MODELS FOR NONSTATIONARY TIME SERIES

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

=

∇

d

Y

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 W_t on the

undifferenced series Y_t? 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 (∇Y_t') = 0 and E(∇Y_t ) = β₁.

For a general ARIMA(p,d,q) model where E (∇d_Y

t) ≠ 0, it can be

argued that Y_t = Y_t' + μ_t, where μ_t is a deterministic polynomial

of degree d and Y_t' is ARIMA(p,d,q) with E Y_t = 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