Robert F. Engle

Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation

Traditional econometric models assume a constant one-period forecast variance. To generalize this implausible assumption, a new class of stochastic processes called autoregressive conditional heteroscedastic (ARCH) processes are introduced in this paper. These are mean zero, serially uncorrelated processes with nonconstant variances conditional on the past, but constant unconditional variances. For such processes, the recent past gives information about the one-period forecast variance. A regression model is then introduced with disturbances following an ARCH process. Maximum likelihood estimators are described and a simple scoring iteration formulated. Ordinary least squares maintains its optimality properties in this set-up, but maximum likelihood is more efficient. The relative efficiency is calculated and can be infinite. To test whether the disturbances follow an ARCH process, the Lagrange multiplier procedure is employed. The test is based simply on the autocorrelation of the squared OLS residuals. This model is used to estimate the means and variances of inflation in the U.K. The ARCH effect is found to be significant and the estimated variances increase substantially during the chaotic seventies.

Co-integration and Error Correction: Representation, Estimation, and Testing

The relationship between cointegration and error correction models, first suggested by Granger, is here extended and used to develop estimation procedures, tests, and empirical examples. A vector of time series is said to be cointegrated with cointegrating vector a if each element is stationary only after differencing while linear combinations a8xt are themselves stationary. A representation theorem connects the moving average , autoregressive, and error correction representations for cointegrated systems. A simple but asymptotically efficient two-step estimator is proposed and applied. Tests for cointegration are suggested and examined by Monte Carlo simulation. A series of examples are presented. Copyright 1987 by The Econometric Society.

Dynamic Conditional Correlation: A Simple Class of Multivariate Generalized Autoregressive Conditional Heteroskedasticity Models

Time varying correlations are often estimated with multivariate generalized autoregressive conditional heteroskedasticity (GARCH) models that are linear in squares and cross products of the data. A new class of multivariate models called dynamic conditional correlation models is proposed. These have the flexibility of univariate GARCH models coupled with parsimonious parametric models for the correlations. They are not linear but can often be estimated very simply with univariate or two-step methods based on the likelihood function. It is shown that they perform well in a variety of situations and provide sensible empirical results.

Multivariate Simultaneous Generalized ARCH

This paper presents theoretical results on the formulation and estimation of multivariate generalized ARCH models within simultaneous equations systems. A new parameterization of the multivariate ARCH process is proposed, and equivalence relations are discussed for the various ARCH parameterizations. Constraints sufficient to guarantee the positive definiteness of the conditional covariance matrices are developed, and necessary and sufficient conditions for covariance stationarity are presented. Identification and maximum likelihood estimation of the parameters in the simultaneous equations context are also covered.

A long memory property of stock market returns and a new model

Abstract A ‘long memory’ property of stock market returns is investigated in this paper. It is found that not only there is substantially more correlation between absolute returns than returns themselves, but the power transformation of the absolute return ¦rt¦d also has quite high autocorrelation for long lags. It is possible to characterize ¦rt¦d to be ‘long memory’ and this property is strongest when d is around 1. This result appears to argue against ARCH type specifications based upon squared returns. But our Monte-Carlo study shows that both ARCH type models based on squared returns and those based on absolute return can produce this property. A new general class of models is proposed which allows the power δ of the heteroskedasticity equation to be estimated from the data.

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.

Estimating Time Varying Risk Premia in the Term Structure: The Arch-M Model

The expectati on of the excess holding yield on a long bond is postulated to depend upon its conditional variance. Engles ARCH model is extended to allow the conditional variance to be a determinant of the mean and is called ARCH-M. Estimation and infer ence procedures are proposed, and the model is applied to three interest rate data sets. In most cases the ARCH process and the time varying risk premium are highly significant. A collection of LM diagnostic tests reveals the robustness of the model to various specification changes such as alternative volatility or ARCH measures, regime changes, and interest rate formulations. The model explains and interprets the recent econometric failures of the expectations hypothesis of the term structure. Copyright 1987 by The Econometric Society.

FORECASTING AND TESTING IN CO-INTEGRATED SYSTEMS*

Abstract This paper examines the behavior of forecasts made from a co-integrated system as introduced by Granger (1981), Granger and Weiss (1983) and Engle and Granger (1987). It is established that a multi-step forecast will satisfy the co-integrating relation exactly and that this particular linear combination of forecasts will have a finite limiting forecast error variance. A simulation study compares the multi-step forecast accuracy of unrestricted vector autoregression with the two-step estimation of the vector autoregression imposing the co-integration restriction. To test whether a system exhibits co-integration, the procedures introduced in Engle and Granger (1987) are extended to allow different sample sizes and numbers of variables.

Autoregressive conditional duration : a new model for irregularly spaced transaction data

This paper proposes a new statistical model for the analysis of data which arrive at irregular intervals. The model treats the time between events as a stochastic process and proposes a new class of point processes with dependent arrival rates. The conditional intensity is developed and compared with other self-exciting processes. Because the model focuses on the expected duration between events, it is called the autoregressive conditional duration (ACD) model. Asymptotic properties of the quasi maximum likelihood estimator are developed as a corollary to ARCH model results. Strong evidence is provided for duration clustering for the financial transaction data analyzed; both deterministic time-of-day effects and stochastic effects are important. The model is applied to the arrival times of trades and therefore is a model of transaction volume, and also to the arrival of other events such as price changes. Models for the volatility of prices are estimated with price-based durations, and examined from a market microstructure point of view.

Dynamic Conditional Correlation : A Simple Class of Multivariate GARCH Models

Time varying correlations are often estimated with Multivariate Garch models that are linear in squares and cross products of returns. A new class of multivariate models called dynamic conditional correlation (DCC) models is proposed. These have the flexibility of univariate GARCH models coupled with parsimonious parametric models for the correlations. They are not linear but can often be estimated very simply with univariate or two step methods based on the likelihood function. It is shown that they perform well in a variety of situations and give sensible empirical results.