 |
Download a pdf copy of this help file here |
|
Download a pdf copy of this help file here |
See also: Time series introduction, Time series modeling in finance, Geometric Brownian Motion models, Markov Chain models, Birth and death models, Time series in ModelRisk
An ever-increasing number of autoregressive models are being developed in the financial area. The ones of more general interest are AR, MA, ARMA, ARCH and GARCH and it is more standard to apply the models to the return r rather than the stock price S. We also give the equations for EGARCH and APARCH.
Be warned though: before being convinced that some subtle variation of the model gives a genuine advantage, try generating a few samples for simpler models that you have fit to the data and see whether they can create scenarios of a similar pattern.
ModelRisk offers functions that fit each of these series to data and produce forecasts. The data can be live linked to historical values which is very convenient for keeping you model automatically up-to-date.
The equation for an autoregressive process of order p, or AR(p), is:
where et are independent Normal(0, s) random variables. Some constraints on the parameters {ai} are needed if one wants to keep the model stationary (meaning the marginal distribution of r is the same for all t) e.g. for an AR(1), |a1|<1. In most situations an AR(1) or AR(2) is sufficiently elaborate, i.e:
AR(1):
AR(2):
You can see that this is just a regression model where rt is the dependent variable and rt-i are the explanatory variables. It is usual, though not essential, that ai>ai+1, i.e. that rt is explained more by more recent values (t-1, t-2, ..) rather than older values (t-10, t-11, ...).
The equation for a moving-average process of order q, or MA(q), is:
This says that the variable rt is Normally distributed about a mean equal to:
where et are independent Normal(0, s) random variables again. In other words the mean of rt is the mean of the process as a whole m plus some weighting of the variation of q previous terms from the mean. Similarly to AR models, it is usual that bi>bi+1, i.e. that rt is explained more by more recent terms (t-1, t-2, ..) rather than older terms (t-10, t-11, ...).
We can put the AR(p) and MA(q) processes together to creates an autoregressive, moving-average model ARMA(p,q) process with mean m that is described by the following equation:
Or
In practice, the ARMA(1,1) is usually sufficiently complex so the equation simplifies to:
ARCH models were
originally developed to account for fat tails by allowing clustering of
periods of volatility (heteroscedastic, or heteroskedastic, means 'having different variances').
One of the assumptions in regression models that were previously used
for analysis of high frequency financial data is that the error terms
have a constant variance. Engle
(1982), who won the 2003 Nobel Memorial Prize for Economics, introduced
the ARCH model, applying it to quarterly U.K. inflation data. ARCH was
later generalized to GARCH by Bollerslev
(1986) which has proven more successful in fitting to financial data.
Let
denote the returns or
return residuals and assume that
, where zt
are independent, Normal(0, 1) distributed. and the 
is
modelled by:
,
where
,
and at least one 
.
Then rt is said to follow an autoregressive conditional
heteroskedastic, ARCH(q), process with mean m. It models
the variance of the current error term as a function of the variance of
previous error terms (rt-i-m). Since each ai >
0, it has the effect of grouping low (or high) volatilities together.
If an autoregressive moving average process (ARMA process) is assumed for the variance, then rt is said to be a generalised autoregressive conditional heteroskedastic GARCH(p,q) process with mean m :
where
p is the order of GARCH terms and q is the order of ARCH
terms, , 
and at least one 
or
.
In practice the model most generally used is a GARCH(1,1):
The asymmetric power autoregressive conditional heteroskedasticity, APARCH(p,q), model was introduced by Ding, Granger, and Engle (1993) and is defined as follows:
where 
and
at lease one or
. d
plays the role of a Box-Cox transformation
of the conditional standard deviation
,
while the
's reflect
the so-called leverage effect. APARCH has proved very promising and is
now quite widespread because it nests several other models as special
cases, e.g. the ARCH(d
= 1, gi = 0, bi = 0)
, GARCH (d = 2, gi = 0), (TS-GARCH (d
= 1, gi = 0), GJR-GARCH (d
= 2), TARCH (d
= 1) and NARCH (bi
= 1, gi
= 0).
In practice the model most generally used is an APARCH(1,1):
The exponential
general autoregressive conditional heteroskedastic, EGARCH(p,q),
model was another form of GARCH model with the purpose of allowing negative
values in the linear error variance equation. The GARCH model imposes
nonnegative constraints on the parameters, and
, while there are no such restrictions
on these parameters in the EGARCH model. In the EGARCH(p,q)
model, the conditional variance, , is formulated by an asymmetric function
of lagged disturbances:
where
when
zt is a standard normal variable.
Again, in practice the model most generally used has p = q =1 i.e. is an EGARCH(1,1):
Read on: Markov Chain models