A nesting framework for Markov-switching GARCH modelling with an application to the German stock market

In this paper, we establish a generalized two-regime Markov-switching GARCH model which enables us to specify complex (symmetric and asymmetric) GARCH equations that may differ considerably in their functional forms across the two Markov regimes. We show how previously proposed collapsing procedures for the Markov-switching GARCH model can be extended to estimate our general specification by means of classical maximum-likelihood methods. We estimate several variants of the generalized Markov-switching GARCH model using daily excess returns of the German stock market index DAX sampled during the last decade. Our empirical study has two major findings. First, our generalized model outperforms all nested specifications in terms of (a) statistical fit (when model selection is based on likelihood ratio tests) and (b) out-of-sample volatility forecasting performance. Second, we find significant Markov-switching structures in German stock market data, with substantially differing volatility equations across the regimes.     

Mixed Normal Conditional Heteroskedasticity

Both unconditional mixed normal distributions and GARCH modelswith fat-tailed conditional distributions have been employedin the literature for modeling financial data. We consider amixed normal distribution coupled with a GARCH-type structure(termed MN-GARCH) which allows for conditional variance in eachof the components as well as dynamic feedback between the components.Special cases and relationships with previously proposed specificationsare discussed and stationarity conditions are derived. For theempirically most relevant GARCH(1,1) case, the conditions forexistence of arbitrary integer moments are given and analyticexpressions of the unconditional skewness, kurtosis, and autocorrelationsof the squared process are derived. Finally, employing dailyreturn data on the NASDAQ index, we provide a detailed empiricalanalysis and compare both the in-sample fit and out-of-sampleforecasting performance of the MN-GARCH as well as recentlyproposed Markov-switching models. We show that the MN-GARCHapproach can generate a plausible disaggregation of the conditionalvariance process in which the components' volatility dynamicshave a clearly distinct behavior, which is, for example, compatiblewith the well-known leverage effect.     

Relative forecasting performance of volatility models: Monte Carlo evidence

Abstract

A Monte Carlo (MC) experiment is conducted to study the forecasting performance of a variety of volatility models under alternative data-generating processes (DGPs). The models included in the MC study are the (Fractionally Integrated) Generalized Autoregressive Conditional Heteroskedasticity models ((FI)GARCH), the Stochastic Volatility model (SV), the Long Memory Stochastic Volatility model (LMSV) and the Markov-switching Multifractal model (MSM). The MC study enables us to compare the relative forecasting performance of the models accounting for different characterizations of the latent volatility process: specifications that incorporate short/long memory, autoregressive components, stochastic shocks, Markov-switching and multifractality. Forecasts are evaluated by means of mean squared errors (MSE), mean absolute errors (MAE) and value-at-risk (VaR) diagnostics. Furthermore, complementarities between models are explored via forecast combinations. The results show that (i) the MSM model best forecasts volatility under any other alternative characterization of the latent volatility process and (ii) forecast combinations provide systematic improvements upon most single misspecified models, but are typically inferior to the MSM model even if the latter is applied to data governed by other processes.     

Stationarity of a Markov-Switching GARCH Model

This article investigates some structural properties of theMarkov-switching GARCH process introduced by Haas, Mittnik,and Paolella. First, a sufficient and necessary condition forthe existence of the weakly stationary solution of the processis presented. The solution is weakly stationary, and the causalexpansion of the Markov-switching GARCH process is also established.Second, the general conditions for the existence of any integer-ordermoment of the square of the process are derived. The techniqueused in this article for the weak stationarity and the high-ordermoments of the process is different from that used by Haas,Mittnik, and Paolella and avoids the assumption that the processstarted in the infinite past with finite variance. Third, asufficient and necessary condition for the strict stationarityof the Markov-switching GARCH process with possibly infinitevariance is given. Finally, the strict stationarity of the so-calledintegrated Markov-switching GARCH process is also discussed.     

Modeling structural changes in the volatility process

GARCH-type models have been very successful in describing the volatility dynamics of financial return series for short periods of time. However, the time-varying behavior of investors, for example, may cause the structure of volatility to change and the assumption of stationarity is no longer plausible. To deal with this issue, the current paper proposes a conditional volatility model with time-varying coefficients based on a multinomial switching mechanism. By giving more weight to either the persistence or shock term in a GARCH model, conditional on their relative ability to forecast a benchmark volatility measure, the switching reinforces the persistent nature of the GARCH model. The estimation of this benchmark volatility targeting or BVT-GARCH model for Dow 30 stocks indicates that the switching model is able to outperform a number of relevant GARCH setups, both in- and out-of-sample, also without any informational advantages.     

