**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 *= W*t 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

*) = β*

_{t}_{1}.

For a general ARIMA(p,d,q) model where E (∇*d _{Y}*

*t) ≠ 0, it can be *

argued that Y* _{t} = Y_{t}*' + μ

*, where μ*

_{t}

_{t}is a deterministic polynomialof 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