Linear GARCH Research Articles

A CLOSED-FORM ESTIMATOR FOR THE GARCH(1,1) MODEL

We propose a closed-form estimator for the linear GARCH(1,1) model. The estimator has the advantage over the often used quasi-maximum likelihood estimator (QMLE) that it can be easily implemented and does not require the use of any numerical optimization procedures or the choice of initial values of the conditional variance process. We derive the asymptotic properties of the estimator, showing T(κ−1)/κ-consistency for some κ ∈ (1,2) when the fourth moment exists and -asymptotic normality when the eighth moment exists. We demonstrate that a finite number of Newton–Raphson iterations using our estimator as starting point will yield asymptotically the same distribution as the QMLE when the fourth moment exists. A simulation study confirms our theoretical results.The first author's research was supported by the Shoemaker Foundation. The second author's research was supported by the Economic and Social Science Research Council of the United Kingdom.

A Closed-form Estimator for the GARCH(1,1)-Model

We propose a closed-form estimator for the linear GARCH(1,1) model. The estimator has the advantage over the often used quasi-maximum-likelihood estimator (QMLE) that it can be easily implemented, and does not require the use of any numerical optimisation procedures or the choice of initial values of the conditional variance process. We derive the asymptotic properties of the estimator, showing T^{(K-1)/K}-consistency for some K(1,2) when the 4th moment exists and the square root of T-asymptotic normality when the 8th moment exists. We demonstrate that a finite number of Newton-Raphson iterations using our estimator as starting point will yield asymptotically the same distribution as the QMLE when the 4th moment exists. A simulation study confirms our theoretical results.

Modelling Stock Returns in the G-7 and in Selected CEE Economies: A Non-Linear GARCH Approach

This paper investigates conditional variance patterns in daily return series of stock market indices in the G-7 and 6 selected economies of Central and Eastern Europe. For this purpose, various linear and asymmetric GARCH models are employed. The analysis is conducted for Canada, France, Germany, Italy, Japan, the UK and the US for which the TSX, CAC-40, DAX-100, BCI, Nikkei-225, FTSE-100 and DJ-30 indices are respectively considered over the period 1987 to 2002. Furthermore, the official indices of Czech, Hungarian, Polish, Russian, Slovak and Slovene stock markets are also studied, i.e. the PX-50, BUX, WIGI, RFS, SAX-16 and SBI, respectively, over 1991/1995 to 2002. The estimation results reveal that the selected stock returns for the G-7 can be reasonably well modelled using linear specifications whereas the overwhelming majority of the stock indices from Central and Eastern Europe can be much better characterised using asymmetric models. In other words, stock markets of the transition economies exhibit much more asymmetry because negative shocks hit much harder these markets than positive news. It also turns out that these changes do not occur in a smooth manner but happen pretty brusquely. This corroborates the usual observation that emerging stock markets may collapse much more suddenly and recover more slowly than G-7 stock markets.

Predictive Ability of Asymmetric Volatility Models At Medium-Term Horizons

Using realized volatility to estimate conditional variance of financial returns, we compare forecasts of volatility from linear GARCH models with asymmetric ones. We consider horizons extending to 30 days. Forecasts are compared using three different evaluation tests. With data from an equity index and two foreign exchange returns, we show that asymmetric models provide statistically significant forecast improvements upon the GARCH model for two of the datasets and improve forecasts for all datasets by means of forecasts combinations. These results extend to about 10 days in the future, beyond which the forecasts are statistically inseparable from each other.

Predictive Ability of Asymmetric Volatility Models at Medium-Term Horizons

Using realized volatility to estimate conditional variance of financial returns, we compare forecasts of volatility from linear GARCH models with asymmetric ones. We consider horizons extending to 30 days. Forecasts are compared using three different evaluation tests. With data from an equity index and two foreign exchange returns, we show that asymmetric models provide statistically significant forecast improvements upon the GARCH model for two of the datasets and improve forecasts for all datasets by means of forecasts combinations. These results extend to about 10 days in the future, beyond which the forecasts are statistically inseparable from each other.

MIXING AND MOMENT PROPERTIES OF VARIOUS GARCH AND STOCHASTIC VOLATILITY MODELS

This paper first provides some useful results on a generalized random coefficient autoregressive model and a generalized hidden Markov model. These results simultaneously imply strict stationarity, existence of higher order moments, geometric ergodicity, and β-mixing with exponential decay rates, which are important properties for statistical inference. As applications, we then provide easy-to-verify sufficient conditions to ensure β-mixing and finite higher order moments for various linear and nonlinear GARCH(1,1), linear and power GARCH(p,q), stochastic volatility, and autoregressive conditional duration models. For many of these models, our sufficient conditions for existence of second moments and exponential β-mixing are also necessary. For several GARCH(1,1) models, our sufficient conditions for existence of higher order moments again coincide with the necessary ones in He and Terasvirta (1999, Journal of Econometrics 92, 173–192).

