Non-seasonal ARIMA Models

Non-seasonal ARIMA Models

ARIMA models, also called Box-Jenkins models, are models that may possibly include autoregressive terms, moving average terms, and differencing operations.  Various abbreviations are used:

• When a model only involves autoregressive terms it may be referred to as an AR model.  When a model only involves moving average terms, it may be referred to as an MA model.
• When no differencing is involved, the abbreviation ARMA may be used.

Note: This week we’re only considering non-seasonal models.  We’ll expand our toolkit to include seasonal models next week.

Specifying the Elements of the Model

In most software programs, the elements in the model are specified in the order (AR order, differencing, MA order).  As examples,

A model with (only) two AR terms would be specified as an ARIMA of order (2,0,0).

A MA(2) model would be specified as an ARIMA of order (0,0,2).

A model with one AR term, a first difference, and one MA term would have order (1,1,1).

For the last model, ARIMA (1,1,1), a model with one AR term and one MA term is being applied to the variable z_t=x_t-x_t-1.  A first difference might be used to account for a linear trend in the data.

The differencing order refers to successive first differences.  For example, for a difference order = 2 the variable analyzed is z_t = (x_t-x_t-1) - (x_t-1-x_t-2), the first difference of first differences.  This type of difference might account for a quadratic trend in the data.

Identifying a Possible Model

Three items should be considered to determine a first guess at an ARIMA model: a time series plot of the data, the ACF, and the PACF.

Time series plot of the observed series.

In Lesson 1.1, we discussed what to look for: possible trend, seasonality, outliers, constant variance or nonconstant variance.

• You won’t be able to spot any particular model by looking at this plot, but you will be able to see the need for various possible actions.
• If there’s an obvious upward or downward linear trend, a first difference may be needed.  A quadratic trend might need a 2^nd order difference (as described above).  We rarely want to go much beyond two.  In those cases, we might want to think about things like smoothing, which we will cover later in the course.  Over-differencing can cause us to introduce unnecessary levels of dependency (difference white noise to obtain a MA(1)–difference again to obtain a MA(2), etc.)
• For data with a curved upward trend accompanied by increasing variance, you should consider transforming the series with either a logarithm or a square root.

Note: Nonconstant variance in a series with no trend may have to be addressed with something like an ARCH model which includes a model for changing variation over time.  We’ll cover ARCH models later in the course.