Modeling and predicting the CBOE market volatility index

This paper performs a thorough statistical examination of the time-series properties of the market volatility index (VIX) from the Chicago Board Options Exchange (CBOE). The motivation lies on the widespread consensus that the VIX is a barometer to the overall market sentiment as to what concerns risk appetite. To assess the statistical behavior of the time series, we run a series of preliminary analyses whose results suggest there is some long-range dependence in the VIX index. This is consistent with the strong empirical evidence in the literature supporting long memory in both options-implied and realized volatilities. We thus resort to linear and nonlinear heterogeneous autoregressive (HAR) processes, including smooth transition and threshold HAR-type models, as well as to smooth transition autoregressive trees (START) for modeling and forecasting purposes. The in-sample results for the HAR-type indicate that they cope with the long-range dependence in the VIX time series as well as the more popular ARFIMA model. In addition, the highly nonlinear START specification also does a god job in controlling for the long memory. The out-of-sample analysis evince that the linear ARMA and ARFIMA models perform very well in the short run and very poorly in the long-run, whereas the START model entails by far the best results for the longer horizon despite of failing at shorter horizons. In contrast, the HAR-type models entail reasonable relative performances in most horizons. Finally, we also show how a simple forecast combination brings about great improvements in terms of predictive ability for most horizons.heterogeneous autoregression, implied volatility, smooth transition, VIX.

Modelling multiple regimes in financial volatility with a flexible coefficient GARCH model

In this paper a flexible GARCH-type model is developed with the aim of describing sign and size asymmetries in financial volatility as well as intermittent dynamics and excess of kurtosis. A sufficient condition for strict stationarity and ergodicity of the model is established and the existence of the second- and fourth-order moments is discussed. It is shown that the model may have explosive regimes and still be strictly stationary and ergodic. Furthermore, estimation of the parameters is carefully addressed and the asymptotic properties of the quasi-maximum likelihood estimator is derived. A modeling cycle based on a sequence of simple and easily implemented Lagrange multiplier tests is discussed in order to avoid the estimation of unidentified models. A Monte-Carlo experiment is designed to evaluate the methodology. Empirical examples are used to illustrate the use of the model in practical situations.Volatility, GARCH models, multiple regimes, nonlinear time series, smooth transition, finance, asymmetry, leverage effect, excess of kurtosis.

Asymmetric effects and long memory in the volatility of Dow Jones stocks

Does volatility reflect a continuous reaction to past shocks or changes in the markets induce shifts in the volatility dynamics? In this paper, we provide empirical evidence that cumulated price variations convey meaningful information about multiple regimes in the realized volatility of stocks, where large falls (rises) in prices are linked to persistent regimes of high (low) variance in stock returns. Incorporating past cumulated daily returns as a explanatory variable in a flexible and systematic nonlinear framework, we estimate that falls of different magnitudes over less than two months are associated with volatility levels 20% and 60% higher than the average of periods with stable or rising prices. We show that this effect accounts for large empirical values of long memory parameter estimates. Finally, we analyze that the proposed model significantly improves out of sample performance in relation to standard methods. This result is more pronounced in periods of high volatility.Realized volatility, long memory, nonlinear models, asymmetric effects, regime switching, regression trees, smooth transition, value-at-risk, forecasting, empirical finance.

Forecasting Realized Volatility with Linear and Nonlinear Models

In this paper we consider a nonlinear model based on neural networks as well as linear models to forecast the daily volatility of the S&P 500 and FTSE 100 indexes. As a proxy for daily volatility, we consider a consistent and unbiased estimator of the integrated volatility that is computed from high frequency intra-day returns. We also consider a simple algorithm based on bagging (bootstrap aggregation) in order to specify the models analyzed in this paper.Financial econometrics, volatility forecasting, neural networks, nonlinear models, realized volatility, bagging.

Linear Programming-Based Estimators in Simple Linear Regression

In this paper we introduce a linear programming estimator (LPE) for the slope parameter in a constrained linear regression model with a single regressor. The LPE is interesting because it can be superconsistent in the presence of an endogenous regressor and, hence, preferable to the ordinary least squares estimator (LSE). Two dierent cases are considered as we investigate the statistical properties of the LPE. In the rst case, the regressor is assumed to be xed in repeated samples. In the second, the regressor is stochastic and potentially endogenous. For both cases the strong consistency and exact nite-sample distribution of the LPE is established. Conditions under which the LPE is consistent in the presence of serially correlated, heteroskedastic errors are also given. Finally, we describe how the LPE can be extended to the case with multiple regressors and conjecture that the extended estimator is consistent under conditions analogous to the ones given herein. Finite-sample properties of the LPE and extended LPE in comparison to the LSE and instrumental variable estimator (IVE) are investigated in a simulation study. One advantage of the LPE is that it does not require an instrument.

