ARCH-type Models Research Articles

Bayesian comparison of bivariate ARCH-type models for the main exchange rates in Poland

We use the daily zloty (PLN) values of the US dollar (USD) and German mark (DEM) to compare various bivariate ARCH-type models through their Bayes factors. We start with three non-nested, conditionally Student t specifications: the VechGARCH(1,1) model, Bollerslev's (Rev. Econom. Statist. 72 (1990) 498) constant conditional correlation model and a latent factor GARCH model. Since only the first model explains the data relatively well, but uses too many parameters, we examine its special cases, including a simple t-BEKK(1,1) specification that has very high Bayes factors against most of other models.

Stochastic volatility: Bayesian computation using automatic differentiation and the extended Kalman filter

Summary Stochastic volatility (SV) models provide more realistic and flexible alternatives to ARCH-type models for describing time-varying volatility exhibited in many financial time series. They belong to the wide class of nonlinear state-space models. As classical parameter estimation for SV models is difficult due to the intractable form of the likelihood, Bayesian approaches using Markov chain Monte Carlo (MCMC) techniques for posterior computations have been suggested. In this paper, an efficient MCMC algorithm for posterior computation in SV models is presented. It is related to the integration sampler of Kim et al. (1998) but does not need an offset mixture of normals approximation to the likelihood. Instead, the extended Kaiman Filter is combined with the Laplace approximation to compute the likelihood function by integrating out all unknown system states. We make use of automatic differentiation in computing the posterior mode and in designing an efficient Metropolis-Hastings algorithm. We compare the new algorithm to the single-update Gibbs sampler and the integration sampler using a well-known time series of pound/dollar exchange rates.

Consistent ranking of volatility models

Abstract We show that the empirical ranking of volatility models can be inconsistent for the true ranking if the evaluation is based on a proxy for the population measure of volatility. For example, the substitution of a squared return for the conditional variance in the evaluation of ARCH-type models can result in an inferior model being chosen as ‘best’ with a probability that converges to one as the sample size increases. We document the practical relevance of this problem in an empirical application and by simulation experiments. Our results provide an additional argument for using the realized variance in out-of-sample evaluations rather than the squared return. We derive the theoretical results in a general framework that is not specific to the comparison of volatility models. Similar problems can arise in comparisons of forecasting models whenever the predicted variable is a latent variable.

Modelling Daily Value-At-Risk Using Realized Volatility and Arch Type Models

In this paper we show how to compute a daily VaR measure for two stock indexes (CAC40 and SP500) using the one-day-ahead forecast of the daily realized volatility. The daily realized volatility is equal to the sum of the squared intraday returns over a given day and thus uses intraday information to define an aggregated daily volatility measure. While the VaR specification based on an ARFIMAX(0,d,1)-skewed Student model for the daily realized volatility provides adequate one-day-ahead VaR forecasts, it does not really improve on the performance of a VaR model based on the skewed Student APARCH model and estimated using daily data. Thus, for the two financial assets considered in an univariate framework, both methods seem to be equivalent. This paper also shows that daily returns standardized by the square root of the one-day-ahead forecast of the daily realized volatility are not normally distributed.

Forecasting Stock Market Volatility: Evidence From Fourteen Countries

This paper evaluates the out-of-sample forecasting accuracy of eleven models for weekly and monthly volatility in fourteen stock markets. Volatility is defined as within-week (within-month) standard deviation of continuously compounded daily returns on the stock market index of each country for the ten-year period 1988 to 1997. The first half of the sample is retained for the estimation of parameters while the second half is for the forecast period. The following models are employed: a random walk model, a historical mean model, moving average models, weighted moving average models, exponentially weighted moving average models, an exponential smoothing model, a regression model, an ARCH model, a GARCH model, a GJR-GARCH model, and an EGARCH model. We first use the standard (symmetric) loss functions to evaluate the performance of the competing models: the mean error, the mean absolute error, the root mean squared error, and the mean absolute percentage error. According to all of these standard loss functions, the exponential smoothing model provides superior forecasts of volatility. On the other hand, ARCH-based models generally prove to be the worst forecasting models. We also employ the asymmetric loss functions to penalize under/over-prediction. When under-predictions are penalized more heavily ARCH-type models provide the best forecasts while the random walk is worst. However, when over-predictions of volatility are penalized more heavily the exponential smoothing model performs best while the ARCH-type models are now universally found to be inferior forecasters.

Recent Theoretical Results for Time Series Models with GARCH Errors

This paper provides a review of some recent theoretical results for time series models with GARCH errors, and is directed towards practitioners. Starting with the simple ARCH model and proceeding to the GARCH model, some results for stationary and nonstationary ARMA–GARCH are summarized. Various new ARCH–type models, including double threshold ARCH and GARCH, ARFIMA–GARCH, CHARMA and vector ARMA–GARCH, are also reviewed.

