Elsevier

Journal of Econometrics

Volume 136, Issue 1, January 2007, Pages 237-249
Journal of Econometrics

Aggregation and memory of models of changing volatility

https://doi.org/10.1016/j.jeconom.2006.03.002Get rights and content

Abstract

In this paper we study the effect of contemporaneous aggregation of an arbitrarily large number of covariance stationary processes featuring short memory dynamic conditional heteroskedasticity, when heterogeneity is allowed for across units. We look at the memory properties of the limit aggregate. General conditions for long memory heteroskedasticity are obtained. More specific results relative to certain stochastic volatility models are also developed, providing some examples of how long memory heteroskedasticity can be obtained by aggregation.

Introduction

Contemporaneous aggregation (in the sense of averaging across units) of stationary heterogeneous autoregressive moving average (ARMA) processes can lead to a limit stationary process displaying long memory (see Definition 1 in the Appendix), when the number of units grows to infinity; see Robinson (1978) and Granger (1980). Relatively recent research in empirical finance indicates that the long memory paradigm represents a valid description of the dependence of volatility of financial asset returns (see Ding et al., 1993, Granger and Ding, 1996, Andersen and Bollerslev, 1997 among others). In most studies the time series of stock indexes, such as the Standard & Poor's 500, have been used to support this empirical evidence, suggesting that the aggregation mechanism could be the ultimate source of long memory in the volatility of portfolio returns. This paper presents a theoretical analysis prompted by this suggestion.

Some specific results already exist according to which aggregation of the generalized autoregressive conditional heteroskedasticity (GARCH) model of Bollerslev (1986) do not lead to long memory in the sense of a non-summable autocovariance function (acf) of the squared aggregate (hereafter long memory for brevity); see Zaffaroni (2000) and Kazakevicius et al. (2004). One might wonder whether this negative finding applies more generally or, instead, whether there exist volatility models for which long memory can be obtained by aggregation. Therefore, we investigate the memory implications of the aggregation mechanism within a large class of volatility models which nests both GARCH and stochastic volatility (henceforth SV) models. The class of square root stochastic autoregressive volatility (SR-SARV) models, introduced by Andersen (1994) and generalized by Meddahi and Renault (1996), appears suitable for this task. A general class of SV models is also proposed by Robinson (2001) but it excludes many ARCH-type models, including GARCH.

For a finite number of units n, aggregation of GARCH has been analyzed by Nijman and Sentana (1996) and generalized by Meddahi and Renault (1996) for aggregation of SR-SARV. In contrast to them, we characterize the conditions under which the aggregate displays different features, namely long memory, from those of the micro units by letting n.

We start by deriving two sets of sufficient conditions for ruling out long memory with respect to the SR-SARV class. The first condition says that the micro-units can be cross-sectionally correlated to the extent that the central limit theorem (clt) works. The second condition formalizes the fact that imposing bounded fourth moment of the micro-units might limit the effect of innovations to the conditional variance of the limit aggregate. Many volatility models used nowadays in empirical finance violate these conditions. We then focus on the exponential SV model of Taylor (1986) and on the nonlinear moving average model (nonlinear MA) of Robinson and Zaffaroni (1998), both belonging to the SR-SARV class. Although the necessary conditions for long memory are satisfied in both cases, we show that long memory is ruled out for the exponential SV but is permitted for the nonlinear MA.

The next section presents a set of necessary conditions for long memory with respect to the SR-SARV class. Section 3, which represents the bulk of the paper, analyzes the effect of aggregation of exponential SV and nonlinear MA. Section 4 concludes. The results are stated in propositions whose proofs are reported in the final Appendix.

Section snippets

Some general results

Summarizing Meddahi and Renault (1996, Definition 3.1), a stationary square integrable process {xt} is called a SR-SARV(1) process with respect to the increasing filtration Jt if xt is Jt-adapted, E(xt|Jt-1)=0 and var(xt|Jt-1)ft-1 satisfyingft=ω+γft-1+vt,with the sequence {vt} satisfying E(vt|Jt-1)=0. ω and γ are constant non-negative coefficients with γ<1. This implies Ext2<.

In order to study the effect of aggregation over an arbitrarily large number of units, we assume that at each point in

Main results

We now fully investigate the effect of aggregation for two particular elements of the SR-SARV class, namely the nonlinear MA and the exponential SV, whose definitions are recalled below. In both cases we establish the limit in mean square of Xn,t and, as a by-product, analyze their memory properties.

Conclusions

We have formally established the result of aggregation for two member of the SR-SARV class, namely the nonlinear MA of Robinson and Zaffaroni (1998) and the exponential SV of Taylor (1986). Although the models share many statistical properties, long memory of the limit aggregate is achieved in the former case but not in latter. Several generalizations are possible. For instance, one can analyze the result of aggregation for more general models, such as conditional heteroskedastic factor models,

Acknowledgments

I thank Liudas Giraitis for useful suggestions and Nour Meddahi for stimulating discussions. I am grateful to the Editor, the Associate Editor and an anonymous referee whose suggestions greatly improved the paper.

References (27)

  • T. Bollerslev

    Generalized autoregressive conditional heteroskedasticity

    Journal of Econometrics

    (1986)
  • F. Breidt et al.

    Modelling long-memory stochastic volatility

    Journal of Econometrics

    (1998)
  • R. Engle et al.

    Measuring and testing the impact of news on volatility

    Journal of Finance

    (1993)
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