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6 Nonstationary
and Seasonal
Time Series Models
6.1 ARIMA Models for Nonstationary Time Series
6.2 Identification Techniques
6.3 Unit Roots in Time Series Models
6.4 Forecasting ARIMA Models
6.5 Seasonal ARIMA Models
6.6 Regression with ARMA Errors
In this chapter we shall examine the problem of finding an appropriate model for a
given set of observations {x1,...,xn} that are not necessarily generated by a stationary
time series. If the data (a) exhibit no apparent deviations from stationarity and (b)
have a rapidly decreasing autocovariance function, we attempt to fit an ARMA model
to the mean-corrected data using the techniques developed in Chapter 5. Otherwise,
we look first for a transformation of the data that generates a new series with the
properties (a) and (b). This can frequently be achieved by differencing, leading us
to consider the class of ARIMA (autoregressive integrated moving-average) models,
defined in Section 6.1. We have in fact already encountered ARIMA processes. The
model fitted in Example 5.1.1 to the Dow Jones Utilities Index was obtained by fitting
an AR model to the differenced data, thereby effectively fitting an ARIMA model to
the original series. In Section 6.1 we shall give a more systematic account of such
models.
In Section 6.2 we discuss the problem of finding an appropriate transformation for
the data and identifying a satisfactory ARMA(p, q) model for the transformed data.
The latter can be handled using the techniques developed in Chapter 5. The sample
ACF and PACF and the preliminary estimators ˆφm and ˆθm of Section 5.1 can provide
useful guidance in this choice. However, our prime criterion for model selection will
be the AICC statistic discussed in Section 5.5.2. To apply this criterion we compute
maximum likelihood estimators of φ, θ, and σ 2 for a variety of competing p and q