MODELS FOR NONSTATIONARY TIME SERIES

MODELS FOR
NONSTATIONAR
Y TIME SERIES
By Eni Sumarminingsih, SSi,
MM

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
Yt and et does not die out—indeed, the weights
applied to Y0 and e1 grow exponentially large

The explosive behavior of such a model is also
reflected in the model’s variance and covariance

functions. These are easily found to be

The same general exponential growth or explosive
behavior will occur for any φ such that |φ| > 1

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
where ∇Yt = Yt – Yt – 1 is the first difference of Yt

ARIMA
AModels
time series {Yt} is said to follow an
integrated autoregressive moving
average model if the dth difference Wt =
∇dYt is a stationary ARMA process
If {Wt} follows an ARMA(p,q) model, we
say that {Yt} is an ARIMA(p,d,q) process

Fortunately, for practical purposes, we can
usually take d = 1 or at most 2.

Consider then an ARIMA(p,1,q) process.
With Wt = Yt − Yt − 1, we have

or, in terms of the observed series,

The IMA(1,1) Model
In difference equation form, the model is

or

After a little rearrangement, we can write

From Equation (5.2.6), we can easily derive variances and
correlations. We have

and

The IMA(2,2) Model
In difference equation form, we have

or

The ARI(1,1) Model

or

Constant Terms in ARIMA Models
For an ARIMA(p,d,q) model, ∇dYt = 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:

Taking expected values on both sides of the latter expression, we

find that

so that

or, conversely, 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
added linear deterministic time trend (t + m + 1)θ0 with slope θ0.

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 (∇dYt) ≠ 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
θ0 ≠ 0, a quadratic trend would be implied.

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
Evaluation of a range of transformations based on a grid of λ
values, say ±1, ±1/2, ±1/3, ±1/4, and 0, will usually suffice