Bootstrapping realized multivariate volatility measures

We propose a bootstrap method for statistics that are a function of multivariate high frequency returns such as realized regression, covariance and correlation coefficients. We show that the finite sample performance of the bootstrap is superior to the existing first-order asymptotic theory. Nevertheless, and contrary to the existing results in the bootstrap literature for regression models subject to error heteroskedasticity, the Edgeworth expansion for the pairs bootstrap that we develop here shows that this method is not second-order accurate. We argue that this is due to the fact that the conditional mean parameters of realized regression models are heterogeneous under stochastic volatility.     

Forecasting multivariate realized stock market volatility

We present a new matrix-logarithm model of the realized covariance matrix of stock returns. The model uses latent factors which are functions of lagged volatility, lagged returns and other forecasting variables. The model has several advantages: it is parsimonious; it does not require imposing parameter restrictions; and, it results in a positive-definite estimated covariance matrix. We apply the model to the covariance matrix of size-sorted stock returns and find that two factors are sufficient to capture most of the dynamics.     

Edgeworth expansions for realized volatility and related estimators

This paper shows that the asymptotic normal approximation is often insufficiently accurate for volatility estimators based on high frequency data. To remedy this, we derive Edgeworth expansions for such estimators. The expansions are developed in the framework of small-noise asymptotics. The results have application to Cornish–Fisher inversion and help setting intervals more accurately than those relying on normal distribution.     

Forecasting the realized volatility of Energy Stock Market: A multimodel comparison

The realized volatility forecasting of energy sector stocks facilitates the establishment of corresponding risk warning mechanisms and investor decisions. In this paper, we collected two different energy sector indices and used different methods, namely principal component analysis (PCA) and sparse principal component analysis (SPCA), to extract features, and combined LSTM and GRU to construct 12 different models. The results show that the SPCA-LSTM model we constructed has the best forecasting performance in the realized volatility forecasting of energy indices, and SPCA has better forecasting results than PCA in the feature extraction stage. The results of the robustness test indicate that our results are robust.     

Jump-robust volatility estimation using nearest neighbor truncation

We propose two new jump-robust estimators of integrated variance that allow for an asymptotic limit theory in the presence of jumps. Specifically, our MedRV estimator has better efficiency properties than the tripower variation measure and displays better finite-sample robustness to jumps and small (“zero”) returns. We stress the benefits of local volatility measures using short return blocks, as this greatly alleviates the downward biases stemming from rapid fluctuations in volatility, including diurnal (intraday) U-shape patterns. An empirical investigation of the Dow Jones 30 stocks and extensive simulations corroborate the robustness and efficiency properties of our nearest neighbor truncation estimators.     

Realized volatility forecasting and market microstructure noise

We extend the analytical results for reduced form realized volatility based forecasting in ABM (2004) to allow for market microstructure frictions in the observed high-frequency returns. Our results build on the eigenfunction representation of the general stochastic volatility class of models developed byMeddahi (2001). In addition to traditional realized volatility measures and the role of the underlying sampling frequencies, we also explore the forecasting performance of several alternative volatility measures designed to mitigate the impact of the microstructure noise. Our analysis is facilitated by a simple unified quadratic form representation for all these estimators. Our results suggest that the detrimental impact of the noise on forecast accuracy can be substantial. Moreover, the linear forecasts based on a simple-to-implement ‘average’ (or ‘subsampled’) estimator obtained by averaging standard sparsely sampled realized volatility measures generally perform on par with the best alternative robust measures.     

Moving average stochastic volatility models with application to inflation forecast

We introduce a new class of models that has both stochastic volatility and moving average errors, where the conditional mean has a state space representation. Having a moving average component, however, means that the errors in the measurement equation are no longer serially independent, and estimation becomes more difficult. We develop a posterior simulator that builds upon recent advances in precision-based algorithms for estimating these new models. In an empirical application involving US inflation we find that these moving average stochastic volatility models provide better in-sample fitness and out-of-sample forecast performance than the standard variants with only stochastic volatility.     

Data-based ranking of realised volatility estimators

This paper presents new methods for comparing the accuracy of estimators of the quadratic variation of a price process. I provide conditions under which the relative accuracy of competing estimators can be consistently estimated (as T→∞), and show that forecast evaluation tests may be adapted to the problem of ranking these estimators. The proposed methods avoid making specific assumptions about microstructure noise, and facilitate comparisons of estimators that would be difficult using methods from the extant literature, such as those based on different sampling schemes. An application to high frequency IBM data between 1996 and 2007 illustrates the new methods.     

