Autoregressive Heteroscedasticity Model and Its Variants 223
are predictable and we can determine explicit formulas to make predictions.
In particular, we can make predictions of volatility, which is a key measure
of risk.
Arch in the Mean Model
The ARCH in the mean model, denoted ARCH-M model, is a variant of the
ARCH model suggested by the consideration that investors require higher
returns to bear higher risk and, therefore, periods when the conditional
variance is higher should be associated with higher returns.^6 The ARCH-M
model is written as follows:
Rdtt=+σ ut (11.5)
utt=σεt (11.6)
σtt^2 =+ca 11 ua^22 −−++ mtum (11.7)
Equation (11.5) states that returns at time t are the sum of two compo-
nents: dσt, and ut. The first component dσt is proportional to volatility at
time t while the second component ut has the same form of the ARCH(m)
model. If we set d = 0 the ARCH in the mean model becomes the standard
ARCH(m) model given by equations (11.3) and (11.4).
Recall that standard ARCH models represent zero-mean returns; we
assume that if returns have a constant mean, the mean has been subtracted.
Because of the addition of the (always positive) term dσt in equation (11.5)
in the ARCH-in-the-mean model the conditional mean of returns is not zero
but is time varying. The conditional mean is bigger when volatility is high,
smaller when volatility is low.
GARCH Model
Figures 11.5 and 11.6 show how adding lags allows one to reduce the fre-
quency of switching between low and high volatility. Of course, in a practi-
cal application, all constants need to be estimated. The need to estimate
constants is the key weakness of models with many parameters: estimation
(^6) Robert F. Engle, David V. Lilien, and Russell P. Robins, “Estimating Time Varying
Risk Premia in the Term Structure: The ARCH-M Model,” Econometrica 55 (1987):
391–407.