Long-range Dependence In Volatility Research Articles

Modelling Australian Dollar Volatility at Multiple Horizons with High-Frequency Data

Long-range dependency of the volatility of exchange-rate time series plays a crucial role in the evaluation of exchange-rate risks, in particular for the commodity currencies. The Australian dollar is currently holding the fifth rank in the global top 10 most frequently traded currencies. The popularity of the Aussie dollar among currency traders belongs to the so-called three G’s—Geology, Geography and Government policy. The Australian economy is largely driven by commodities. The strength of the Australian dollar is counter-cyclical relative to other currencies and ties proximately to the geographical, commercial linkage with Asia and the commodity cycle. As such, we consider that the Australian dollar presents strong characteristics of the commodity currency. In this study, we provide an examination of the Australian dollar–US dollar rates. For the period from 18:05, 7th August 2019 to 9:25, 16th September 2019 with a total of 8481 observations, a wavelet-based approach that allows for modelling long-memory characteristics of this currency pair at different trading horizons is used in our analysis. Findings from our analysis indicate that long-range dependence in volatility is observed and it is persistent across horizons. However, this long-range dependence in volatility is most prominent at the horizon longer than daily. Policy implications have emerged based on the findings of this paper in relation to the important determinant of volatility dynamics, which can be incorporated in optimal trading strategies and policy implications.

Markov switch smooth transition HYGARCH model: Stability and estimation

HYGARCH model is basically used to model long-range dependence in volatility. We propose Markov switch smooth-transition HYGARCH model, where the volatility in each state is a time-dependent convex combination of GARCH and FIGARCH. This model provides a flexible structure to capture different levels of volatilities and also short and long memory effects. The necessary and sufficient condition for the asymptotic stability is derived. Forecast of conditional variance is studied by using all past information through a parsimonious way. Bayesian estimations based on Gibbs sampling are provided. A simulation study has been given to evaluate the estimations and model stability. The competitive performance of the proposed model is shown by comparing it with the HYGARCH and smooth-transition HYGARCH models for some period of the S&P500 and Dow Jones industrial average indices based on volatility and value-at-risk forecasts.

Volatility clustering: A nonlinear theoretical approach

This paper verifies the endogenous mechanism and economic intuition on volatility clustering using the coexistence of two locally stable attractors proposed by Gaunersdorfer et al. (2008). By considering a simple asset pricing model with two types of boundedly rational traders, fundamentalists and trend followers, and noise traders, we provide theoretical conditions on the coexistence of a locally stable steady state and a locally stable invariant circle of the underlying nonlinear deterministic financial market model and show numerically that the interaction of the coexistence of the deterministic dynamics and noise processes can endogenously generate volatility clustering and long range dependence in volatility observed in financial markets. Economically, volatility clustering occurs when neither the fundamental nor trend following traders dominate the market and when traders switch more often between the two strategies.

Volatility Clustering: A Nonlinear Theoretical Approach

This paper verifies the endogenous mechanism and economic intuition on volatility clustering using the coexistence of two locally stable attractors proposed by Gaunersdorfer, Hommes and Wagener (2008). By considering a simple asset pricing model with two types of boundedly rational traders, fundamentalists and trend followers, and noise traders, we provide conditions on the coexistence of locally stable steady state and invariant cycle of the underlying nonlinear deterministic financial market model and show numerically that the interaction of the coexistence of the deterministic dynamics and noise processes can endogenously generate volatility clustering and long range dependence in volatility observed in financial markets. Economically, volatility clustering occurs when neither the fundamental nor trend following traders dominate the market and when traders switch more often between the two strategies.

Financial power laws: Empirical evidence, models, and mechanisms

Financial markets (share markets, foreign exchange markets and others) are all characterized by a number of universal power laws. The most prominent example is the ubiquitous finding of a robust, approximately cubic power law characterizing the distribution of large returns. A similarly robust feature is long-range dependence in volatility (i.e., hyperbolic decline of its autocorrelation function). The recent literature adds temporal scaling of trading volume and multi-scaling of higher moments of returns. Increasing awareness of these properties has recently spurred attempts at theoretical explanations of the emergence of these key characteristics form the market process. In principle, different types of dynamic processes could be responsible for these power-laws. Examples to be found in the economics literature include multiplicative stochastic processes as well as dynamic processes with multiple equilibria. Though both types of dynamics are characterized by intermittent behavior which occasionally generates large bursts of activity, they can be based on fundamentally different perceptions of the trading process. The present paper reviews both the analytical background of the power laws emerging from the above data generating mechanisms as well as pertinent models proposed in the economics literature.

