David Ardia

Bayesian estimation of a Markov-switching threshold asymmetric GARCH model with Student-t innovations

A Bayesian estimation of a regime-switching threshold asymmetric GARCH model is proposed. The specification is based on a Markov-switching model with Student-t innovations and K separate GJR(1,1) processes whose asymmetries are located at free non-positive threshold parameters. The model aims at determining whether or not: (i) structural breaks are present within the volatility dynamics; (ii) asymmetries (leverage effects) are present, and are different between regimes and (iii) the threshold parameters (locations of bad news) are similar between regimes. A novel MCMC scheme is proposed which allows for a fully automatic Bayesian estimation of the model. The presence of two distinct volatility regimes is shown in an empirical application to the Swiss Market Index log-returns. The posterior results indicate no differences with regards to the asymmetries and their thresholds when comparing highly volatile periods with the milder ones. Comparisons with a single-regime specification indicates a better in-sample fit and a better forecasting performance for the Markov-switching model. Copyright The Author(s). Journal compilation Royal Economic Society 2008

Financial Risk Management with Bayesian Estimation of GARCH Models

B where B is a K × K diagonal matrix with β in its diagonal. Note that this mistake is also present in Ardia (2009, page 125). Many thanks to Alex Finke for pointing out this error. 1 γ 2 instead of 1 γ for the first term in the curly bracket. References – The reference Bauwens et al. (2004) should be the working paper Bauwens et al. (2002).

A comparative study of Monte Carlo methods for efficient evaluation of marginal likelihood

Strategic choices for efficient and accurate evaluation of marginal likelihoods by means of Monte Carlo simulation methods are studied for the case of highly non-elliptical posterior distributions. A comparative analysis is presented of possible advantages and limitations of different simulation techniques; of possible choices of candidate distributions and choices of target or warped target distributions; and finally of numerical standard errors. The importance of a robust and flexible estimation strategy is demonstrated where the complete posterior distribution is explored. Given an appropriately yet quickly tuned adaptive candidate, straightforward importance sampling provides a computationally efficient estimator of the marginal likelihood (and a reliable and easily computed corresponding numerical standard error) in the cases investigated, which include a non-linear regression model and a mixture GARCH model. Warping the posterior density can lead to a further gain in efficiency, but it is more important that the posterior kernel be appropriately wrapped by the candidate distribution than that it is warped.

Jump-Diffusion Calibration Using Differential Evolution

The estimation of a jump-diffusion model via Differential Evolution is presented. Finding the maximum likelihood estimator for such processes is a tedious task due to the multimodality of the likelihood function. The performance of the Differential Evolution algorithm is compared to standard optimization techniques.

Moments of Standardized Fernandez-Steel Skewed Distributions: Applications to the Estimation of GARCH-Type Models

We provide general expressions for obtaining raw, absolute and conditional moments for a standardized version of the class of skewed distributions proposed by Fernandez and Steel (1998). We show that these expressions are readily programmable in addition of greatly reducing the computational cost. We discuss several applications that are relevant for the purpose of estimating asymmetric conditional volatility models under skewed distributions.

Density Prediction of Stock Index Returns Using GARCH Models: Frequentist or Bayesian Estimation?

Using well-known GARCH models for density prediction of daily S&P 500 and Nikkei 225 index returns, a comparison is provided between frequentist and Bayesian estimation. No significant difference is found between the qualities of the forecasts of the whole density, whereas the Bayesian approach exhibits significantly better left-tail forecast accuracy.Using GARCH models for density prediction of stock index returns, a comparison is provided between frequentist and Bayesian estimation. No significant difference is found between qualities of whole density forecasts, whereas the Bayesian approach exhibits significantly better left-tail forecast accuracy.

The Impact of Parameter and Model Uncertainty on Market Risk Predictions from GARCH-Type Models

We study the impact of parameter and model uncertainty on the left-tail of predictive densities and in particular on VaR forecasts. To this end, we evaluate the predictive performance of several GARCH-type models estimated via Bayesian and maximum likelihood techniques. In addition to individual models, several combination methods are considered such as Bayesian model averaging and (censored) optimal pooling for linear, log or beta linear pools. Daily returns for a set of stock market indexes are predicted over about 13 years from the early 2000s. We find that Bayesian predictive densities improve the VaR backtest at the 1% risk level for single models and for linear and log pools. We also find that the robust VaR backtest exhibited by linear and log pools is better than the one of single models at the 5% risk level. Finally, the equally-weighted linear pool of Bayesian predictives tends to be the best VaR forecaster in a set of 42 forecasting techniques.

Implied returns and the choice of mean-variance efficient portfolio proxy

Implied expected returns are the expected returns for which a supposedly mean–variance efficient portfolio is effectively efficient, given a covariance matrix. The authors analyze the properties of monthly implied expected stock returns and study their sensitivity to the choice of mean–variance efficient portfolio proxy. For the universe of S&P 100 stocks over the period from 1984 to 2014, they find that using as risk-based portfolio proxy with respect to a market capitalization or fundamental value portfolio brings its biggest gains in return forecasts’ stability and precision. For all the proxies considered, they report that the implied expected returns outperform forecasts based on a time-series model in stability and precision.

The Impact of Covariance Misspecification in Risk-Based Portfolios

The equal-risk-contribution, inverse-volatility weighted, maximum-diversification and minimum-variance portfolio weights are all direct functions of the estimated covariance matrix. We perform a Monte Carlo study to assess the impact of covariance matrix misspecification to these risk-based portfolios. Our results show that the equal-risk-contribution and inverse-volatility weighted portfolio weights are relatively robust to covariance misspecification, but that the minimum-variance and maximum-diversification portfolios are highly sensitive to errors in the estimated variance and correlation, respectively.

Value-at-Risk Prediction in R with the GAS Package

Financial risk managers routinely use non-linear time series models to predict the downside risk of the capital under management. They also need to evaluate the adequacy of their model using so-called backtesting procedures. The latter involve hypothesis testing and evaluation of loss functions. This paper shows how the R package GAS can be used for both the dynamic prediction and the evaluation of downside risk. Emphasis is given to the two key financial downside risk measures: Value-at-Risk (VaR) and Expected Shortfall (ES). High-level functions for: (i) prediction, (ii) backtesting, and (iii) model comparison are discussed, and code examples are provided. An illustration using the series of log-returns of the Dow Jones Industrial Average constituents is reported.GAS models have been recently proposed in time-series econometrics as valuable tools for signal extraction and prediction. This paper details how financial risk managers can use GAS models for Value-at-Risk (VaR) prediction using the novel GAS package for R. Details and code snippets for prediction, comparison and backtesting with GAS models are presented. An empirical application considering Dow Jones Index constituents investigates the VaR forecasting performance of GAS models.