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10.4.1 Rationale
In the previous chapter we considered the random walk model, where each term, x t is dependent
solely upon the previous term, x t−1 and a stochastic white noise term, w t :
x t = x t−1 + w t (10.2)
The autoregressive model is simply an extension of the random walk that includes terms
further back in time. The structure of the model is linear, that is the model depends linearly on
the previous terms, with coefficients for each term. This is where the "regressive" comes from in
"autoregressive". It is essentially a regression model where the previous terms are the predictors.
Definition 10.4.1. Autoregressive Model of order p. A time series model, {x t }, is an autoregressive
model of order p, AR(p), if:
x t = α 1 x t−1 + . . . + α p x t−p + w t (10.3)
p∑
= α i x t−i + w t (10.4)
i=1
Where {w t } is white noise and α i ∈ R, with α p ≠ 0 for a p-order autoregressive process.
If we consider the Backward Shift Operator, B, then we can rewrite the above as a function
θ of B:
θ p (B)x t = (1 − α 1 B − α 2 B 2 − . . . − α p B)x t = w t (10.5)
Perhaps the first thing to notice about the AR(p) model is that a random walk is simply
AR(1) with α 1 equal to unity. As we stated above, the autogressive model is an extension of the
random walk, so this makes sense.
It is straightforward to make predictions with the AR(p) model for any time t. Once the α i
coefficients are determined the estimate simply becomes:
ˆx t = α 1 x t−1 + . . . + α p x t−p (10.6)
Hence we can make n-step ahead forecasts by producing ˆx t , ˆx t+1 , ˆx t+2 , . . . up to ˆx t+n . In fact,
once we consider the ARMA models later in the chapter, we will use the R predict function to
create forecasts (along with standard error confidence interval bands) that will help us produce
trading signals.
10.4.2 Stationarity for Autoregressive Processes
One of the most important aspects of the AR(p) model is that it is not always stationary. Indeed
the stationarity of a particular model depends upon the parameters. I have touched on this before
in my other book, Successful Algorithmic Trading.