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on Econometric Time Series |
| By: | Mark J. Jensen (Federal Reserve Bank of Atlanta, USA); John M. Maheu (Department of Economics, University of Toronto, Canada; RCEA, Italy) |
| Abstract: | This paper proposes a Bayesian nonparametric modeling approach for the return distribution in multivariate GARCH models. In contrast to the parametric literature the return distribution can display general forms of asymmetry and thick tails. An innite mixture of multivariate normals is given a exible Dirichlet process prior. The GARCH functional form enters into each of the components of this mixture. We discuss conjugate methods that allow for scale mixtures and nonconjugate methods which provide mixing over both the location and scale of the normal components. MCMC methods are introduced for posterior simulation and computation of the predictive density. Bayes factors and density forecasts with comparisons to GARCH models with Student-t innovations demonstrate the gains from our exible modeling approach. |
| JEL: | C11 C14 C32 C58 |
| Date: | 2012–06 |
| URL: | http://d.repec.org/n?u=RePEc:rim:rimwps:48_12&r=ets |
| By: | Mark J. Jensen (Federal Reserve Bank of Atlanta, USA); John M. Maheu (University of Toronto, Canada; RCEA, Italy) |
| Abstract: | In this paper we extend the parametric, asymmetric, stochastic volatility model (ASV), where returns are correlated with volatility, by flexibly modeling the bivariate distribution of the return and volatility innovations nonparametrically. Its novelty is in modeling the joint, conditional, return-volatility, distribution with a infinite mixture of bivariate Normal distributions with mean zero vectors, but having unknown mixture weights and covariance matrices. This semiparametric ASV model nests stochastic volatility models whose innovations are distributed as either Normal or Student-t distributions, plus the response in volatility to unexpected return shocks is more general than the fixed asymmetric response with the ASV model. The unknown mixture parameters are modeled with a Dirichlet Process prior. This prior ensures a parsimonious, finite, posterior, mixture that bests represents the distribution of the innovations and a straightforward sampler of the conditional posteriors. We develop a Bayesian Markov chain Monte Carlo sampler to fully characterize the parametric and distributional uncertainty. Nested model comparisons and out-of-sample predictions with the cumulative marginal-likelihoods, and one-day-ahead, predictive log-Bayes factors between the semiparametric and parametric versions of the ASV model shows the semiparametric model forecasting more accurate empirical market returns. A major reason is how volatility responds to an unexpected market movement. When the market is tranquil, expected volatility reacts to a negative (positive) price shock by rising (initially declining, but then rising when the positive shock is large). However, when the market is volatile, the degree of asymmetry and the size of the response in expected volatility is muted. In other words, when times are good, no news is good news, but when times are bad, neither good nor bad news matters with regards to volatility. |
| Keywords: | Bayesian nonparametrics, cumulative Bayes factor, Dirichlet process mixture, infinite mixture model, leverage effect, marginal likelihood, MCMC, non-normal, stochastic volatility, volatility-return relationship |
| JEL: | C11 C14 C53 C58 |
| Date: | 2012–06 |
| URL: | http://d.repec.org/n?u=RePEc:rim:rimwps:45_12&r=ets |
| By: | Martin Burda (Department of Economics, University of Toronto, Canada; IES, Charles University, Czech Republic); John M. Maheu (Department of Economics, University of Toronto, Canada; RCEA, Italy) |
| Abstract: | Hamiltonian Monte Carlo (HMC) is a recent statistical procedure to sample from complex distributions. Distant proposal draws are taken in a sequence of steps following the Hamiltonian dynamics of the underlying parameter space, often yielding superior mixing properties of the resulting Markov chain. However, its performance can deteriorate sharply with the degree of irregularity of the underlying likelihood due to its lack of local adaptability in the parameter space. Riemann Manifold HMC (RMHMC), a locally adaptive version of HMC, alleviates this problem, but at a substantially increased computational cost that can become prohibitive in high-dimensional scenarios. In this paper we propose the Adaptively Updated HMC (AUHMC), an alternative inferential method based on HMC that is both fast and locally adaptive, combining the advantages of both HMC and RMHMC. The benefits become more pronounced with higher dimensionality of the parameter space and with the degree of irregularity of the underlying likelihood surface. We show that AUHMC satisfies detailed balance for a valid MCMC scheme and provide a comparison with RMHMC in terms of effective sample size, highlighting substantial efficiency gains of AUHMC. Simulation examples and an application of the BEKK GARCH model show the practical usefulness of the new posterior sampler. |
| Keywords: | High-dimensional joint sampling, Markov chain Monte Carlo |
| JEL: | C01 C11 C15 C32 |
| Date: | 2012–06 |
| URL: | http://d.repec.org/n?u=RePEc:rim:rimwps:46_12&r=ets |
| By: | Xin Jin (Department of Economics, University of Toronto, Canada); John M. Maheu (Department of Economics, University of Toronto, Canada; RCEA, Italy) |
| Abstract: | This paper proposes new dynamic component models of returns and realized covariance (RCOV) matrices based on time-varying Wishart distributions. Bayesian estimation and model comparison is conducted with a range of multivariate GARCH models and existing RCOV models from the literature. The main method of model comparison consists of a term-structure of density forecasts of returns for multiple forecast horizons. The new joint return-RCOV models provide superior density forecasts for returns from forecast horizons of 1 day to 3 months ahead as well as improved point forecasts for realized covariances. Global minimum variance portfolio selection is improved for forecast horizons up to 3 weeks out. |
| Keywords: | Wishart distribution, predictive likelihoods, density forecasts, realized covariance targeting, MCMC |
| JEL: | C11 C32 C53 G17 |
| Date: | 2012–06 |
| URL: | http://d.repec.org/n?u=RePEc:rim:rimwps:49_12&r=ets |
| By: | Satya N. Majumdar; Gregory Schehr; Gregor Wergen |
| Abstract: | We study the statistics of records of a one-dimensional random walk of n steps, starting from the origin, and in presence of a constant bias c. At each time-step the walker makes a random jump of length \eta drawn from a continuous distribution f(\eta) which is symmetric around a constant drift c. We focus in particular on the case were f(\eta) is a symmetric stable law with a L\'evy index 0 < \mu \leq 2. The record statistics depends crucially on the persistence probability which, as we show here, exhibits different behaviors depending on the sign of c and the value of the parameter \mu. Hence, in the limit of a large number of steps n, the record statistics is sensitive to these parameters (c and \mu) of the jump distribution. We compute the asymptotic mean record number <R_n> after n steps as well as its full distribution P(R,n). We also compute the statistics of the ages of the longest and the shortest lasting record. Our exact computations show the existence of five distinct regions in the (c, 0 < \mu \leq 2) strip where these quantities display qualitatively different behaviors. We also present numerical simulation results that verify our analytical predictions. |
| Date: | 2012–06 |
| URL: | http://d.repec.org/n?u=RePEc:arx:papers:1206.6972&r=ets |
| By: | Simwaka, Kisu |
| Abstract: | Fractional cointegration has attracted interest in time series econometrics in recent years (see among others, Dittmann 2004). According to Engle and Granger (1987), the concept of fractional cointegration was introduced to generalize the traditional cointegration to the long memory framework. Although cointegration tests have been developed for the traditional cointegration framework, these tests do not take into account fractional cointegration. This paper proposes a bootstrap procedure to test for time-varying fractional cointegration. |
| Keywords: | Time-varying fractional cointegration; bootstrap procedure |
| JEL: | C52 C15 C22 |
| Date: | 2012–06–26 |
| URL: | http://d.repec.org/n?u=RePEc:pra:mprapa:39698&r=ets |