Model Uncertainty and Forecast Combination in High‐Dimensional Multivariate Volatility Prediction

In multivariate volatility prediction, identifying the optimal forecasting model is not always a feasible task. This is mainly due to the curse of dimensionality typically affecting multivariate volatility models. In practice only a subset of the potentially available models can be effectively estimated, after imposing severe constraints on the dynamic structure of the volatility process. It follows that in most applications the working forecasting model can be severely misspecified. This situation leaves scope for the application of forecast combination strategies as a tool for improving the predictive accuracy. The aim of the paper is to propose some alternative combination strategies and compare their performances in forecasting high‐dimensional multivariate conditional covariance matrices for a portfolio of US stock returns. In particular, we will consider the combination of volatility predictions generated by multivariate GARCH models, based on daily returns, and dynamic models for realized covariance matrices, built from intra‐daily returns. Copyright © 2015 John Wiley & Sons, Ltd.     

Single‐index and portfolio models for forecasting value‐at‐risk thresholds

The variance of a portfolio can be forecast using a single index model or the covariance matrix of the portfolio. Using univariate and multivariate conditional volatility models, this paper evaluates the performance of the single index and portfolio models in forecasting value‐at‐risk (VaR) thresholds of a portfolio. Likelihood ratio tests of unconditional coverage, independence and conditional coverage of the VaR forecasts suggest that the single‐index model leads to excessive and often serially dependent violations, while the portfolio model leads to too few violations. The single‐index model also leads to lower daily Basel Accord capital charges. The univariate models which display correct conditional coverage lead to higher capital charges than models which lead to too many violations. Overall, the Basel Accord penalties appear to be too lenient and favour models which have too many violations. Copyright © 2008 John Wiley & Sons, Ltd.     

Low and high prices can improve covariance forecasts: The evidence based on currency rates

In this paper we introduce a new specification of the BEKK model, where its parameters are estimated with the use of closing and additionally low and high prices. In an empirical application, we show that the use of additional information related to low and high prices in the formulation of the BEKK model improved the estimation of the covariance matrix of returns and increased the accuracy of covariance and variance forecasts based on this model, compared with using closing prices only. This analysis was performed for the following three most heavily traded currency pairs in the Forex market: EUR/USD, USD/JPY, and GBP/USD. The main result obtained in this study is robust to the applied forecast evaluation criterion. This issue is important from a practical viewpoint, because daily low and high prices are available with closing prices for most financial series.     

Bayes Linear Variance Adjustment for Locally Linear DLMs

This paper exhibits quadratic products of linear combinations of observables which identify the covariance structure underlying the univariate locally linear time series dynamic linear model. The first- and second-order moments for the joint distribution over these observables are given, allowing Bayes linear learning for the underlying covariance structure for the time series model. An example is given which illustrates the methodology and highlights the practical implications of the theory. © 1997 John Wiley & Sons, Ltd.     

Average conditional correlation and tree structures for multivariate GARCH models

We propose a simple class of multivariate GARCH models, allowing for time‐varying conditional correlations. Estimates for time‐varying conditional correlations are constructed by means of a convex combination of averaged correlations (across all series) and dynamic realized (historical) correlations. Our model is very parsimonious. Estimation is computationally feasible in very large dimensions without resorting to any variance reduction technique. We back‐test the models on a six‐dimensional exchange‐rate time series using different goodness‐of‐fit criteria and statistical tests. We collect empirical evidence of their strong predictive power, also in comparison to alternative benchmark procedures. Copyright © 2006 John Wiley & Sons, Ltd.     

Forecasting time‐varying covariance with a robust Bayesian threshold model

This paper proposes a robust multivariate threshold vector autoregressive model with generalized autoregressive conditional heteroskedasticities and dynamic conditional correlations to describe conditional mean, volatility and correlation asymmetries in financial markets. In addition, the threshold variable for regime switching is formulated as a weighted average of endogenous variables to eliminate excessively subjective belief in the threshold variable decision and to serve as the proxy in deciding which market should be the price leader. The estimation is performed using Markov chain Monte Carlo methods. Furthermore, several meaningful criteria are introduced to assess the forecasting performance in the conditional covariance matrix. The proposed methodology is illustrated using daily S&P500 futures and spot prices. Copyright © 2010 John Wiley & Sons, Ltd.     