Markov-switching in target stocks during takeover bids

This paper examines shifts in the market betas and the conditional volatility of stock prices of takeover targets. Using daily stock prices of five European and American targets, we find that adequately specified Markov-switching GARCH models are capable of detecting statistically significant regime-switches in all takeover deal-types (in cash bids, pure share-exchange bids, mixed bids). In particular, conditional volatility regime-switches are found to be most clear-cut for cash bids. Our econometric findings have implications for a broad range of financial applications such as the valuation of target stock options.     

Boom–Bust Cycles and the Forecasting Performance of Linear and Non-Linear Models of House Prices

The tremendous rise in house prices over the last decade has been both a national and a global phenomenon. The growth of secondary mortgage holdings and the increased impact of house prices on consumption and other components of economic activity imply ever-greater importance for accurate forecasts of home price changes. Given the boom–bust nature of housing markets, nonlinear techniques seem intuitively very well suited to forecasting prices, and better, for volatile markets, than linear models which impose symmetry of adjustment in both rising and falling price periods. Accordingly, Crawford and Fratantoni (Real Estate Economics 31:223–243, 2003) apply a Markov-switching model to U.S. home prices, and compare the performance with autoregressive-moving average (ARMA) and generalized autoregressive conditional heteroscedastic (GARCH) models. While the switching model shows great promise with excellent in-sample fit, its out-of-sample forecasts are generally inferior to more standard forecasting techniques. Since these results were published, some researchers have discovered that the Markov-switching model is particularly ill-suited for forecasting. We thus consider other non-linear models besides the Markov switching, and after evaluating alternatives, employ the generalized autoregressive (GAR) model. We find the GAR does a better job at out-of-sample forecasting than ARMA and GARCH models in many cases, especially in those markets traditionally associated with high home-price volatility.     

Forecasting volatility of crude oil futures using a GARCH–RNN hybrid approach

Volatility is an important element for various financial instruments owing to its ability to measure the risk and reward value of a given financial asset. Owing to its importance, forecasting volatility has become a critical task in financial forecasting. In this paper, we propose a suite of hybrid models for forecasting volatility of crude oil under different forecasting horizons. Specifically, we combine the parameters of generalized autoregressive conditional heteroscedasticity (GARCH) and Glosten–Jagannathan–Runkle (GJR)-GARCH with long short-term memory (LSTM) to create three new forecasting models named GARCH–LSTM, GJR-LSTM, and GARCH-GJRGARCH LSTM in order to forecast crude oil volatility of West Texas Intermediate on different forecasting horizons and compare their performance with the classical volatility forecasting models. Specifically, we compare the performances against existing methodologies of forecasting volatility such as GARCH and found that the proposed hybrid models improve upon the forecasting accuracy of Crude Oil: West Texas Intermediate under various forecasting horizons and perform better than GARCH and GJR-GARCH, with GG–LSTM performing the best of the three proposed models at 7-, 14-, and 21-day-ahead forecasts in terms of heteroscedasticity-adjusted mean square error and heteroscedasticity-adjusted mean absolute error, with significance testing conducted through the model confidence set showing that GG–LSTM is a strong contender for forecasting crude oil volatility under different forecasting regimes and rolling-window schemes. The contribution of the paper is that it enhances the forecasting ability of crude oil futures volatility, which is essential for trading, hedging, and purposes of arbitrage, and that the proposed model dwells upon existing literature and enhances the forecasting accuracy of crude oil volatility by fusing a neural network model with multiple econometric models.     

Estimating stochastic volatility models using integrated nested Laplace approximations

Volatility in financial time series is mainly analysed through two classes of models; the generalized autoregressive conditional heteroscedasticity (GARCH) models and the stochastic volatility (SV) ones. GARCH models are straightforward to estimate using maximum-likelihood techniques, while SV models require more complex inferential and computational tools, such as Markov Chain Monte Carlo (MCMC). Hence, although provided with a series of theoretical advantages, SV models are in practice much less popular than GARCH ones. In this paper, we solve the problem of inference for some SV models by applying a new inferential tool, integrated nested Laplace approximations (INLAs). INLA substitutes MCMC simulations with accurate deterministic approximations, making a full Bayesian analysis of many kinds of SV models extremely fast and accurate. Our hope is that the use of INLA will help SV models to become more appealing to the financial industry, where, due to their complexity, they are rarely used in practice.