Principal Volatility Component Analysis

Many empirical time series such as asset returns and traffic data exhibit the characteristic of time-varying conditional covariances, known as volatility or conditional heteroscedasticity. Modeling multivariate volatility, however, encounters several difficulties, including the curse of dimensionality. Dimension reduction can be useful and is often necessary. The goal of this article is to extend the idea of principal component analysis to principal volatility component (PVC) analysis. We define a cumulative generalized kurtosis matrix to summarize the volatility dependence of multivariate time series. Spectral analysis of this generalized kurtosis matrix is used to define PVCs. We consider a sample estimate of the generalized kurtosis matrix and propose test statistics for detecting linear combinations that do not have conditional heteroscedasticity. For application, we applied the proposed analysis to weekly log returns of seven exchange rates against U.S. dollar from 2000 to 2011 and found a linear combination among the exchange rates that has no conditional heteroscedasticity.     

Markov Switching in GARCH Processes and Mean-Reverting Stock-Market Volatility

This article introduces four models of conditional heteroscedasticity that contain Markov-switching parameters to examine their multiperiod stock-market volatility forecasts as predictions of options-implied volatilities. The volatility model that best predicts the behavior of the options-implied volatilities allows the Student-t degrees-of-freedom parameter to switch such that the conditional variance and kurtosis are subject to discrete shifts. The half-life of the most leptokurtic state is estimated to be a week, so expected market volatility reverts to near-normal levels fairly quickly following a spike.     

The Message in Daily Exchange Rates: A Conditional-Variance Tale

Formal testing procedures confirm the presence of a unit root in the autoregressive polynomial of the univariate time series representation of daily exchange-rate data. The first differences of the logarithms of daily spot rates are approximately uncorrelated through time, and a generalized autoregressive conditional heteroscedasticity model with daily dummy variables and conditionally t-distributed errors is found to provide a good representation to the leptokurtosis and time-dependent conditional heteroscedasticity. The parameter estimates and characteristics of the models are found to be very similar for six different currencies. These apparent stylized facts carry over to weekly, fortnightly, and monthly data in which the degree of leptokurtosis and time-dependent heteroscedasticity is reduced as the length of the sampling interval increases.     

Modeling heteroscedastic age-period-cohort cancer data

The extent to which cancer will be a burden on the Canadian health-care system will be determined by future cancer rates and future population levels in the high-risk age groups. Parametric models of incidence and mortality rates for various cancers may be used to obtain medium-term forecasts of rates, which then can be used in conjunction with population projections to obtain forecasts of total incidence and mortality. Age-period-cohort cancer data often exhibit marked heteroscedasticity, which complicates the modeling of the data. Methods to allow for the effects of this heteroscedasticity on residual processes are developed and discussed in the context of modeling Canadian female breast-cancer incidence data.     

Semiparametric ARCH Models

This article introduces a semiparametric autoregressive conditional heteroscedasticity (ARCH) model that has conditional first and second moments given by autoregressive moving average and ARCH parametric formulations but a conditional density that is assumed only to be sufficiently smooth to be approximated by a nonparametric density estimator. For several particular conditional densities, the relative efficiency of the quasi-maximum likelihood estimator is compared with maximum likelihood under correct specification. These potential efficiency gains for a fully adaptive procedure are compared in a Monte Carlo experiment with the observed gains from using the proposed semiparametric procedure, and it is found that the estimator captures a substantial proportion of the potential. The estimator is applied to daily stock returns from small firms that are found to exhibit conditional skewness and kurtosis and to the British pound to dollar exchange rate.     

