MODELS FOR NONSTATIONARY TIME SERIESBy 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 ModelsA time series {Yt} is said to follow an integrated autoregressive moving average model if the dth difference Wt = ∇dYt is a stationary ARMA processIf {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 meanA 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 valuesIf 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 transformationWe can consider λ as an additional parameter in the model to be estimated from the observed dataEvaluation 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