WORKING PAPERS SERIES
Aggregation and Memory of
Models of Changing Volatility
Aggregation and memory of
models of changing volatility
Paolo Zaffaroni ∗
Banca d’ Italia
In this paper we study the effect of contemporaneous aggregation of
an arbitrarily large number of processes featuring dynamic conditional
heteroskedasticity with short memory when heterogeneity across units
is allowed for. We look at the memory properties of the limit aggre-
gate. General, necessary, conditions for long memory are derived.
More specific results relative to certain stochastic volatility models
are also developed, providing some examples of how long memory
volatility can be obtained by aggregation.
JEL classification: C43
Keywords: stochastic volatility, contemporaneous aggregation, long memory.
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Contemporaneous aggregation, in the sense of averaging across units, of sta-
tionary heterogeneous autoregressive moving average (ARMA) processes can
lead to a limit stationary process displaying long memory, in the sense of
featuring non summable autocovariance function, 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 volatil-
ity of financial asset returns (see Ding, Granger, and Engle (1993), Granger
and Ding (1996) and Andersen and Bollerslev (1997) among others). In most
studies the time series of stock indexes has been used, such as the Standard
& Poor’s 500, to support this empirical evidence, naturally suggesting that
the aggregation mechanism could be the ultimate source of long memory in
the volatility of portfolio returns.
The strong analogies of the generalized autoregressive conditional