Autoregressive models

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.

AR

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, ...).

MA

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, ...).

ARMA

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/GARCH

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):

APARCH

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):

EGARCH

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

 

 

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