Tim Bollerslev

Generalized autoregressive conditional heteroskedasticity

A natural generalization of the ARCH (Autoregressive Conditional Heteroskedastic) process introduced in Engle (1982) to allow for past conditional variances in the current conditional variance equation is proposed. Stationarity conditions and autocorrelation structure for this new class of parametric models are derived. Maximum likelihood estimation and testing are also considered. Finally an empirical example relating to the uncertainty of the inflation rate is presented.

ARCH modeling in finance: A review of the theory and empirical evidence

Although volatility clustering has a long history as a salient empirical regularity characterizing high-frequency speculative prices, it was not until recently that applied researchers in finance have recognized the importance of explicitly modeling time-varying second-order moments. Instrumental in most of these empirical studies has been the Autoregressive Conditional Heteroskedasticity (ARCH) model introduced by Engle (1982). This paper contains an overview of some of the developments in the formulation of ARCH models and a survey of the numerous empirical applications using financial data. Several suggestions for future research, including the implementation and tests of competing asset pricing theories, market microstructure models, information transmission mechanisms, dynamic hedging strategies, and the pricing of derivative assets, are also discussed.

Quasi-maximum likelihood estimation and inference in dynamic models with time-varying covariances

We study the properties of the quasi-maximum likelihood estimator (QMLE) and related test statistics in dynamic models that jointly parameterize conditional means and conditional covariances, when a normal log-likelihood os maximized but the assumption of normality is violated. Because the score of the normal log-likelihood has the martingale difference property when the forst two conditional moments are correctly specified, the QMLE is generally Consistent and has a limiting normal destribution. We provide easily computable formulas for asymptotic standard errors that are valid under nonnormality. Further, we show how robust LM tests for the adequacy of the jointly parameterized mean and variance can be computed from simple auxiliary regressions. An appealing feature of these robyst inference procedures is that only first derivatives of the conditional mean and variance functions are needed. A monte Carlo study indicates that the asymptotic results carry over to finite samples. Estimation of several AR a...

A Capital Asset Pricing Model with Time Varying Covariances

The capital asset pricing model provides a theoretical structure for the pricing of assets with uncertain returns. The premium to induce risk-averse investors to bear risk is proportional to the nondiversifiable risk, which is measured by the covariance of the asset return with the market portfolio return. In this paper a multivariate generalized autoregressive conditional heteroscedastic process is estimated for returns to bills, bonds, and stock where the expected return is proportional to the conditional convariance of each return with that of a fully diversified or market portfolio. It is found that the conditional covariances are quite variable over time and are a significant determinant of time-varying risk premia. The implied betas are also time-varying and forecastable. However, there is evidence that other variables including innovations in consumption should also be considered in the investors information set when estimating the conditional distribution of returns.

Modelling the Coherence in Short-run Nominal Exchange Rates: A Multivariate Generalized ARCH Model

A multivariate time series model with time varying conditional variances and covariances, but constant conditional correlations is proposed. In a multivariate regression framework, the model is readily interpreted as an extension of the Seemingly Unrelated Regression (SUR) model allowing for heteroskedasticity. Parameterizing each of the conditional variances as a univariate Generalized Autoregressive Conditional Heteroskedastic (GARCH) process, the descriptive validity of the model is illustrated for a set of five nominal European U.S. dollar exchange rates following the inception of the European Monetary System (EMS). When compared to the pre- EMS free float period, the comovements between the currenciess are found to be significantly higher over the later period. Copyright 1990 by MIT Press.

A Conditionally Heteroskedastic Time Series Model for Speculative Prices and Rates of Return

The distribution of speculative price changes and rates of return data tend to be uncorrelated over time but characterized by volatile and tranquil periods. A simple time series model designed to capture this dependence is presented. The model is an extension of the Autoregressive Conditional Heteroskedastic (ARCH) and Generalized ARCH (GARCH) models obtained by allowing for conditionally t-distributed errors. The model can be derived as a simple subordinate stochastic process by including an additive unobservable rror term in the conditional variance equation. The descriptive validity of the model is illustrated for a set of foreign exchange rates and stock price indices.

Answering the Skeptics: Yes, Standard Volatility Models Do Provide Accurate Forecasts

A voluminous literature has emerged for modeling the temporal dependencies in financial market volatility using ARCH and stochastic volatility models. While most of these studies have documented highly significant in-sample parameter estimates and pronounced intertemporal volatility persistence, traditional ex post forecast evaluation criteria suggest that the models provide seemingly poor volatility forecasts. Contrary to this contention, the authors show that volatility models produce strikingly accurate interdaily forecasts for the latent volatility factor that would be of interest in most financial applications. New methods for improved ex post interdaily volatility measurements based on high-frequency intradaily data are also discussed. Copyright 1998 by Economics Department of the University of Pennsylvania and the Osaka University Institute of Social and Economic Research Association.

Fractionally integrated generalized autoregressive conditional heteroskedasticity

Abstract The new class of Fractionally Integrated Generalized AutoRegressive Conditionally Heteroskedastic (FIGARCH) processes is introduced. The conditional variance of the process implies a slow hyperbolic rate of decay for the influence of lagged squared innovations. Unlike (I(d) processes for the mean, Maximum Likelihood Estimates (MLE) of the FIGARCH parameters are argued to be T 1 2 - consistent . The small-sample behavior of an approximate MLE procedure is assessed through a simulation study, which also documents how the estimation of a standard GARCH model tends to produce integrated, or IGARCH, like estimates. An empirical example with daily Deutschmark — U.S. dollar exchange rates illustrates the practical relevance of the new FIGARCH specification.

The distribution of realized stock return volatility

Abstract We examine “realized” daily equity return volatilities and correlations obtained from high-frequency intraday transaction prices on individual stocks in the Dow Jones Industrial Average. We find that the unconditional distributions of realized variances and covariances are highly right-skewed, while the realized logarithmic standard deviations and correlations are approximately Gaussian, as are the distributions of the returns scaled by realized standard deviations. Realized volatilities and correlations show strong temporal dependence and appear to be well described by long-memory processes. Finally, there is strong evidence that realized volatilities and correlations move together in a manner broadly consistent with latent factor structure.

The Distribution of Realized Exchange Rate Volatility

Using high-frequency data on deutschemark and yen returns against the dollar, we construct model-free estimates of daily exchange rate volatility and correlation that cover an entire decade. Our estimates, termed realized volatilities and correlations, are not only model-free, but also approximately free of measurement error under general conditions, which we discuss in detail. Hence, for practical purposes, we may treat the exchange rate volatilities and correlations as observed rather than latent. We do so, and we characterize their joint distribution, both unconditionally and conditionally. Noteworthy results include a simple normality-inducing volatility transformation, high contemporaneous correlation across volatilities, high correlation between correlation and volatilities, pronounced and persistent dynamics in volatilities and correlations, evidence of long-memory dynamics in volatilities and correlations, and remarkably precise scaling laws under temporal aggregation.