A multivariate volatility vine copula model

This article proposes a dynamic framework for modeling and forecasting of realized covariance matrices using vine copulas to allow for more flexible dependencies between assets. Our model automatically guarantees positive definiteness of the forecast through the use of a Cholesky decomposition of the realized covariance matrix. We explicitly account for long-memory behavior by using fractionally integrated autoregressive moving average (ARFIMA) and heterogeneous autoregressive (HAR) models for the individual elements of the decomposition. Furthermore, our model incorporates non-Gaussian innovations and GARCH effects, accounting for volatility clustering and unconditional kurtosis. The dependence structure between assets is studied using vine copula constructions, which allow for nonlinearity and asymmetry without suffering from an inflexible tail behavior or symmetry restrictions as in conventional multivariate models. Further, the copulas have a direct impact on the point forecasts of the realized covariances matrices, due to being computed as a nonlinear transformation of the forecasts for the Cholesky matrix. Beside studying in-sample properties, we assess the usefulness of our method in a one-day-ahead forecasting framework, comparing recent types of models for the realized covariance matrix based on a model confidence set approach. Additionally, we find that in Value-at-Risk (VaR) forecasting, vine models require less capital requirements due to smoother and more accurate forecasts.     

Periodic Autoregressive Conditional Heteroscedasticity

Most high-frequency asset returns exhibit seasonal volatility patterns. This article proposes a new class of models featuring periodicity in conditional heteroscedasticity explicitly designed to capture the repetitive seasonal time variation in the second-order moments. This new class of periodic autoregressive conditional heteroscedasticity, or P-ARCH, models is directly related to the class of periodic autoregressive moving average (ARMA) models for the mean. The implicit relation between periodic generalized ARCH (P-GARCH) structures and time-invariant seasonal weak GARCH processes documents how neglected autoregressive conditional heteroscedastic periodicity may give rise to a loss in forecast efficiency. The importance and magnitude of this informational loss are quantified for a variety of loss functions through the use of Monte Carlo simulation methods. Two empirical examples with daily bilateral Deutschemark/British pound and intraday Deutschemark/U.S. dollar spot exchange rates highlight the practical relevance of the new P-GARCH class of models. Extensions to discrete-time periodic representations of stochastic volatility models subject to time deformation are briefly discussed.     

Combination forecasting for directional accuracy: An application to survey interest rate forecasts

Using published interest rates forecasts issued by professional economists, two combination forecasts designed to improve the directional accuracy of interest rate forecasting are constructed. The first combination forecast takes a weighted average of the individual forecasters' predictions. The more successful the forecaster was in past forecasts at predicting the direction of change in interest rates, the greater is the weight given to his/her current forecast. The second combination forecast is simply the forecast issued by the forecaster who had the greatest success rate at predicting the direction of change in interest rates in previous forecasts. In cases where two or more forecasters tie for best historic directional accuracy track record, the arithmetic mean of these forecasters is used. The study finds that neither combination forecasting method performs better than coin-flipping at predicting the direction of change in interest rates. Nor does either method beat the simple arithmetic mean of the predictions of all the forecasters surveyed at predicting the direction of change in interest rates.     

Forecasting power-transformed time series data

When there is an interest in forecasting the growth rates as well as the levels of a single macro-economic time series, a practitioner faces the question of whether a forecasting model should be constructed for growth rates, for levels, or for both. In this paper, we investigate this issue for 10 US (un-)employment series, where we evaluate the forecasts from a non-linear time series model for power-transformed data. Our main finding is that models for growth rates (levels) do not automatically result in the most accurate forecasts of growth rates (levels).     

Conditional Heteroscedasticity in Qualitative Response Models of Time Series: A Gibbs-Sampling Approach to the Bank Prime Rate

Previous time series applications of qualitative response models have ignored features of the data, such as conditional heteroscedasticity, that are routinely addressed in time series econometrics of financial data. This article addresses this issue by adding Markov-switching heteroscedasticity to a dynamic ordered probit model of discrete changes in the bank prime lending rate and estimating via the Gibbs sampler. The dynamic ordered probit model of Eichengreen, Watson, and Grossman allows for serial autocorrelation in probit analysis of a time series, and this article demonstrates the relative simplicity of estimating a dynamic ordered probit using the Gibbs sampler instead of the Eichengreen et al. maximum likelihood procedure. In addition, the extension to regime-switching parameters and conditional heteroscedasticity is easy to implement under Gibbs sampling. The article compares tests of goodness of fit between dynamic ordered probit models of the prime rate that have constant variance and conditional heteroscedasticity.     

