Abstract not found

In this paper we introduce a new class of models, called OSV, by combining an ordinal response model and the idea of stochastic volatility. Corresponding time series occur in high-frequency finance when the stocks are traded on a coarse grid. For parameter estimation we develop an efficient Grouped Move Multigrid Monte Carlo (GM-MGMC) sampler. This sampler is based on a scale transformation group, whose elements operate on the random samples of a certain conditional distribution. Also volatility estimates are provided. For illustration, we apply our new model class to price changes of the IBM stock. Dependencies on covariates are quantified and compared with theoretical results for such processes.

Ordinal stochastic volatility (OSV) models were recently developed and fitted by Müller and Czado (2008) to account for the discreteness of financial price changes, while allowing for stochastic volatility (SV). The model allows for exogenous factors both on the mean and volatility level. A Bayesian approach using Markov Chain Monte Carlo (MCMC) is followed to facilitate estimation in these parameter driven models. In this paper the applicability of the OSV model to financial stocks with different levels of trading activity is investigated and the influence of time between trades, volume, day time and the number of quotes between trades is determined. In a second focus we compare the performance of OSV models to SV models by developing model selection criteria. This analysis shows that the discreteness of price changes should not be ignored.

Abstract: Financial return series of su ¢ ciently high frequency display stylized facts such as volatility clustering, high kurtosis, low starting and slow-decaying autocorrelation function of squared returns and the so-called Taylor e¤ect. In order to evaluate the capacity of volatility models to reproduce these facts, we apply both standard and robust measures of kurtosis and autocorrelation of squares to …rst-order GARCH, EGARCH and ARSV models. Robust measures provide a fresh view on stylized facts which is useful because many …nancial time series are contaminated with outliers.

Quantiles provide a comprehensive description of the properties of a variable and tracking changes in quantiles over time using signal extraction methods can be informative. It is shown here how stationarity tests can be generalized to test the null hypothesis that a particular quantile is constant over time by using weighted indicators. Corresponding tests based on expectiles are also proposed; these might be expected to be more powerful for distributions that are not heavy-tailed. Tests for changing dispersion and asymmetry may be based on contrasts between particular quantiles or expectiles. We report Monte Carlo experiments investigating the e¤ectiveness of the proposed tests and then move on to consider how to test for relative time invariance, based on residuals from …tting a time-varying level or trend. Empirical examples, using stock returns and U.S. in‡ation, provide an indication of the practical importance of the tests.

Economic time series display features such as trend, seasonal, and cycle that we do not observe directly from the data. The cycle is of particular interest to economists as it is a measure of the fluctuations in economic activity. An unobserved components model attempts to capture the features of a time series by assuming that they follow stochastic processes that, when put together, yield the observations. The aim of this article is thus to illustrate the use of unobserved components models in economics and finance and to show how they can be used for forecasting and policy making. Setting up models in terms of components of interest helps in model building; see the discussions in [1] and [2] for a comparison with alternative approaches. A detailed treatment of unobserved components models is given in [3]. The statistical treatment of unobserved components models is based on the state-space form. The unobserved Digital Object Identifier 10.1109/MCS.2009.934465 components, which depend on the state vector, are related to the observations by a measurement equation. The Kalman filter is the basic recursion for estimating the state, and hence the unobserved components, in a linear state-space model (see “Kalman Filter”). The estimates, which are based on current and past observations, can be used to make predictions. Backward recursions yield smoothed estimates of components at each point in time based on past, current, and future observations. A set of one-step-ahead prediction errors, called innovations, is produced by the Kalman filter. In a Gaussian model, the innovations can be used to construct a likelihood function that can be maximized numerically with respect to unknown parameters in the system; see [4]. Once the parameters are estimated, the innovations can be used to construct test statistics that are designed to assess how well the model fits. The STAMP package [5] embodies a model-building procedure in which test statistics are produced as part of the output.

Time reversal invariance can be summarised as follows: no difference can be measured if a sequence of events is run forward or backward in time. Because price time series are dominated by a randomness that hides possible structures and orders, the existence of time reversal invariance requires care to be investigated. Different statistics are constructed with the property to be zero for time series which are time reversal invariant; they all show that high-frequency empirical foreign exchange prices are not invariant. The same statistics are applied to mathematical processes that should mimic empirical prices. Monte Carlo simulations show that only some ARCH processes with a multi-timescales structure can reproduce the empirical findings. A GARCH(1,1) process can only reproduce some asymmetry. On the other hand, all the stochastic volatility type processes are time reversal invariant. This clear difference related to the process structures gives some strong selection criterion for processes.