G@RCH 2.2: An Ox Package for Estimating and Forecasting Various ARCH Models

This paper discusses and documents G@RCH 2.2, an Ox package dedicated to the estimation and forecast of various univariate ARCH–type models including GARCH, EGARCH, GJR, APARCH, IGARCH, FIGARCH, HYGARCH, FIEGARCH and FIAPARCH specifications of the conditional variance and an AR(FI)MA specification of the conditional mean.These models can be estimated by Approximate (Quasi) Maximum Likelihood under four assumptions: normal, Student–t, GED or skewed Student errors. Explanatory variables can enter both the conditional mean and the conditional variance equations. h–step–ahead forecasts of both the conditional mean and the conditional variance are available as well as many mispecification tests.We first propose an overview of the package’s features, with the presentation of the different specifications of the conditional mean and conditional variance. Then further explanations are given about the estimation methods. Measures of the accuracy of the procedures are also given and the GARCH features provided by G@RCH are compared with those of nine other econometric softwares. Finally, a concrete application of G@RCH 2.2 is provided.

Nonstationary nonlinear heteroskedasticity

In this paper, we consider time series with the conditional heteroskedasticities that are given by nonlinear functions of integrated processes. Such time series are said to have nonlinear nonstationary heteroskedasticity (NNH), and the functions generating conditional heterogeneity are called heterogeneity generating functions (HGF's). Various statistical properties of time series with NNH are investigated for a wide class of HGF's. For NNH models with a variety of HGF's, volatility clustering and leptokurtosis, which are common features of ARCH type models, are manifest. In particular, it is shown that the sample autocorrelations of their squared processes vanish only very slowly, or do not even vanish at all, in the limit. Volatility clustering is therefore well expected. The NNH models with certain types of HGF's indeed have sample characteristics that are very similar to those of ARCH type models. Moreover, the sample kurtosis of the NNH model either diverges or has a stable limiting distribution with support truncated on the left by the kurtosis of the innovations. This would well explain the presence of leptokurtosis in many observed time series data. To illustrate the empirical relevancy of our model, we analyze the spreads between the forward and spot rates of USD/DM exchange rates. It is found that the conditional variances of the spreads can be well modelled as a nonlinear function of the levels of the spot rates.

Forecasting volatility in the New Zealand stock market

This study evaluates the performance of nine alternative models for predicting stock price volatility using daily New Zealand data. The competing models contain both simple models such as the random walk and smoothing models and complex models such as ARCH-type models and a stochastic volatility model. Four different measures are used to evaluate the forecasting accuracy. The main results are the following: (1) the stochastic volatility model provides the best performance among all the candidates; (2) ARCH-type models can perform well or badly depending on the form chosen: the performance of the GARCH(3,2) model, the best model within the ARCH family, is sensitive to the choice of assessment measures; and (3) the regression and exponentially weighted moving average models do not perform well according to any assessment measure, in contrast to the results found in various markets.

ARCH versus information-based variances: evidence from the Tokyo Stock Market

By using both the individual stock prices quoted on the Tokyo Stock Exchange and their price index (TOPIX), this paper examines whether the conditional variance of stock returns is characterized by the auto-regressive-conditional-heteroskedasticity (ARCH) effect or the information-based effect. The paper finds that the inclusion of the trading volume in both generalized ARCH (GARCH) and exponential GARCH (EGARCH) specifications eliminates the ARCH effect for individual stocks and the TOPIX. The paper explains the reasons for these results. The findings suggest strong support for the information-based variance model which gives a parallel explanation to the ARCH-type models.

Heterogeneous volatility cascade in financial markets

Using high frequency data, we have studied empirically the change of volatility, also called volatility derivative, for various time horizons. In particular, the correlation between the volatility derivative and the volatility realized in the next time period is a measure of the response function of the market participants. This correlation shows explicitly the heterogeneous structure of the market according to the characteristic time horizons of the different agents. It reveals a volatility cascade from long to short time horizons, with a structure different from the one observed in turbulence. Moreover, we have developed a new ARCH-type model which incorporates the different groups of agents, with their characteristic memory. This model reproduces well the empirical response function, and allows us to quantify the importance of each group.

An examination of nonlinear dependence in exchange rates, using recent methods from chaos theory

Interest in the relevance of nonlinear dynamics to finance and economics has spurred the evolution of new ways to analyze time series data. Tests for chaos, based on a metric approach which measures spatial correlations, led to the development of the correlation dimension test for chaos and the BDS test for nonlinearity. More recently, a topological method has been introduced into the scientific literature which employs a simple qualitative test for chaos that is adaptable to the characteristics of financial data. A quantitative version is also presented here. Conflicting evidence exists about the presence of chaotic behavior in exchange-rate data. The qualitative topological test does not support evidence of a chaotic generating mechanism in these series. The quantitative form finds nonlinear dependence and is a useful diagnostic to determine the adequacy of ARCH-type models for this nonlinear structure.