Breaks or long range dependence in the energy futures volatility: Out-of-sample forecasting and VaR analysis

The aim of this paper is to propose an empirical strategy that allows the discrimination between true and spurious long memory behaviors. That strategy is based on the comparison between the estimated long memory parameter before and after filtering out the breaks. To date the breaks, we use the probability smoothing of the Markov Switching GARCH model of Haas et al. (2004). Application of this strategy to the crude oil, heating oil, RBOB regular gasoline and the propane futures energy with the one, two, three and four months maturities show strong evidence for the presence of long range dependence in all futures energy prices volatility11As proxy of volatility, we use the squared returns. Others proxies are used in the empirical studies such as the absolute value of returns. Our empirical studies show there are strong differences between these two proxies. time series. This result of long range dependence in the volatility is confirmed by the superiority of the FIGARCH and FIEGARCH models compared with the Markov switching GARCH models in terms of out-of-sample forecasting and value at risk (VaR) performances. Moreover, we show that the proposed empirical strategy is robust to different data frequency. Practical implications of the results for market participants are proposed and discussed.

How Can Long Memory Affect Investors’ Decisions: An Empirical Evidence Using Copulas Theory

This paper empirically investigates the extent to which classical Markowitz’s frontier instability may be driven by conditional short and long range dependence in volatility. We resort to the Mean-Variance program in the portfolio management using concordance measures and copula function. Then, we propose a new approach based on CVaR-Copula. The empirical results suggest that our approach outperforms the classical Markowitz’s program and it depends on the selected diversification measure.

A new hyperbolic GARCH model

There are two commonly used hyperbolic GARCH processes, the FIGARCH and HYGARCH processes, in modeling the long-range dependence in volatility. However, the FIGARCH process always has infinite variance, and the HYGARCH model has a more complicated form. This paper builds a simple bridge between a common GARCH model and an integrated GARCH model, and hence a new hyperbolic GARCH model along the lines of FIGARCH models. The new model remedies the drawback of FIGARCH processes by allowing the existence of finite variance as in HYGARCH models, while it has a form nearly as simple as the FIGARCH model. Two inference tools, including the Gaussian QMLE and a portmanteau test for the adequacy of the fitted model, are derived, and an easily implemented test for hyperbolic memory is also constructed. Their finite sample performances are evaluated by simulation experiments, and an empirical example gives further support to our new model.

Heterogeneous beliefs and adaptive behaviour in a continuous-time asset price model

This paper extends the analysis of the seminal work of Brock and Hommes (1997, 1998) on heterogeneous beliefs and rational routes to randomness in discrete-time models to a continuous-time model of asset pricing. The resulting model characterised mathematically by a system of stochastic delay differential equations provides a unified approach to deal with adaptive behaviour of heterogeneous agents and market stability impact of lagged price used by chartists to form their expectations. For the underlying deterministic model, we show not only that the result of Brock and Hommes on rational routes to market instability in discrete-time holds in continuous-time but also a double edged effect of an increase in lagged price used by the chartists on market stability. For the stochastic model, we demonstrate that the interaction and boundedly rational behaviour of heterogeneous agents can generate various market phenomena such as bubbles and crashes and replicate stylised facts including volatility clustering, and long range dependence in volatility.

Score Tests for Hyperbolic GARCH Models

Davidson (2004) recently proposed the hyperbolic GARCH model to capture the phenomenon of long-range dependence in volatility, with the extent of such dependence measured by the geometric or hyperbolic decay of the coefficients in an ARCH(∞) model. In this article, we reinterpret the hyperbolic GARCH model by building a link with the common GARCH model, and construct a simplified score test to check the presence of the hyperbolic decay. We derive the asymptotic of the test statistic under the null hypothesis and the local alternatives. We conduct Monte Carlo simulation experiments to study the performance of this test, and report an illustration on two log return sequences. This article has supplementary material online.

