Database Reference
In-Depth Information
Using the same dataset plotted in
Figure 8.2
,
the plot of the ACF is provided in
Figure 8.3
Autocorrelation function (ACF)
By convention, the quantity
h
in the ACF is referred to as the
lag
, the difference
between the time points t and t + h. At lag 0, the ACF provides the correlation of
every point with itself. So ACF(0) always equals 1. According to the ACF plot, at lag
1 the correlation between is approximately 0.9, which is very close to 1.
So appears to be a good predictor of the value of . Because ACF(2) is around
0.8, also appears to be a good predictor of the value of . A similar argument
could be made for lag 3 to lag 8. (All the autocorrelations are greater than 0.6.) In
other words, a model can be considered that would express as a linear sum of its
previous 8 terms. Such a model is known as an autoregressive model of order 8.
8.2.2 Autoregressive Models
For a stationary time series, , an
autoregressive model of
order p
, denoted AR(p), is expressed as shown in
Equation 8.5
:
where
is a constant for a nonzero-centered time series:
is a constant for j = 1, 2, …, p
is the value of the time series at time
