Autoregressive integrated moving average

In statistics and econometrics, and in particular in time series analysis, an autoregressive integrated moving average (ARIMA) model is a generalization of an autoregressive moving average (ARMA) model. These models are fitted to time series data either to better understand the data or to predict future points in the series (forecasting). They are applied in some cases where data show evidence of non-stationarity, where an initial differencing step (corresponding to the "integrated" part of the model) can be applied to remove the non-stationarity.

The model is generally referred to as an ARIMA(p,d,q) model where p, d, and q are non-negative integers that refer to the order of the autoregressive, integrated, and moving average parts of the model respectively. ARIMA models form an important part of the Box-Jenkins approach to time-series modelling.

When one of the terms is zero, it's usual to drop AR, I or MA. For example, an I(1) model is ARIMA(0,1,0), and a MA(1) model is ARIMA(0,0,1).

Definition

Given a time series of data $X_t$ where $t$ is an integer index and the $X_t$ are real numbers, then an ARMA(p,q) model is given by:

$\left( 1 - \sum_{i=1}^{p} \alpha_i L^i \right) X_t = \left( 1 + \sum_{i=1}^q \theta_i L^i \right) \varepsilon_t \,$

where $L$ is the lag operator, the $\alpha_i$ are the parameters of the autoregressive part of the model, the $\theta_i$ are the parameters of the moving average part and the $\varepsilon_t$ are error terms. The error terms $\varepsilon_t$ are generally assumed to be independent, identically distributed variables sampled from a normal distribution with zero mean.

Assume now that the polynomial $\left( 1 - \sum_{i=1}^{p} \alpha_i L^i \right)$ has a unitary root of multiplicity d. Then it can be rewritten as:

$\left( 1 - \sum_{i=1}^p \alpha_i L^i \right) = \left( 1 + \sum_{i=1}^{p-d} \phi_i L^i \right) \left( 1 - L \right)^{d} .$

An ARIMA(p,d,q) process expresses this polynomial factorisation property, and is given by:

$\left( 1 - \sum_{i=1}^p \phi_i L^i \right) \left( 1-L \right)^d X_t = \left( 1 + \sum_{i=1}^q \theta_i L^i \right) \varepsilon_t \,$

and thus can be thought as a particular case of an ARMA(p+d,q) process having the auto-regressive polynomial with some roots in the unity. For this reason every ARIMA model with d>0 is not wide sense stationary.

Other special forms

The explicit identification of the factorisation of the autoregression polynomial into factors as above, can be extended to other cases, firstly to apply to the moving average polynomial and secondly to include other special factors. For example, having a factor $\left( 1 - L^s \right)$ in a model is one way of including a non-stationary seasonality of period s into the model. Another example is the factor $\left( 1 -\sqrt{3} L + L^2 \right)$, which includes a (non-stationary) seasonality of period 12. The effect of the first type of factor is to allow each season's value to drift separately over time, whereas with the second type values for adjacent seasons move together.

Identification and specification of appropriate factors in an ARIMA model can be an important step in modelling as it can allow a reduction in the overall number of parameters to be estimated, while allowing the imposition on the model of types of behaviour that logic and experience suggest should be there.

Forecasts using ARIMA models

ARIMA models are used for observable non-stationary processes $X_t$ that have some clearly identifiable trends:

• a constant trend (i.e. zero average) is modeled by $d = 0$
• a linear trend (i.e. linear growth behavior) is modeled by $d = 1$
• a quadratic trend (i.e. quadratic growth behavior) is modeled by $d = 2$

In these cases the ARIMA model can be viewed as a "cascade" of two models. The first is non-stationary:

$Y_t = \left( 1-L \right)^d X_t$

while the second is wide-sense stationary:

$\left( 1 - \sum_{i=1}^p \phi_i L^i \right) Y_t = \left( 1 + \sum_{i=1}^q \theta_i L^i \right) \varepsilon_t \, .$

Now standard forecasts techniques can be formulated for the process $Y_t$, and then (having the sufficient number of initial conditions) $X_t$ can be forecast via opportune integration steps.

Examples

Some well-known special cases arise naturally. For example, an ARIMA(0,1,0) model is given by:

$X_t = X_{t-1} + \varepsilon_t$

which is simply a random walk.

A number of variations on the ARIMA model are commonly used. For example, if multiple time series are used then the $X_t$ can be thought of as vectors and a VARIMA model may be appropriate. Sometimes a seasonal effect is suspected in the model. For example, consider a model of daily road traffic volumes. Weekends clearly exhibit different behaviour from weekdays. In this case it is often considered better to use a SARIMA (seasonal ARIMA) model than to increase the order of the AR or MA parts of the model. If the time-series is suspected to exhibit long-range dependence then the $d$ parameter may be replaced by certain non-integer values in an autoregressive fractionally integrated moving average model, which is also called a Fractional ARIMA (FARIMA or ARFIMA) model.

Implementations in statistics packages

Various packages that apply methodology like Box-Jenkins parameter optimization are available to find the right parameters for the ARIMA model.

• In R, the stats package includes an arima function. The function is documented in "ARIMA Modelling of Time Series". Besides the ARIMA(p,d,q) part, the function also includes seasonal factors, an intercept term, and exogenous variables (xreg, called "external regressors").
• The "forecast" package in R can automatically select an ARIMA model for a given time series with the auto.arima() function. The package can also simulate seasonal and non-seasonal ARIMA models with its simulate.Arima() function. It also has a function Arima(), which is a wrapper for the arima from the "stats" package.
• SAS(R) of "SAS Institute Inc." includes extensive ARIMA processing in its Econometric and Time Series Analysis system: SAS/ETS.

See also

• Autocorrelation
• ARMA
• X-12-ARIMA
• Partial autocorrelation

References

• Mills, Terence C. (1990) Time Series Techniques for Economists. Cambridge University Press
• Percival, Donald B. and Andrew T. Walden. (1993) Spectral Analysis for Physical Applications. Cambridge University Press.

External links

• The US Census Bureau uses ARIMA for "seasonally adjusted" data (programs, docs, and papers here)