GARCH models without positivity constraints: Exponential or log GARCH?

doi:10.1016/j.jeconom.2013.05.004

Journal of Econometrics

Volume 177, Issue 1, November 2013, Pages 34-46

https://doi.org/10.1016/j.jeconom.2013.05.004 Get rights and content

Abstract

This paper provides a probabilistic and statistical comparison of the log-GARCH and EGARCH models, which both rely on multiplicative volatility dynamics without positivity constraints. We compare the main probabilistic properties (strict stationarity, existence of moments, tails) of the EGARCH model, which are already known, with those of an asymmetric version of the log-GARCH. The quasi-maximum likelihood estimation of the log-GARCH parameters is shown to be strongly consistent and asymptotically normal. Similar estimation results are only available for the EGARCH (1,1) model, and under much stronger assumptions. The comparison is pursued via simulation experiments and estimation on real data.

Section snippets

Preliminaries

Since their introduction by Engle (1982) and Bollerslev (1986), GARCH models have attracted much attention and have been widely investigated in the literature. Many extensions have been suggested and, among them, the EGARCH (Exponential GARCH) introduced and studied by Nelson (1991) is very popular. In this model, the log-volatility is expressed as a linear combination of its past values and past values of the positive and negative parts of the innovations. Two main reasons for the success of

Stationarity, moments and asymmetries of the log-GARCH

We start by studying the existence of solutions to Model (1.1).

Tail properties of the log-GARCH

In this section, we investigate differences between the EGARCH and the log-GARCH in terms of tail properties.

Estimating the log-GARCH by QML

We now consider the statistical inference. Let $ϵ_{1}, \dots, ϵ_{n}$ be observations of the stationary solution of (1.1), where $θ$ is equal to an unknown value $θ_{0}$ belonging to some parameter space $Θ \subset R^{d}$ , with $d = 2 q + p + 1$ . A QMLE of $θ_{0}$ is defined as any measurable solution ${\hat{θ}}_{n}$ of ${\hat{θ}}_{n} = \underset{θ \in Θ}{arg min} {\tilde{Q}}_{n} (θ),$ with ${\tilde{Q}}_{n} (θ) = n^{- 1} \sum_{t = r_{0} + 1}^{n} {\tilde{ℓ}}_{t} (θ), {\tilde{ℓ}}_{t} (θ) = \frac{ϵ_{t}^{2}}{{\tilde{σ}}_{t}^{2} (θ)} + log {\tilde{σ}}_{t}^{2} (θ),$ where $r_{0}$ is a fixed integer and $log {\tilde{σ}}_{t}^{2} (θ)$ is recursively defined, for $t = 1, 2, \dots, n$ , by $log {\tilde{σ}}_{t}^{2} (θ) = ω + \sum_{i = 1}^{q} (α_{i +} 1_{{ϵ_{t - i} > 0}} + α_{i -} 1_{{ϵ_{t - i} < 0}}) log ϵ_{t - i}^{2} + \sum_{j = 1}^{p} β_{j} log {\tilde{σ}}_{t - j}^{2} (θ$

Asymmetric log-ACD model for duration data

The dynamics of duration between stock price changes has attracted much attention in the econometrics literature. Engle and Russell (1998) proposed the Autoregressive Conditional Duration (ACD) model, which assumes that the duration between price changes has the dynamics of the square of a GARCH. Bauwens and Giot, 2000, Bauwens and Giot, 2003 introduced logarithmic versions of the ACD, that do not constrain the sign of the coefficients (see also Bauwens et al., 2004 and Allen et al., 2008).

An application to exchange rates

We consider returns series of the daily exchange rates of the American Dollar (USD), the Japanese Yen (JPY), the British Pound (BGP), the Swiss Franc (CHF) and Canadian Dollar (CAD) with respect to the Euro. The observations cover the period from January 5, 1999 to January 18, 2012, which corresponds to 3344 observations. The data were obtained from the web site

http://www.ecb.int/stats/exchange/eurofxref/html/index.en.html.

Table 1 displays the estimated log-GARCH(1, 1) and EGARCH (1, 1) models

Proof of Theorem 2.1

Since the random variable $‖ C_{0} ‖$ is bounded, we have $E {log}^{+} ‖ C_{0} ‖ < \infty$ . The moment condition on $η_{t}$ entails that we also have $E {log}^{+} ‖ b_{0} ‖ < \infty$ . When $γ (C) < 0$ , Cauchy’s root test shows that, almost surely (a.s.), the series $z_{t} = b_{t} + \sum_{n = 0}^{\infty} C_{t} C_{t - 1} \dots C_{t - n} b_{t - n - 1}$ converges absolutely for all $t$ and satisfies (2.2). A strictly stationary solution to model (1.1) is then obtained as $ϵ_{t} = exp {\frac{1}{2} z_{2 q + 1, t}} η_{t}$ , where $z_{i, t}$ denotes the $i$ -th element of $z_{t}$ . This solution is non anticipative and ergodic, as a measurable function of ${η_{u}, u \leq t$

Conclusion

In this paper, we investigated the probabilistic properties of the log- $GARCH (p, q)$ model. We found sufficient conditions for the existence of moments and log-moments of the strictly stationary solutions. We analyzed the dependence structure through the leverage effect and the regular variation properties, and we compared this structure with that of the EGARCH model.

As far as the estimation is concerned, it should be emphasized that the log-GARCH model appears to be much more tractable than the

Acknowledgments

We are most thankful to the Co-Editor, Yacine Ait-Sahalia, and two anonymous referees for insightful comments and suggestions. We are also grateful to the Agence Nationale de la Recherche (ANR), which supported this work via the Project ECONOM&RISK (ANR 2010 blanc 1804 03).

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View full text

GARCH models without positivity constraints: Exponential or log GARCH?

Abstract

Section snippets

Preliminaries

Stationarity, moments and asymmetries of the log-GARCH

Tail properties of the log-GARCH

Estimating the log-GARCH by QML

Asymmetric log-ACD model for duration data

An application to exchange rates

Proof of Theorem 2.1

Conclusion

Acknowledgments

Journal of Econometrics

Journal of Econometrics

Journal of Econometrics

Journal of Econometrics

Finite sample properties of the QMLE for the log-ACD model: application to Australian stocks

Journal of Econometrics

Asymptotic normality of the quasi maximum likelihood estimator for multidimensional causal processes

Annals of Statistics

The moments of log-ACD models

Quantitative and Qualitative Analysis in Social Sciences

The logarithmic ACD model: an application to the bidask quote process of three NYSE stocks

Annales d’Economie et de Statistique

Asymmetric ACD models: introducing price information in ACD models

Empirical Economics

A comparison of financial duration models via density forecast

International Journal of Forecasting

GARCH processes: structure and estimation

Bernoulli

The Lindeberg-Levy theorem for martingales

Proceedings of the American Mathematical Society

Strict stationarity of generalized autoregressive processes

Annals of Probability

On some limit theorems similar to the arc-sin law

Theory of Probability and its Applications

Temporal aggregation of GARCH processes

Econometrica

Autoregressive conditional heteroskedasticity with estimates of the variance of UK inflation

Econometrica