Asymmetries, breaks, and long-range dependence: An estimation framework for daily realized volatility

We study the simultaneous occurrence of long memory and nonlinear effects, such as structural breaks and thresholds, in autoregressive moving average (ARMA) time series models and apply our modeling framework to series of daily realized volatility. Asymptotic theory for the quasi-maximum likelihood estimator is developed and a sequence of model specification tests is described. Our framework allows for general nonlinear functions, including smoothly changing intercepts. The theoretical results in the paper can be applied to any series with long memory and nonlinearity. We apply the methodology to realized volatility of individual stocks of the Dow Jones Industrial Average during the period 1995 to 2005. We find strong evidence of nonlinear effects and explore different specifications of the model framework. A forecasting exercise demonstrates that allowing for nonlinearities in long memory models yields significant performance gains.Realized volatility, structural breaks, smooth transitions, nonlinear models, long memory, persistence.

Forecasting realized volatility models:the benefits of bagging and nonlinear specifications

We forecast daily realized volatilities with linear and nonlinear models and evaluate the benefits of bootstrap aggregation (bagging) in producing more precise forecasts. We consider the linear autoregressive (AR) model, the Heterogeneous Autoregressive model (HAR), and a non-linear HAR model based on a neural network specification that allows for logistic transition effects (NNHAR). The models and the bagging schemes are applied to the realized volatility time series of the S&P500 index from 3-Jan-2000 through 30-Dec-2005. Our main findings are: (1) For the HAR model, bagging successfully averages over the randomness of variable selection; however, when the NN model is considered, there is no clear benefit from using bagging; (2) including past returns in the models improves the forecast precision; and (3) the NNHAR model outperforms the linear alternatives.

Modeling and Forecasting Short-term Interest Rates: The Benefits of Smooth Regimes, Macroeconomic Variables, and Bagging

In this paper we propose a smooth transition tree model for both the conditional mean and variance of the short-term interest rate process. The estimation of such models is addressed and the asymptotic properties of the quasi-maximum likelihood estimator are derived. Model specification is also discussed. When the model is applied to the US short-term interest rate we find (1) leading indicators for inflation and real activity are the most relevant predictors in characterizing the multiple regimes’ structure; (2) the optimal model has three limiting regimes. Moreover, we provide empirical evidence of the power of the model in forecasting the first two conditional moments when it is used in connection with bootstrap aggregation (bagging).short-term interest rate, regression tree, smooth transition, conditional variance, bagging, asymptotic theory

Realized volatility: a review

This paper reviews the exciting and rapidly expanding literature on realized volatility. After presenting a general univariate framework for estimating realized volatilities, a simple discrete time model is presented in order to motivate the main results. A continuous time specification provides the theoretical foundation for the main results in this literature. Cases with and without microstructure noise are considered, and it is shown how microstructure noise can cause severe problems in terms of consistent estimation of the daily realized volatility. Independent and dependent noise processes are examined. The most important methods for providing consistent estimators are presented, and a critical exposition of different techniques is given. The finite sample properties are discussed in comparison with their asymptotic properties. A multivariate model is presented to discuss estimation of the realized covariances. Various issues relating to modelling and forecasting realized volatilities are considered. The main empirical findings using univariate and multivariate methods are summarized.

Modelling and forecasting short-term electricity load: a two step methodology

The goal of this paper is to develop a forecasting model of the hourly electricity load demand in the area covered by an utility company located in the southeast of Brazil. A di®erent model is constructed for each hour of day, thus there are 24 di®erent models. Each model is based on a decomposition of the daily series of each hour in two components. The ¯rst component is purely deterministic and is related to trends, seasonality, and special days e®ect. The second one is stochastic and follows a linear autoregressive model. The multi-step forecasting performance of the proposed methodology is compared with a benchmark model and the results indicate that our proposal is a useful tool for electricity load forecasting.