MODELS FOR
NONSTATIONAR
Y TIME SERIES
Stationarity Through
Differencing
Consider again the AR(1) model
Consider in particular the equation
Iterating into the past as we have done before
yields
We see that the influence of distant past values of
The explosive behavior of such a model is also
reflected in the model’s variance and covariance
functions. These are easily found to be
A more reasonable type of nonstationarity obtains
when φ = 1. If φ = 1, the AR(1) model equation is
This is the relationship satisfied by the random
walk process. Alternatively, we can rewrite this as
ARIMA
Models
A time series {
Y
t} is said to follow an
integrated autoregressive moving
average
model if the
d
th difference
W
t=
∇
dY
t
is a stationary ARMA process
If {
W
t} follows an ARMA(
p
,
q
) model, we
say that {
Y
t} is an ARIMA(
p
,
d
,
q
) process
Consider then an ARIMA(
p
,1,
q
) process.
With
W
t=
Y
t−
Y
t − 1, we have
The IMA(1,1) Model
In difference equation form, the model is
or
From Equation (5.2.6), we can easily derive variances and correlations. We have
The IMA(2,2) Model
In difference equation form, we have
The ARI(1,1) Model
Constant Terms in ARIMA Models
For an ARIMA(p,d,q) model, ∇dY
t = Wt is a stationary ARMA(p,q)
process. Our standard assumption is that stationary models have a zero mean
A nonzero constant mean, μ, in a stationary ARMA model {Wt} can be accommodated in either of two ways. We can assume that
Alternatively, we can introduce a constant term θ0 into the model as follows:
so that
What will be the effect of a nonzero mean for Wt on the
undifferenced series Yt? Consider the IMA(1,1) case with a constant
term. We have
or
by iterating into the past, we find that
Comparing this with Equation (5.2.6), we see that we have an
An equivalent representation of the process would then be
Where Y’t is an IMA(1,1) series with E (∇Yt') = 0 and E(∇Yt ) = β1.
For a general ARIMA(p,d,q) model where E (∇dY
t) ≠ 0, it can be
argued that Yt = Yt' + μt, where μt is a deterministic polynomial
of degree d and Yt' is ARIMA(p,d,q) with E Yt = 0. With d = 2 and
Power Transformations
A flexible family of transformations, the power
transformations, was introduced by Box and Cox (1964). For a given value of the parameter λ, the transformation is defined by
The power transformation applies only to positive data values If some of the values are negative or zero, a positive constant may be added to all of the values to make them all positive before doing the power transformation
We can consider λ as an additional parameter in the model to be estimated from the observed data
