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10.6.3 Rationale
To date we have considered autoregressive processes and moving average processes.
The former model considers its own past behaviour as inputs for the model. It attempts to
capture market participant effects such as momentum and mean-reversion in stock trading. The
latter model is used to characterise "shock" information to a series such as a surprise earnings
announcements. A good example of "shock" news would be the BP Deepwater Horizon oil spill.
An ARMA model attempts to capture both of these aspects when modelling financial time
series. Note however that it does not take into account volatility clustering, which is a key
empirical phenomena of many financial time series. It is not a conditional heteroskedastic model.
For that we will need to wait for the ARCH and GARCH models.
10.6.4 Definition
The ARMA(p,q) model is a linear combination of two linear models and thus is still linear:
Definition 10.6.3. Autoregressive Moving Average Model of order p, q. A time series model,
{x t }, is an autoregressive moving average model of order p, q, ARMA(p,q), if:
x t = α 1 x t−1 + α 2 x t−2 + . . . + w t + β 1 w t−1 + β 2 w t−2 . . . + β q w t−q (10.18)
Where {w t } is white noise with E(w t ) = 0 and variance σ 2 .
If we consider the Backward Shift Operator, B then we can rewrite the above as a function θ
and φ of B:
θ p (B)x t = φ q (B)w t (10.19)
We can straightforwardly see that by setting p ≠ 0 and q = 0 we recover the AR(p) model.
Similarly if we set p = 0 and q ≠ 0 we recover the MA(q) model.
One of the key features of the ARMA model is that it is parsimonious and redundant in its
parameters. That is, an ARMA model will often require fewer parameters than an AR(p) or
MA(q) model alone. In addition if we rewrite the equation in terms of the BSO then the θ and
φ polynomials can sometimes share a common factor leading to a simpler model.
10.6.5 Simulations and Correlograms
As with the autoregressive and moving average models we will now simulate various ARMA
series and then attempt to fit ARMA models to these realisations. We carry this out because we
want to ensure that we understand the fitting procedure, including how to calculate confidence
intervals for the models, as well as ensure that the procedure does actually recover reasonable
estimates for the original ARMA parameters.
In the previous sections we manually constructed the AR and MA series by drawing N
samples from a normal distribution and then crafting the specific time series model using lags of
these samples.
However, there is a more straightforward way to simulate AR, MA, ARMA and even ARIMA
data, simply by using the arima.sim method in R.