FORECASTING VOLATILITY IN THE PRESENCE OF MODEL INSTABILITY

Recent advances in financial econometrics have allowed for the construction of efficient ex post measures of daily volatility. This paper investigates the importance of instability in models of realised volatility and their corresponding forecasts. Testing for model instability is conducted with a subsampling method. We show that removing structurally unstable data of a short duration has a negligible impact on the accuracy of conditional mean forecasts of volatility. In contrast, it does provide a substantial improvement in a model's forecast density of volatility. In addition, the forecasting performance improves, often dramatically, when we evaluate models on structurally stable data.     

Forecasting the spot price of gold:combined forecast approaches versus a composite forecast approach

The fluctuation of the gold price has significant impact on the economic and social aspects of a society. In the literature, most authors have employed fundamental analysis approach in forecast model building. The basic principle underlying this approach is that it is the supply and the demand which simultaneously determines the gold price. However, due to the lack of data of quantity supplied and quantity demanded, simultaneous econometric approach seems unsuccessful. In this paper, combined and composite time series forecasting techniques are proposed. The effects of various economic factors towards spot price of gold are also examined. Among the combined forecasting models, it seems that the odds-matrix method of assigning weights provides the most accurate forecasts of spot price of gold. For the economic factors considered, the futures price of gold and and the exchange rate seem to be most informative in forecasting the spot price of gold.     

Testing for Common Features

This article introduces a class of statistical tests for the hypothesis that some feature that is present in each of several variables is common to them. Features are data properties such as serial correlation, trends, seasonality, heteroscedasticity, autoregressive conditional hetero-scedasticity, and excess kurtosis. A feature is detected by a hypothesis test taking no feature as the null, and a common feature is detected by a test that finds linear combinations of variables with no feature. Often, an exact asymptotic critical value can be obtained that is simply a test of overidentifying restrictions in an instrumental variable regression. This article tests for a common international business cycle.     

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.     

Zero-inflated compound Poisson distributions in integer-valued GARCH models

In this paper we introduce a wide class of integer-valued stochastic processes that allows to take into consideration, simultaneously, relevant characteristics observed in count data namely zero inflation, overdispersion and conditional heteroscedasticity. This class includes, in particular, the compound Poisson, the zero-inflated Poisson and the zero-inflated negative binomial INGARCH models, recently proposed in literature. The main probabilistic analysis of this class of processes is here developed. Precisely, first- and second-order stationarity conditions are derived, the autocorrelation function is deduced and the strict stationarity is established in a large subclass. We also analyse in a particular model the existence of higher-order moments and deduce the explicit form for the first four cumulants, as well as its skewness and kurtosis.     

Conditional Correlation Models of Autoregressive Conditional Heteroscedasticity With Nonstationary GARCH Equations

In this article, we investigate the effects of careful modeling the long-run dynamics of the volatilities of stock market returns on the conditional correlation structure. To this end, we allow the individual unconditional variances in conditional correlation generalized autoregressive conditional heteroscedasticity (CC-GARCH) models to change smoothly over time by incorporating a nonstationary component in the variance equations such as the spline-GARCH model and the time-varying (TV)-GARCH model. The variance equations combine the long-run and the short-run dynamic behavior of the volatilities. The structure of the conditional correlation matrix is assumed to be either time independent or to vary over time. We apply our model to pairs of seven daily stock returns belonging to the S&P 500 composite index and traded at the New York Stock Exchange. The results suggest that accounting for deterministic changes in the unconditional variances improves the fit of the multivariate CC-GARCH models to the data. The effect of careful specification of the variance equations on the estimated correlations is variable: in some cases rather small, in others more discernible. We also show empirically that the CC-GARCH models with time-varying unconditional variances using the TV-GARCH model outperform the other models under study in terms of out-of-sample forecasting performance. In addition, we find that portfolio volatility-timing strategies based on time-varying unconditional variances often outperform the unmodeled long-run variances strategy out-of-sample. As a by-product, we generalize news impact surfaces to the situation in which both the GARCH equations and the conditional correlations contain a deterministic component that is a function of time.     

