH,jumps, and mean reversion

In examining stochastic models for commodity prices, central questions often revolve around time‐varying trend, stochastic convenience yield and volatility, and mean reversion. This paper seeks to assess and compare alternative approaches to modelling these effects, with focus on forecast performance. Three specifications are considered: (i) random‐walk models with GARCH and normal or Student‐t innovations; (ii) Poisson‐based jump‐diffusion models with GARCH and normal or Student‐t innovations; and (iii) mean‐reverting models that allow for uncertainty in equilibrium price. Our empirical application makes use of aluminium spot and futures price series at daily and weekly frequencies. Results show: (i) models with stochastic convenience yield outperform all other competing models, and for all forecast horizons; (ii) the use of futures prices does not always yield lower forecast error values compared to the use of spot prices; and (iii) within the class of (G)ARCH random‐walk models, no model uniformly dominates the other. Copyright © 2008 John Wiley & Sons, Ltd.     

Assessing inefficiency in the s&p 500 futures market

We analyse the price movement of the S&P 500 futures market for violations of the efficient market hypothesis on a short-term basis. To assess market inefficiency we construct a model and find that the returns, i.e. the difference in the logarithm of closing prices on consecutive days, exhibit the usual conditional heteroscedasticity behaviour typical of long series of financial data. To account for this non-linear behaviour we scale the returns by a volatility factor which depends on the daily high, low, and closing price. The rescaled series, which may be interpreted as the trend-countertrend component of the time series, is modelled using Box and Jenkins techniques. The resulting model is an ARMA(1,1). The scale factors are assumed to form a time series and are modelled using a semi-non-parametric method which avoids the restrictive assumptions of most ARCH or GARCH models. Using the combined model we perform 1000 simulations of market data, each simulation comprising 250 days (approximately one year). We then formulate a naive trading strategy which is based on the ratio of the one-day-ahead expected return to its one-day-ahead expected conditional standard deviation. The trading strategy has four adjustable parameters which are set to maximize profits for the simulation data. Next, we apply the trading strategy to one year of recent out-of-sample data. Our conclusion is that the S&P 500 futures market exhibits only slight inefficiencies, but that there exist, in principle, better trading strategies which take account of risk than the benchmark strategy of buy-and-hold. We have also constructed a linear model for the return series. Using the linear model, we have simulated returns and determined the optimum values for the adjustable parameters of the trading strategy. In this case, the optimum trading strategy is the same as the benchmark strategy, buy-and-hold. Finally, we have compared the profitability of the optimized trading strategy, based on the non-linear model, to three ad hoc trading strategies using the out-of-sample data. The three ad hoc strategies are more profitable than the optimized strategy.     

Autoregressive gamma processes

We introduce a class of autoregressive gamma processes with conditional distributions from the family of noncentred gamma (up to a scale factor). The paper provides the stationarity and ergodicity conditions for ARG processes of any autoregressive order p, including long memory, and closed‐form expressions of conditional moments. The nonlinear state space representation of an ARG process is used to derive the filtering, smoothing and forecasting algorithms. The paper also presents estimation and inference methods, illustrated by an application to interquote durations data on an infrequently traded stock listed on the Toronto Stock Exchange (TSX). Copyright © 2006 John Wiley & Sons, Ltd.     

Estimating and Forecasting APARCH‐Skew‐t Model by Wavelet Support Vector Machines

This paper concentrates on comparing estimation and forecasting ability of quasi‐maximum likelihood (QML) and support vector machines (SVM) for financial data. The financial series are fitted into a family of asymmetric power ARCH (APARCH) models. As the skewness and kurtosis are common characteristics of the financial series, a skew‐t distributed innovation is assumed to model the fat tail and asymmetry. Prior research indicates that the QML estimator for the APARCH model is inefficient when the data distribution shows departure from normality, so the current paper utilizes the semi‐parametric‐based SVM method and shows that it is more efficient than the QML under the skewed Student's‐t distributed error. As the SVM is a kernel‐based technique, we further investigate its performance by applying separately a Gaussian kernel and a wavelet kernel. The results suggest that the SVM‐based method generally performs better than QML for both in‐sample and out‐of‐sample data. The outcomes also highlight the fact that the wavelet kernel outperforms the Gaussian kernel with lower forecasting error, better generation capability and more computation efficiency. Copyright © 2014 John Wiley & Sons, Ltd.     

