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\centerline{\bf STANFORD UNIVERSITY}
\centerline{\bf DEPARTMENT OF STATISTICS}
\centerline{\big DEPARTMENTAL SEMINAR}
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\centerline{4:15 p.m., Tuesday, May 20, 2003}
\centerline{Sequoia Hall Room 200}
\centerline{(Cookies at 3:45 in 1st Floor Lounge)}

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\centerline{\sl Eric Hillebrand}
\centerline{\sl Stanford University}
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\centerline{\bf Unknown Parameter Changes in GARCH and ARMA Models}
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Abstract:

A common finding in the empirical financial literature is that the
volatility of financial data exhibits high persistence, or slow mean
reversion of the order of months. In GARCH models, this is represented
by the fact that the sum of the estimates of the autoregressive
parameters is close to one when sample periods are considered that cover
several years.

It has been documented in simulations, however, that parameter changes
in the data-generating process lead to exactly this phenomenon. It has
also been reported that segmentations of financial time series and local
GARCH estimations on the segments considerably reduce the estimated
persistence, that is, the sum of the estimated autoregressive parameters
is below one. One possible explanation is that the data-generating
persistence within segments of constant parameters is relatively low and
that the globally measured high persistence is caused by the parameter
changes.

I will show that in GARCH models of orders up to GARCH(2,2), unknown
parameter changes in the data-generating process in fact cause the sum
of the estimated autoregressive parameters to be close to one. This is a
consequence of the geometry of the estimation problem, not of the
statistical properties of the estimators, about which little is known.
This particular estimation problem is not confined to GARCH models, the
arguments are readily generalized to ARMA models of orders up to
ARMA(2,2). I will illustrate this effect using synthetic and market
data.

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