Forecasting volatility of emerging stock markets: linear versus non-linear GARCH models

ARCH and GARCH models are substantially used for modelling volatility of time series data. It is proven by many studies that if variables are significantly skewed, linear versions of these models are not sufficient for both explaining the past volatility and forecasting the future volatility. In this paper, we compare the linear(GARCH(1,1)) and non-linear(EGARCH) versions of GARCH model by using the monthly stock market returns of seven emerging countries from February 1988 to December 1996. We find that for emerging stock markets GARCH(1,1) model performs better than EGARCH model, even if stock market return series display skewed distributions. Copyright © 2000 John Wiley & Sons, Ltd.

Forecasting unemployment

Results are reported of a forecasting competition between linear autoregressive, GARCH, threshold autoregressive and neural network models of the UK monthly unemployment rate series. Evidence is uncovered suggesting the existence of nonlinearities in the series.

Forecast intervals in ARCH models: bootstrap versus parametric methods

This paper proposes an alternative bootstrap method for constructing prediction intervals for an ARMA process, when its innovations are represented by a linear GARCH model. The potential of this bootstrap method is assessed through a study which compares the proposed technique with a standard one.

The Efficacy of Neural Networks in Predicting Returns on Stock and Bond Indices*

ABSTRACTThis paper uses two recently developed tests to identify neglected nonlinearity in the relationship between excess returns on four asset classes and several economic and financial variables. Having found some evidence of possible nonlinearity, it was then investigated whether the predictive power of these variables could be enhanced by using neural network models instead of linear regression or GARCH models.Some evidence of nonlinearity in the relationships between the explanatory variables and large stocks and corporate bonds was found. It was also found that the GARCH models are conditionally efficient with respect to neural network models, but the neural network models outperform GARCH models if financial performance measures are used. In resonance with the results reported for the tests for neglected nonlinearity, it was found that the neural network forecasts are conditionally efficient with respect to linear regression models for large stocks and corporate bonds, whereas the evidence is not statistically significant for small stocks and intermediate‐term government bonds. This difference persists even when financial performance measures for individual asset classes are used for comparison.

Common Persistence in Conditional Variances

Since the introduction of the autoregressive conditional heteroskedastic (ARCH) model in Engle (1982), numerous applications of this modeling strategy have already appeared. A common finding in many of these studies with high frequency financial or monetary data concerns the presence of an approximate unit root in the autoregressive polynomial in the univariate time series representation for the conditional second order moments of the process, as in the so-called integrated generalized ARCH (IGARCH) class of models proposed in Engle and Bollerslev (1986). In the IGARCH models shocks to the conditional variance are persistent, in the sense that they remain important for forecasts of all horizons. This idea is readily extended to a multivariate framework. Even though many time series may exhibit persistence in variance, it is likely that several different variables share the same common long-run component. In that situation, the variables are naturally defined to be co-persistent in variance, and the co-persistent linear combination is interpretable as a long-run relationship. Conditions for co-persistence to occur in the multivariate linear GARCH model are presented. These conditions parallel the conditions for linear co-integration in the mean, as developed by Engle and Granger (1987). The presence of co-persistence has important implications for asset pricing relationships and in optimal portfolio allocation decisions. An empirical example relating to the time series properties of nominal U.S. dollar exchange rates for the deutschemark and the British pound provides a simple illustration of the ideas.

Nonlinear time-series analysis of stock volatilities

SUMMARY The absolute value of the mean-corrected excess return is used in this paper to measure the volatility of stock returns. We apply various nonlinearity tests available in the literature to show that such volatility series are strongly nonlinear. We then explore the use of threshold autoregressive (TAR) models in describing monthly volatility series. The models built suggest that the volatility series exhibit significant lower-order serial correlations when the volatility is large, indicating certain volatility clustering in stock returns. Out-of-sample forecasts are used to compare the TAR models with linear ARMA models and nonlinear GARCH and EGARCH models. Based on mean squared error and average absolute deviation, the comparisons show that (a) the TAR models consistently outperform the linear ARMA models in multi-step ahead forecasts for large stocks, (b) the TAR models provide better forecasts than the GARCH and EGARCH models also for the volatilities of large stock returns, and (c) the EGARCH model gives the best long-horizon volatility forecasts for small stock returns.

Prediction in dynamic models with time-dependent conditional variances

This paper considers forecasting the conditional mean and variance from a single-equation dynamic model with autocorrelated disturbances following an ARMA process, and innovations with time-dependent conditional heteroskedasticity as represented by a linear GARCH process. Expressions for the minimum MSE predictor and the conditional MSE are presented. We also derive the formula for all the theoretical moments of the prediction error distribution from a general dynamic model with GARCH(1, 1) innovations. These results are then used in the construction of ex ante prediction confidence intervals by means of the Cornish-Fisher asymptotic expansion. An empirical example relating to the uncertainty of the expected depreciation of foreign exchange rates illustrates the usefulness of the results.

Are the GARCH models best in out-of-sample performance?

Abstract Out-of-sample performance of exchange rate volatility model depends on criteria used to measure it. The linear GARCH models cannot generally outperform the nonlinear models in the RMSE criterion. Furthermore, the nonparametric kernel model is best in the MAE criterion.