Covariance estimation for multivariate conditionally Gaussian dynamic linear models

In multivariate time series, estimation of the covariance matrix of observation innovations plays an important role in forecasting as it enables computation of standardized forecast error vectors as well as the computation of confidence bounds of forecasts. We develop an online, non‐iterative Bayesian algorithm for estimation and forecasting. It is empirically found that, for a range of simulated time series, the proposed covariance estimator has good performance converging to the true values of the unknown observation covariance matrix. Over a simulated time series, the new method approximates the correct estimates, produced by a non‐sequential Monte Carlo simulation procedure, which is used here as the gold standard. The special, but important, vector autoregressive (VAR) and time‐varying VAR models are illustrated by considering London metal exchange data consisting of spot prices of aluminium, copper, lead and zinc. Copyright © 2007 John Wiley & Sons, Ltd.     

Forecast Combinations in a DSGE‐VAR Lab

We explore the benefits of forecast combinations based on forecast‐encompassing tests compared to simple averages and to Bates–Granger combinations. We also consider a new combination algorithm that fuses test‐based and Bates–Granger weighting. For a realistic simulation design, we generate multivariate time series samples from a macroeconomic DSGE‐VAR (dynamic stochastic general equilibrium–vector autoregressive) model. Results generally support Bates–Granger over uniform weighting, whereas benefits of test‐based weights depend on the sample size and on the prediction horizon. In a corresponding application to real‐world data, simple averaging performs best. Uniform averages may be the weighting scheme that is most robust to empirically observed irregularities. Copyright © 2016 John Wiley & Sons, Ltd.     

Prediction intervals for exponential smoothing using two new classes of state space models

Three general classes of state space models are presented, using the single source of error formulation. The first class is the standard linear model with homoscedastic errors, the second retains the linear structure but incorporates a dynamic form of heteroscedasticity, and the third allows for non‐linear structure in the observation equation as well as heteroscedasticity. These three classes provide stochastic models for a wide variety of exponential smoothing methods. We use these classes to provide exact analytic (matrix) expressions for forecast error variances that can be used to construct prediction intervals one or multiple steps ahead. These formulas are reduced to non‐matrix expressions for 15 state space models that underlie the most common exponential smoothing methods. We discuss relationships between our expressions and previous suggestions for finding forecast error variances and prediction intervals for exponential smoothing methods. Simpler approximations are developed for the more complex schemes and their validity examined. The paper concludes with a numerical example using a non‐linear model. Copyright © 2005 John Wiley & Sons, Ltd.     

On the Modelling and Forecasting of Multivariate Realized Volatility: Generalized Heterogeneous Autoregressive (GHAR) Model

Recent multivariate extensions of the popular heterogeneous autoregressive model (HAR) for realized volatility leave substantial information unmodelled in residuals. We propose to employ a system of seemingly unrelated regressions to model and forecast a realized covariance matrix to capture this information. We find that the newly proposed generalized heterogeneous autoregressive (GHAR) model outperforms competing approaches in terms of economic gains, providing better mean–variance trade‐off, while, in terms of statistical precision, GHAR is not substantially dominated by any other model. Our results provide a comprehensive comparison of the performance when realized covariance, subsampled realized covariance and multivariate realized kernel estimators are used. We study the contribution of the estimators across different sampling frequencies, and show that the multivariate realized kernel and subsampled realized covariance estimators deliver further gains compared to realized covariance estimated on a 5‐minute frequency. In order to show economic and statistical gains, a portfolio of various sizes is used. Copyright © 2016 John Wiley & Sons, Ltd.     

A comparison between linear and nonlinear forecasts for nonlinear AR models

In this paper the relative forecast performance of nonlinear models to linear models is assessed by the conditional probability that the absolute forecast error of the nonlinear forecast is smaller than that of the linear forecast. The comparison probability is explicitly expressed and is shown to be an increasing function of the distance between nonlinear and linear forecasts under certain conditions. This expression of the comparison probability may not only be useful in determining the predictor, which is either a more accurate or a simpler forecast, to be used but also provides a good explanation for an odd phenomenon discussed by Pemberton. The relative forecast performance of a nonlinear model to a linear model is demonstrated to be sensitive to its forecast origins. A new forecast is thus proposed to improve the relative forecast performance of nonlinear models based on forecast origins. © 1997 John Wiley & Sons, Ltd.     