Heterogeneous Volatility Cascade in Financial Markets

Using high frequency data, we have studied empirically the change of volatility, also called volatility derivative, for various time horizons. In particular, the correlation between the volatility derivative and the volatility realized in the next time period is a measure of the response function of the market participants. This correlation shows explicitly the heterogeneous structure of the market according to the characteristic time horizons of the differents agents. It reveals a volatility cascade from long to short time horizons, with a structure different from the one observed in turbulence. Moreover, we have developed a new ARCH-type model which incorporates the different groups of agents, with their characteristic memory. This model reproduces well the empirical response function, and allows us to quantify the importance of each group.

A Forecast Comparison of Volatility Models: Does Anything Beat a GARCH(1,1)?

We compare 330 ARCH-type models in terms of their ability to describe the conditional variance. The models are compared out-of-sample using DM-$ exchange rate data and IBM return data, where the latter is based on a new data set of realized variance. We find no evidence that a GARCH(1,1) is outperformed by more sophisticated models in our analysis of exchange rates, whereas the GARCH(1,1) is clearly inferior to models that can accommodate a leverage effect in our analysis of IBM returns. The models are compared with the test for superior predictive ability (SPA) and the reality check for data snooping (RC). Our empirical results show that the RC lacks power to an extent that makes it unable to distinguish 'good' and 'bad' models in our analysis.

Estimating non-linear ARMA models using Fourier coefficients

Linear time-series models are often inadequate to capture the presence of asymmetric adjustment and/or conditional volatility. Parametric models of asymmetric adjustment and ARCH-type models necessitate specifying the nature of the non-linear coefficient. If there is little a priori information concerning the actual form of the non-linearity, the estimated model can suffer from a misspecification error. We show that a non-linear time-series can be represented by a deterministic time-dependent coefficient model without first specifying the nature of the non-linearity. The methodology is applied to real GDP and the NYSE Transportation Index.

Excess kurtosis of conditional distribution for daily stock returns: the case of Japan

Not only the unconditional distribution but also the conditional distribution of daily asset returns are known to be leptokurtic. Thus, some authors have suggested using ARCH-type models with leptokurtic distributions such as the t-distribution and the generalized error distribution (GED) for the conditional distribution. The purpose of this paper examines what distribution is fit for the conditional distribution of daily Japanese stock returns. In particular, we estimate an exponential GARCH (EGARCH) model developed by Nelson (1991) jointly with the generalized t-distribution, which nests both the Student's t-distribution and the generalized error distribution (GED) employed by other researchers. It is shown that the Student's t-distribution is suited for capturing the excess kurtosis of conditional distribution for daily Japanese stock returns, which does not depend on sample period and is consistent with what Bollerslev et al. (1994) have found by fitting a similar model to daily US stock returns.

Volatility of real GDP: some evidence from the United States, the United Kingdom and Japan

This paper presents an empirical analysis of the volatility of real growth rates for the United States, the United Kingdom and Japan. Three ARCH-type models (GARCH, T-GARCH and E-GARCH) were estimated utilizing the maximum likelihood method. The GARCH version provided the best statistical fit, suggesting that volatility is variable and is symmetric than asymmetric to real growth rates in GDP.

HARCH Processes are Heavy Tailed

Heterogeneous Auto-Regressive Conditional Heteroskedastic processes were introduced by Mu¨ller, Dacarogna, Dave´, Olsen, Pictet and von Weizsa¨cker in order to improve traditional ARCH-type models describing financial time series. In a later paper Embrechts, Samorodnitsky, Dacorogna and Mu¨ller asked how heavy the tails of stationary processes of this type are. We provide a partial answer to this question, using mainly monotonicity arguments to compare HARCH processes to other processes with a simpler recursive structure.

Power transformation and forecasting the magnitude of exchange rate changes

This paper examines the impact of power transformations on the forecasting performance of AR models with respect to the magnitude of change in Australian bilateral exchange rates. The results obtained in this study suggest that the use of squared returns as per ARCH type models are generally inferior to other power transformations. The optimal power transformation is sensitive to the measure of forecasting performance chosen although it is close to the returns themselves as per a Taylor/Schwert type model.

Finite sample properties of the ARCH class of models with stochastic volatility

This paper shows that ARCH-type models provide adequate estimates of conditional volatility when the data are generated under stochastic volatility. Under reasonable scenarios, ARCH estimates out-perform method-of moments and quasi-maximum likelihood estimators of the true model. © 1997 Elsevier Science S.A.