Semi-parametric expected shortfall forecasting in financial markets

Bayesian methods have proved effective for quantile estimation, including for financial Value-at-Risk forecasting. Expected shortfall (ES) is a competing tail risk measure, favoured by the Basel Committee, that can be semi-parametrically estimated via asymmetric least squares. An asymmetric Gaussian density is proposed, allowing a likelihood to be developed, that facilitates both pseudo-maximum likelihood and Bayesian semi-parametric estimation, and leads to forecasts of quantiles, expectiles and ES. Further, the conditional autoregressive expectile class of model is generalised to two fully nonlinear families. Adaptive Markov chain Monte Carlo sampling schemes are developed for the Bayesian estimation. The proposed models are favoured in an empirical study forecasting eight financial return series: evidence of more accurate ES forecasting, compared to a range of competing methods, is found, while Bayesian estimated models tend to be more accurate. However, during a financial crisis period most models perform badly, while two existing models perform best.     

Quantile Regression for Location‐Scale Time Series Models with Conditional Heteroscedasticity

This paper considers quantile regression for a wide class of time series models including autoregressive and moving average (ARMA) models with asymmetric generalized autoregressive conditional heteroscedasticity errors. The classical mean‐variance models are reinterpreted as conditional location‐scale models so that the quantile regression method can be naturally geared into the considered models. The consistency and asymptotic normality of the quantile regression estimator is established in location‐scale time series models under mild conditions. In the application of this result to ARMA‐generalized autoregressive conditional heteroscedasticity models, more primitive conditions are deduced to obtain the asymptotic properties. For illustration, a simulation study and a real data analysis are provided.     

Space-time short- to medium-term wind speed forecasting

Accurate wind power forecasts depend on reliable wind speed forecasts. Numerical weather predictions utilize huge amounts of computing time, but still have rather low spatial and temporal resolution. However, stochastic wind speed forecasts perform well in rather high temporal resolution settings. They consume comparably little computing resources and return reliable forecasts, if forecasting horizons are not too long. In the recent literature, spatial interdependence is increasingly taken into consideration. In this paper we propose a new and quite flexible multivariate model that accounts for neighbouring weather stations’ information and as such, exploits spatial data at a high resolution. The model is applied to forecasting horizons of up to 1 day and is capable of handling a high resolution temporal structure. We use a periodic vector autoregressive model with seasonal lags to account for the interaction of the explanatory variables. Periodicity is considered and is modelled by cubic B-splines. Due to the model’s flexibility, the number of explanatory variables becomes huge. Therefore, we utilize time-saving shrinkage methods like lasso and elastic net for estimation. Particularly, a relatively newly developed iteratively re-weighted lasso and elastic net is applied that also incorporates heteroscedasticity. We compare our model to several benchmarks. The out-of-sample forecasting results show that the exploitation of spatial information increases the forecasting accuracy tremendously, in comparison to models in use so far.     

Cellular automata and Riccati equation models for diffusion of innovations

Innovation diffusion represents a central topic both for researchers and for managers and policy makers. Traditionally, it has been examined using the successful Bass models (BM, GBM), based on an aggregate differential approach, which assures flexibility and reliable forecasts. More recently, the rising interest towards adoptions at the individual level has suggested the use of agent based models, like Cellular Automata models (CA), that are generally implemented through computer simulations. In this paper we present a link between a particular kind of CA and a separable non autonomous Riccati equation, whose general structure includes the Bass models as a special case. Through this link we propose an alternative to direct computer simulations, based on real data, and a new aggregate model, which simultaneously considers birth and death processes within the diffusion. The main results, referred to the closed form solution, the identification and the statistical analysis of our new model, may be both of theoretical and empirical interest. In particular, we examine two applied case studies, illustrating some forecasting improvements obtained.