Unemployment variation over the business cycles: a comparison of forecasting models

Asymmetry has been well documented in the business cycle literature. The asymmetric business cycle suggests that major macroeconomic series, such as a country's unemployment rate, are non‐linear and, therefore, the use of linear models to explain their behaviour and forecast their future values may not be appropriate. Many researchers have focused on providing evidence for the non‐linearity in the unemployment series. Only recently have there been some developments in applying non‐linear models to estimate and forecast unemployment rates. A major concern of non‐linear modelling is the model specification problem; it is very hard to test all possible non‐linear specifications, and to select the most appropriate specification for a particular model. Artificial neural network (ANN) models provide a solution to the difficulty of forecasting unemployment over the asymmetric business cycle. ANN models are non‐linear, do not rely upon the classical regression assumptions, are capable of learning the structure of all kinds of patterns in a data set with a specified degree of accuracy, and can then use this structure to forecast future values of the data. In this paper, we apply two ANN models, a back‐propagation model and a generalized regression neural network model to estimate and forecast post‐war aggregate unemployment rates in the USA, Canada, UK, France and Japan. We compare the out‐of‐sample forecast results obtained by the ANN models with those obtained by several linear and non‐linear times series models currently used in the literature. It is shown that the artificial neural network models are able to forecast the unemployment series as well as, and in some cases better than, the other univariate econometrics time series models in our test. Copyright © 2004 John Wiley & Sons, Ltd.     

An evaluation of bayesian forecasting

‘Bayesian forecasting’ is a time series method of forecasting which (in the United Kingdom) has become synonymous with the state space formulation of Harrison and Stevens (1976). The approach is distinct from other time series methods in that it envisages changes in model structure. A disjoint class of models is chosen to encompass the changes. Each data point is retrospectively evaluated (using Bayes theorem) to judge which of the models held. Forecasts are then derived conditional on an assumed model holding true. The final forecasts are weighted sums of these conditional forecasts. Few empirical evaluations have been carried out. This paper reports a large scale comparison of time series forecasting methods including the Bayesian. The approach is two fold: a simulation study to examine parameter sensitivity and an empirical study which contrasts Bayesian with other time series methods.     

Forecast comparisons of four models of U.S. interest rates

In this paper we compare the out of sample forecasts from four alternative interest rate models based on expanding information sets. The random walk model is the most restrictive. The univariate time series model allows for a richer dynamic pattern and more conditioning information on own rates. The multivariate time series model permits a flexible dynamic pattern with own- and cross-series information. Finally, the forecasts from the MPS econometric model depend on the full model structure and information set. In theory, more information is preferred to less. In practice, complicated misspecified models can perform much worse than simple (also probably misspecified) models. For forecasts evaluated over the volatile 1970s the multivariate time series model forecasts are considerably better than those from simpler models which use less conditioning information, as well as forecasts from the MPS model which uses substantially more conditioning information but also imposes ‘structural’ economic restrictions.     

Forecasting Multivariate Time Series with the Theta Method

In this study building on earlier work on the properties and performance of the univariate Theta method for a unit root data‐generating process we: (a) derive new theoretical formulations for the application of the method on multivariate time series; (b) investigate the conditions for which the multivariate Theta method is expected to forecast better than the univariate one; (c) evaluate through simulations the bivariate form of the method; and (d) evaluate this latter model in real macroeconomic and financial time series. The study provides sufficient empirical evidence to illustrate the suitability of the method for vector forecasting; furthermore it provides the motivation for further investigation of the multivariate Theta method for higher dimensions. Copyright © 2015 John Wiley & Sons, Ltd.     