A non‐Gaussian generalization of the Airline model for robust seasonal adjustment

In their seminal book Time Series Analysis: Forecasting and Control, Box and Jenkins (1976) introduce the Airline model, which is still routinely used for the modelling of economic seasonal time series. The Airline model is for a differenced time series (in levels and seasons) and constitutes a linear moving average of lagged Gaussian disturbances which depends on two coefficients and a fixed variance. In this paper a novel approach to seasonal adjustment is developed that is based on the Airline model and that accounts for outliers and breaks in time series. For this purpose we consider the canonical representation of the Airline model. It takes the model as a sum of trend, seasonal and irregular (unobserved) components which are uniquely identified as a result of the canonical decomposition. The resulting unobserved components time series model is extended by components that allow for outliers and breaks. When all components depend on Gaussian disturbances, the model can be cast in state space form and the Kalman filter can compute the exact log‐likelihood function. Related filtering and smoothing algorithms can be used to compute minimum mean squared error estimates of the unobserved components. However, the outlier and break components typically rely on heavy‐tailed densities such as the t or the mixture of normals. For this class of non‐Gaussian models, Monte Carlo simulation techniques will be used for estimation, signal extraction and seasonal adjustment. This robust approach to seasonal adjustment allows outliers to be accounted for, while keeping the underlying structures that are currently used to aid reporting of economic time series data. Copyright © 2006 John Wiley & Sons, Ltd.     

Non-linearity and exchange rates

The conditional heteroscedastic models (CHM) are commonly used to describe the dynamics of nominal exchange rates. However, some investigations have already pointed out that the CHMs are not able to fully explain all non-linearities exhibited by the exchange rate series. This paper analyses the performance of univariate CHMs in modelling the non-linearities of nominal exchange rate series vis-à-vis the US dollar. Twelve currencies are examined on a weekly basis: The Belgian, Swiss and French francs; the Canadian dollar; the German mark; the Danish and Norwegian kroners; the British and Irish pounds; the Italian lira; the Japanese yen and the Dutch guilder. The CHMs captured in a satisfactory way the volatility clustering of the series and show volatility peaks in historically nervous periods of the international market. Moreover, the results of the BDS tests for whiteness applied to the standardized residuals show the good specification of the models. Copyright © 1998 John Wiley & Sons, Ltd.     

Evaluating volatility and interval forecasts

A widely used approach to evaluating volatility forecasts uses a regression framework which measures the bias and variance of the forecast. We show that the associated test for bias is inappropriate before introducing a more suitable procedure which is based on the test for bias in a conditional mean forecast. Although volatility has been the most common measure of the variability in a financial time series, in many situations confidence interval forecasts are required. We consider the evaluation of interval forecasts and present a regression‐based procedure which uses quantile regression to assess quantile estimator bias and variance. We use exchange rate data to illustrate the proposal by evaluating seven quantile estimators, one of which is a new non‐parametric autoregressive conditional heteroscedasticity quantile estimator. The empirical analysis shows that the new evaluation procedure provides useful insight into the quality of quantile estimators. Copyright © 1999 John Wiley & Sons, Ltd.     

Forecasting with latent structure time series models: an application to nominal interest rates

In this paper we develop a latent structure extension of a commonly used structural time series model and use the model as a basis for forecasting. Each unobserved regime has its own unique slope and variances to describe the process generating the data, and at any given time period the model predicts a priori which regime best characterizes the data. This is accomplished by using a multinomial logit model in which the primary explanatory variable is a measure of how consistent each regime has been with recent observations. The model is especially well suited to forecasting series which are subject to frequent and/or major shocks. An application to nominal interest rates shows that the behaviour of the three‐month US Treasury bill rate is adequately explained by three regimes. The forecasting accuracy is superior to that produced by a traditional single‐regime model and a standard ARIMA model with a conditionally heteroscedastic error. Copyright © 1999 John Wiley & Sons, Ltd.     

Time Series of Zero‐Inflated Counts and their Coherent Forecasting

Poisson integer‐valued auto‐regressive process of order 1 (PINAR(1)) due to Al‐Osh and Alzaid (Journal of Time Series Analysis 1987; 8 (3): 261–275) and McKenzie (Advances in Applied Probability 1988; 20 (4): 822–835) has received a significant attention in modelling low‐count time series during the last two decades because of its simplicity. But in many practical scenarios, the process appears to be inadequate, especially when data are overdispersed in nature. This overdispersion occurs mainly for three reasons: presence of some extreme values, large number of zeros, and presence of both extreme values with a large number of zeros. In this article, we develop a zero‐inflated Poisson INAR(1) process as an alternative to the PINAR(1) process when the number of zeros in the data is larger than the expected number of zeros by the Poisson process. We investigate some important properties such as stationarity, ergodicity, autocorrelation structure, and conditional distribution, with a detailed study on h‐step‐ahead coherent forecasting. A comparative study among different methods of parameter estimation is carried out using some simulated data. One real dataset is analysed for practical illustration. Copyright © 2015 John Wiley & Sons, Ltd.