A semiparametric stochastic volatility model

In this paper the correlation structure in the classical leverage stochastic volatility (SV) model is generalized based on a linear spline. In the new model the correlation between the return and volatility innovations is time varying and depends nonparametrically on the type of news arrived to the market. Theoretical properties of the proposed model are examined. The model estimation and comparison are conducted by Bayesian methods. The performance of the estimates are examined in simulations. The new model is fitted to daily and weekly US data and compared with the classical SV and GARCH models in terms of their in-sample and out-of-sample performances. Empirical results suggest evidence in favor of the proposed model. In particular, the new model finds strong evidence of time varying leverage effect in individual stocks when the classical model fails to identify the leverage effect.     

Efficient estimation of a multivariate multiplicative volatility model

We propose a multivariate generalization of the multiplicative volatility model of Engle and Rangel (2008), which has a nonparametric long run component and a unit multivariate GARCH short run dynamic component. We suggest various kernel-based estimation procedures for the parametric and nonparametric components, and derive the asymptotic properties thereof. For the parametric part of the model, we obtain the semiparametric efficiency bound. Our method is applied to a bivariate stock index series. We find that the univariate model of Engle and Rangel (2008) appears to be violated in the data whereas our multivariate model is more consistent with the data.     

Volatility forecast comparison using imperfect volatility proxies

The use of a conditionally unbiased, but imperfect, volatility proxy can lead to undesirable outcomes in standard methods for comparing conditional variance forecasts. We motivate our study with analytical results on the distortions caused by some widely used loss functions, when used with standard volatility proxies such as squared returns, the intra-daily range or realised volatility. We then derive necessary and sufficient conditions on the functional form of the loss function for the ranking of competing volatility forecasts to be robust to the presence of noise in the volatility proxy, and derive some useful special cases of this class of “robust” loss functions. The methods are illustrated with an application to the volatility of returns on IBM over the period 1993 to 2003.     

Adaptive estimation of the dynamics of a discrete time stochastic volatility model

This paper is concerned with the discrete time stochastic volatility model _Yi=exp(_Xi/2)_ηi$Y_i=exp(X_i/2){\eta }_i$, _Xi+1=b(_Xi)+σ(_Xi)_ξi+1$X_{i+1}=b\left(X_i\right)+\sigma \left(X_i\right){\xi }_{i+1}$, where only (_Yi)$\left(Y_i\right)$ is observed. The model is rewritten as a particular hidden model: _Zi=_Xi+_εi$Z_i=X_i+{\epsilon }_i$, _Xi+1=b(_Xi)+σ(_Xi)_ξi+1$X_{i+1}=b\left(X_i\right)+\sigma \left(X_i\right){\xi }_{i+1}$, where (_ξi)$\left({\xi }_i\right)$ and (_εi)$\left({\epsilon }_i\right)$ are independent sequences of i.i.d. noise. Moreover, the sequences (_Xi)$\left(X_i\right)$ and (_εi)$\left({\epsilon }_i\right)$ are independent and the distribution of ε$\epsilon$ is known. Then, our aim is to estimate the functions b$b$ and σ²${\sigma }^2$ when only observations _Z1,…,_Zn$Z_1,\dots ,Z_n$ are available. We propose to estimate bf$bf$ and (b²+σ²)f$(b^2+{\sigma }^2)f$ and study the integrated mean square error of projection estimators of these functions on automatically selected projection spaces. By ratio strategy, estimators of b$b$ and σ²${\sigma }^2$ are then deduced. The mean square risk of the resulting estimators are studied and their rates are discussed. Lastly, simulation experiments are provided: constants in the penalty functions defining the estimators are calibrated and the quality of the estimators is checked on several examples.     

Asymptotically distribution-free tests for the volatility function of a diffusion

This paper develops two tests for parametric volatility function of a diffusion model based on Khmaladze (1981)’s martingale transformation. The tests impose no restrictions on the functional form of the drift function and are shown to be asymptotically distribution-free. The tests are consistent against a large class of fixed alternatives and have nontrivial power against a class of root-n$n$ local alternatives. The paper also extends the tests of volatility to testing for joint specifications of drift and volatility. Monte Carlo simulations show that the tests perform well in finite samples. The proposed tests are then applied to testing models of short-term interest, using data of Treasury bill rate and Eurodollar deposit rate. The empirical results show that the commonly used CKLS volatility function of Chan et al. (1992) fits volatility function poorly and none of the parametric interest rate models considered in the paper fit data well.     