On approximate pseudo-maximum likelihood estimation for LARCH-processes

Linear ARCH (LARCH) processes were introduced by Robinson [J. Econometrics 47 (1991) 67--84] to model long-range dependence in volatility and leverage. Basic theoretical properties of LARCH processes have been investigated in the recent literature. However, there is a lack of estimation methods and corresponding asymptotic theory. In this paper, we consider estimation of the dependence parameters for LARCH processes with non-summable hyperbolically decaying coefficients. Asymptotic limit theorems are derived. A central limit theorem with $\sqrt{n}$-rate of convergence holds for an approximate conditional pseudo-maximum likelihood estimator. To obtain a computable version that includes observed values only, a further approximation is required. The computable estimator is again asymptotically normal, however with a rate of convergence that is slower than $\sqrt{n}.$

Testing for long-range dependence in the Brazilian term structure of interest rates

This paper presents empirical evidence of fractional dynamics in interest rates for different maturities for Brazil. A variation of a newly developed test for long-range dependence, the V/ S statistic, with a post-blackening bootstrap is employed. Results suggest that Brazilian interest rates possess strong long-range dependence in volatility, even when considering the structural break in 1999. These findings imply that the development of policy models that give rise to long-range dependence in interest rates’ volatility could be very useful. The long–short-term interest rates spread has strong long-range dependence, which suggests that traditional tests of expectation hypothesis of the term structure of interest rates may be misspecified.

How long memory in volatility affects true dependence structure

Long memory in volatility is a stylized fact found in most financial return series. This paper empirically investigates the extent to which interdependence in emerging markets may be driven by conditional short and long range dependence in volatility. We fit copulas to pairs of raw and filtered returns, analyse the observed changes in the dependence structure may be driven by volatility, and discuss whether or not asymmetries on propagation of crisis may be interpreted as intrinsic characteristics of the markets. We also use the findings to construct portfolios possessing desirable expected behavior such as dependence at extreme positive levels.

Weighted Averages and Local Polynomial Estimation for Fractional Linear ARCH Processes

Abstract We consider local polynomial regression estimation for time series defined by Yi = μ(ti ) + Ui where Ui is a stationary zero mean process with Wold decomposition Ui = C(B)Zi . The innovations Zi are assumed to be generated by a LARCH process, and may thus exhibit long-range correlations in volatility. The linear dependence structure, defined by the filter C(B), includes short memory, long memory and antipersistence. A central limit theorem for weighted averages, with triangular arrays of weights, and a limit theorem for local polynomial estimates of μ(v)(ti ) are derived. The asymptotic distribution of μ(v)(ti ) turns out to be unaffected by long-range dependence in volatility. The question of optimal regression weights is also addressed.

Power-law behaviour, heterogeneity, and trend chasing

Long-range dependence in volatility is one of the most prominent examples in financial market research involving universal power laws. Its characterization has recently spurred attempts to provide some explanations of the underlying mechanism. This paper contributes to this recent line of research by analyzing a simple market fraction asset pricing model with two types of traders – fundamentalists who trade on the price deviation from estimated fundamental value and trend followers whose conditional mean and variance of the trend are updated through a geometric learning process. Our analysis shows that agent heterogeneity, risk-adjusted trend chasing through the geometric learning process, and the interplay of noisy fundamental and demand processes and the underlying deterministic dynamics can be the source of power-law distributed fluctuations. In particular, the noisy demand plays an important role in the generation of insignificant autocorrelations (ACs) on returns, while the significant decaying AC patterns of the absolute returns and squared returns are more influenced by the noisy fundamental process. A statistical analysis based on Monte Carlo simulations is conducted to characterize the decay rate. Realistic estimates of the power-law decay indices and the (FI)GARCH parameters are presented.

Testing for time-varying long-range dependence in volatility for emerging markets

This paper tests whether volatility for equity returns for emerging markets possesses long-range dependence. Furthermore, the assertion of whether long-range dependence is time-varying is checked through a rolling sample approach. The empirical results suggest that there exists long-range dependence in emerging equity returns' volatility and also that it is time-varying. This assertion also holds true for Japan and the US, which are considered more developed markets. Moreover, these results are robust to “shuffling” the data to eliminate short-term autocorrelation. Therefore, they suggest that the class of GARCH processes, which are currently employed to analyze volatility of financial time series, is misspecified.

Ranking efficiency for emerging equity markets II☆

Abstract This paper employs a “rolling sample” approach to estimate Hurst exponents for emerging markets squared and absolute returns. The findings suggests that these markets possess strong long-range dependence in volatility. Empirical results suggest that Asian equity markets are more efficient than those of Latin America and that the US is the most efficient country.

Structural change and long-range dependence in volatility of exchange rates: either, neither or both?

In this paper, we test for the existence of long memory and structural breaks in the realized variance process for the DM/US$ and Yen/US$ exchange rates. While long memory is evident in the actual processes, a structural break analysis reveals that this feature is partially explained by unaccounted changes in regime. We then compare the forecasting performance of Markov switching models with that of an ARFIMA model. The results indicate that neglecting the break process is not important for very short term forecasting once it is allowed for a long memory component in the model, but that superior forecasts can be obtained at longer horizons by modelling both long memory and structural change.