Many ways exist to measure and model financial asset volatility. In principle, as the frequency of the data increases, the quality of forecasts should improve. Yet, there is no consensus about a “true ” or "best " measure of volatility. In this paper we propose to jointly consider absolute daily returns, daily high-low range and daily realized volatility to develop a forecasting model based on their conditional dynamics. As all are non-negative series, we develop a multiplicative error model that is consistent and asymptotically normal under a wide range of specifications for the error density function. The estimation results show significant interactions between the indicators. We also show that one-month-ahead forecasts match well (both in and out of sample) the market-based volatility measure provided by an average of implied volatilities of index options as measured by VIX.

Abstract—For financial time series, the generation of error bars on the point prediction is important in order to estimate the corresponding risk. The Bayesian evidence framework, already successfully applied to design of multilayer perceptrons, is applied in this paper to least squares support vector machine (LS-SVM) regression in order to infer nonlinear models for predicting a time series and the related volatility. On the first level of inference, a statistical framework is related to the LS-SVM formulation which allows to include the time-varying volatility of the market by an appropriate choice of several hyperparameters. By the use of equality constraints and a 2-norm, the model parameters of the LS-SVM are obtained from a linear Karush-Kuhn-Tucker system in the dual space. Error bars on the model predictions are obtained by marginalizing over the model parameters. The hyperparameters of the model are inferred on the second level of inference. The inferred hyperparameters, related to the volatility, are used to construct a volatility model within the evidence framework. Model comparison is performed on the third level of inference in order to automatically tune the parameters of the kernel function and to select the relevant inputs. The LS-SVM formulation allows to derive analytic expressions in the feature space and practical expressions are obtained in the dual space replacing the inner product by the related kernel function using Mercer’s theorem. The one step ahead prediction performances obtained on the prediction of the weekly 90-day T-bill rate and the daily DAX30 closing prices show that significant out of sample sign predictions can be made with respect to the Pesaran-Timmerman test statistic. Index Terms—Bayesian inference, financial time series prediction, hyperparameter selection, least squares support vector machines (LS-SVMs), model comparison, volatility modeling. I.

In the framework of general semimartingale models we provide limit theorems for variational sums including the p-th power variation, i.e. the sum of p-th absolute powers of increments of a process. This gives new insight in the use of quadratic and realised power variation as an estimate for the integrated volatility in finance. It also provides a criterion to decide from high frequency data, whether a jump component should be included in the model. Furthermore, results on the asymptotic behaviour of integrals with respect to Levy processes, estimates for integrals with respect to Levy measures and non-parametric estimation for Levy processes will be derived and viewed in the framework of variational sums.

In the framework of stochastic volatility models we examine estimators for the integrated volatility based on the p-th power variation, i.e. the sum of p-th absolute powers of the log-returns. We derive consistency and distributional results for the estimators given high frequency data, especially taking into account what kind of process we may add to our model without e#ecting the estimate of the integrated volatility. This may on the one hand be interpreted as a possible flexibility in modelling, e.g. adding jumps or even leaving the framework of semimartingales by adding a fractional Brownian motion, or on the other hand as robustness against model misspecification.

A time-varying quantile can be …tted by formulating a time series model for the corresponding population quantile and iteratively applying a suitably modi…ed state space signal extraction algorithm. It is shown that such quantiles satisfy the de…ning property of …xed quantiles in having the appropriate number of observations above and below. Like quantiles, time-varying expectiles can be estimated by a state space signal extraction algorithm and they satisfy properties that generalize the moment conditions associated with …xed expectiles. Because the state space form can handle irregularly spaced observations, the proposed algorithms can be adapted to provide a viable means of computing spline-based non-parametric quantile and expectile regressions.

Abstract: In this paper, we study the transmission of an asymptotic bias in two-stage regressions with non-iid errors and random regressors in which the possible endogeneity of some explanatory variables is treated via first-stage predictive equations. In particular, we characterise the transmission of an asymptotic bias in the first-stage estimates to the second-stage estimates. As an example, we fully develop the case of two-stage quantile regressions. Finally, Monte Carlo simulation results illustrate the occurrence of the bias in small samples.

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Least absolute deviation estimation for fractionally

We consider the problem of modelling the dependence between financial markets. In financial economics, the classical tool is the Pearson (or linear correlation) to compare the dependence structure. We show that this coefficient does not give a precise information on the dependence structure. Instead, we propose a conceptual framework based on copulas. Two applications are proposed. The first one concerns the study of extreme dependence between international equity markets. The second one concerns the analysis of the East Asian crisis. 1