Autoregressive Heteroscedasticity Model and Its Variants 233
The form of the process is similar to the univariate ARCH in equation
(11.1) but the two residuals εε 12 tt, are now characterized by different vari-
ances σσ 12 tt,^22 plus a covariance term σ 12 t. Analogous to the univariate ARCH,
we assume that variances and covariances are time-dependent autoregressive
processes. But in this case every term depends on all lagged squared returns
and all products of lagged returns. For example, for one lag we can write:
σ^21 tt=+ca 1111 Ra^2 ,,−− 1 ++ 12 RR 11 tt 21 ,,−−aR 13 22 t 1112 221112
σ tt=+caRa,,−−++ 22 RR 11 tt 21 ,,− aR 23 2 ,,,,t
ttcaRaRRtta−
=+ −−++−1222
331112
σ 32 1121 33321
R^2
,t−(11.15)
Hence, the simplest bivariate ARCH model with one lag implies estimat-
ing 12 parameters. For three time series and one lag it is necessary to esti-
mate 42 parameters. Clearly the number of parameters to estimate becomes
prohibitive for any except bivariate processes.
In practice, the number of parameters to estimate is too large to allow
models to be estimated with samples of the size available in financial
time series, implying that the VEC model can be used only in the case
of two series. Therefore simplifications are needed. Several simplify-
ing approaches have been proposed.^16 A popular approach proposed by
Bollerslev assumes that conditional correlations are constant and models
the variance of each series with a GARCH model.^17 With this approach,
we need to estimate a constant correlation matrix plus the GARCH
parameters for each series.
Key Points
■ (^) Variance is a measure of volatility. Because in many financial appli-
cations variance is commonly used as a proxy measure for risk, an
important application of financial econometrics is the forecasting of the
variance.
(^16) Other approaches are described in Luc Bauwens, Sébastien Laurent, and Jeroen V.
K. Rombouts, “Multivariate GARCH Models: A Survey,” Journal of Applied Econo-
metrics 21 (2006): 79–109; and Robert F. Engle, Sergio M. Focardi, and Frank J.
Fabozzi, “ARCH/GARCH Models in Applied Financial Econometrics,” in Hand-
book of Finance, ed. Frank J. Fabozzi, vol. 3 (Hoboken, NJ: John Wiley & Sons,
2008): 689–700.
(^17) Tim Bollerslev, “Modeling the Coherence in Short-Run Nominal Exchange Rates:
A Multivariate Generalized ARCH Approach,” Review of Economics and Statistics
72 (1990): 498–505.