A fractal forecasting model for financial time series

Financial market time series exhibit high degrees of non‐linear variability, and frequently have fractal properties. When the fractal dimension of a time series is non‐integer, this is associated with two features: (1) inhomogeneity—extreme fluctuations at irregular intervals, and (2) scaling symmetries—proportionality relationships between fluctuations over different separation distances. In multivariate systems such as financial markets, fractality is stochastic rather than deterministic, and generally originates as a result of multiplicative interactions. Volatility diffusion models with multiple stochastic factors can generate fractal structures. In some cases, such as exchange rates, the underlying structural equation also gives rise to fractality. Fractal principles can be used to develop forecasting algorithms. The forecasting method that yields the best results here is the state transition‐fitted residual scale ratio (ST‐FRSR) model. A state transition model is used to predict the conditional probability of extreme events. Ratios of rates of change at proximate separation distances are used to parameterize the scaling symmetries. Forecasting experiments are run using intraday exchange rate futures contracts measured at 15‐minute intervals. The overall forecast error is reduced on average by up to 7% and in one instance by nearly a quarter. However, the forecast error during the outlying events is reduced by 39% to 57%. The ST‐FRSR reduces the predictive error primarily by capturing extreme fluctuations more accurately. Copyright © 2004 John Wiley & Sons, Ltd.     

A threshold factor multivariate stochastic volatility model

A new multivariate stochastic volatility model is developed in this paper. The main feature of this model is to allow threshold asymmetry in a factor covariance structure. The new model provides a parsimonious characterization of volatility and correlation asymmetry in response to market news. Statistical inferences are drawn from Markov chain Monte Carlo methods. We introduce news impact analysis to analyze volatility asymmetry with a factor structure. This analysis helps us to study different responses of volatility to historical market information in a multivariate volatility framework. Our model is successful when applied to an extensive empirical study of twenty stocks. Copyright © 2009 John Wiley & Sons, Ltd.     

Long‐memory forecasting of US monetary indices

Several studies have tested for long‐range dependence in macroeconomic and financial time series but very few have assessed the usefulness of long‐memory models as forecast‐generating mechanisms. This study tests for fractional differencing in the US monetary indices (simple sum and divisia) and compares the out‐of‐sample fractional forecasts to benchmark forecasts. The long‐memory parameter is estimated using Robinson's Gaussian semi‐parametric and multivariate log‐periodogram methods. The evidence amply suggests that the monetary series possess a fractional order between one and two. Fractional out‐of‐sample forecasts are consistently more accurate (with the exception of the M3 series) than benchmark autoregressive forecasts but the forecasting gains are not generally statistically significant. In terms of forecast encompassing, the fractional model encompasses the autoregressive model for the divisia series but neither model encompasses the other for the simple sum series. Copyright © 2006 John Wiley & Sons, Ltd.     

Modeling interval trendlines: Symbolic singular spectrum analysis for interval time series

In this article we propose an extension of singular spectrum analysis for interval-valued time series. The proposed methods can be used to decompose and forecast the dynamics governing a set-valued stochastic process. The resulting components on which the interval time series is decomposed can be understood as interval trendlines, cycles, or noise. Forecasting can be conducted through a linear recurrent method, and we devised generalizations of the decomposition method for the multivariate setting. The performance of the proposed methods is showcased in a simulation study. We apply the proposed methods so to track the dynamics governing the Argentina Stock Market (MERVAL) in real time, in a case study over a period of turbulence that led to discussions of the government of Argentina with the International Monetary Fund.     

Predicting Bid–Ask Spreads Using Long‐Memory Autoregressive Conditional Poisson Models

We introduce a long‐memory autoregressive conditional Poisson (LMACP) model to model highly persistent time series of counts. The model is applied to forecast quoted bid–ask spreads, a key parameter in stock trading operations. It is shown that the LMACP nicely captures salient features of bid–ask spreads like the strong autocorrelation and discreteness of observations. We discuss theoretical properties of LMACP models and evaluate rolling‐window forecasts of quoted bid–ask spreads for stocks traded at NYSE and NASDAQ. We show that Poisson time series models significantly outperform forecasts from AR, ARMA, ARFIMA, ACD and FIACD models. The economic significance of our results is supported by the evaluation of a trade schedule. Scheduling trades according to spread forecasts we realize cost savings of up to 14 % of spread transaction costs. Copyright © 2013 John Wiley & Sons, Ltd.