Threshold Arch Models and Asymmetries in Volatility

SUMMARY
This paper attempts to enlarge the class of Threshold Heteroscedastic ~ o d e l (sT ARCH) introduced by
Zakoi'an (1991a). We show that it is possible to relax the positivity constraints on the parameters of the
conditional variance. Unconstrained models provide a greater generality of the paths allowing for nonlinearities
in the volatility. Cyclical behaviour is permitted as well as different relative impacts of positive
and negative shocks on volatility, depending on their size. We give empirical evidence using French stock
returns.

1. INTRODUCTION
One major characteristic of financial time series is that volatility is changing over time. For
example, Schwert (1989) shows that the variations of volatility for monthly stock returns on
the period 1857-1987 range from a low of 2% in the early 1960s to a high of 20% in the early
1930s. Taking this feature into account appeared crucial for many areas of the financial
literature: continuous time models and options pricing, CAPM, investment theory etc. Since
the early 1980s, two parallel approaches have been developed. First, diffusion models with
stochastic volatility (Hull and White, 1987; Wiggins, 1987; Chesney and Scott, 1989). Second,
discrete time ARCH models (Autoregressive Conditionally Heteroscedastic), introduced by
Engle (1982) and generalized by Bollerslev (1986) (GARCH) and Engle et al. (1987) (GARCHM).
Nelson (1990~) connected the two approaches, showing that a GARCH process can be
interpreted as a discrete time approximation of a diffusion model with stochastic volatility.
In their most general form, univariate GARCH models make the conditional variance at
time t a function of exogenous and lagged endogenous variables, past residuals and conditional
variances, time, parameters. Formally, let (ct) be a sequence of prediction errors, o a vector
of parameters, xt a vector of exogenous and lagged endogenous variables, and a? the variance
of ct given information at time t:
ct = at Zt
(Zt) i.i.d with E(Zt) = 0, var(Zt) = 1.
2 2
U? = h(&t-1, '5-2, ...,Of-1,Uf-2, ...,Xt, t, W ) (3)
0883-7252/93/01003 1-19$14.50 Received May 1991
O 1993 by John Wiley & Sons, Ltd. Revised October 1992