Realized volatility forecasting and option pricing

A growing literature advocates the use of microstructure noise-contaminated high-frequency data for the purpose of volatility estimation. This paper evaluates and compares the quality of several recently-proposed estimators in the context of a relevant economic metric, i.e., profits from option pricing and trading. Using forecasts obtained by virtue of alternative volatility estimates, agents price short-term options on the S&P 500 index before trading with each other at average prices. The agents’ average profits and the Sharpe ratios of the profits constitute the criteria used to evaluate alternative volatility estimates and the corresponding forecasts. For our data, we find that estimators with superior finite sample Mean-squared-error properties generate higher average profits and higher Sharpe ratios, in general. We confirm that, even from a forecasting standpoint, there is scope for optimizing the finite sample properties of alternative volatility estimators as advocated by Bandi and Russell [Bandi, F.M., Russell, J.R., 2005. Market microstructure noise, integrated variance estimators, and the accuracy of asymptotic approximations. Working Paper; Bandi, F.M., Russell, J.R., 2008b. Microstructure noise, realized variance, and optimal sampling. Review of Economic Studies 75, 339–369] in recent work.     

Forecasting the Value-at-Risk of REITs using realized volatility jump models

This paper examines jump risk in the time series of Real Estate Investment Trusts (REITs). Using high-frequency index-level and firm-level data, the econometric model in this paper integrates jumps into the volatility forecast by estimating jump augmented Heterogeneous Autoregressive (HAR) models of realized volatility. To assess the information value of these specifications, their forecasting accuracies for generating one-step ahead daily Value-at-Risk are also compared with other VaR specifications, including those generated from historical returns, bootstrap technique, and severity loss distribution.     

Online forecast combinations of distributions: Worst case bounds

This paper considers forecasts with distribution functions that may vary through time. The forecast is achieved by time varying combinations of individual forecasts. We derive theoretical worst case bounds for general algorithms based on multiplicative updates of the combination weights. The bounds are useful for studying properties of forecast combinations when data are non-stationary and there is no unique best model.     

Asymptotics for duration-driven long range dependent processes

We consider processes with second order long range dependence resulting from heavy tailed durations. We refer to this phenomenon as duration-driven long range dependence (DDLRD), as opposed to the more widely studied linear long range dependence based on fractional differencing of an i.i.d. process. We consider in detail two specific processes having DDLRD, originally presented in Taqqu and Levy [1986. Using renewal processes to generate long-range dependence and high variability. Dependence in Probability and Statistics. Birkhauser, Boston, pp. 73–89], and Parke [1999. What is fractional integration? Review of Economics and Statistics 81, 632–638]. For these processes, we obtain the limiting distribution of suitably standardized discrete Fourier transforms (DFTs) and sample autocovariances. At low frequencies, the standardized DFTs converge to a stable law, as do the standardized sample autocovariances at fixed lags. Finite collections of standardized sample autocovariances at a fixed set of lags converge to a degenerate distribution. The standardized DFTs at high frequencies converge to a Gaussian law. Our asymptotic results are strikingly similar for the two DDLRD processes studied. We calibrate our asymptotic results with a simulation study which also investigates the properties of the semiparametric log periodogram regression estimator of the memory parameter.     

Threshold bipower variation and the impact of jumps on volatility forecasting

This study reconsiders the role of jumps for volatility forecasting by showing that jumps have a positive and mostly significant impact on future volatility. This result becomes apparent once volatility is separated into its continuous and discontinuous components using estimators which are not only consistent, but also scarcely plagued by small sample bias. With the aim of achieving this, we introduce the concept of threshold bipower variation, which is based on the joint use of bipower variation and threshold estimation. We show that its generalization (threshold multipower variation) admits a feasible central limit theorem in the presence of jumps and provides less biased estimates, with respect to the standard multipower variation, of the continuous quadratic variation in finite samples. We further provide a new test for jump detection which has substantially more power than tests based on multipower variation. Empirical analysis (on the S&P500 index, individual stocks and US bond yields) shows that the proposed techniques improve significantly the accuracy of volatility forecasts especially in periods following the occurrence of a jump.     

Distribution-free tests of stochastic monotonicity

This article proposes a nonparametric test of monotonicity for conditional distributions and its moments. Unlike previous proposals, our method does not require smooth estimation of the derivatives of nonparametric curves. Distinguishing features of our approach are that critical values are pivotal under the null in finite samples and that the test is invariant to any monotonic continuous transformation of the explanatory variable. The test statistic is the sup-norm of the difference between the empirical copula function and its least concave majorant with respect to the explanatory variable coordinate. The resulting test is able to detect local alternatives converging to the null at the parametric rate n^−1/2$n^{-1/2}$, with n$n$ the sample size. The finite sample performance of the test is examined by means of a Monte Carlo experiment and an application to testing intergenerational income mobility.