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 reduce the non-stationarity.^[1]

Non-seasonal ARIMA models are generally denoted $ARIMA(p, d, q)$ where parameters p, d, and q are non-negative integers, $p$ is the order of the Autoregressive model, $d$ is the degree of differencing, and $q$ is the order of the Moving-average model. Seasonal ARIMA models are usually denoted $ARIMA(p, d, q)(P, D, Q)_m$ , where $m$ refers to the number of periods in each season, and the uppercase $P, D, Q$ refer to the autoregressive, differencing, and moving average terms for the seasonal part of the ARIMA model.^[2]^[3] ARIMA models form an important part of the Box-Jenkins approach to time-series modelling.

When two out of the three terms are zeros, the model may be referred to based on the non-zero parameter, dropping "AR", "I" or "MA" from the acronym describing the model. For example, ARIMA (1,0,0) is AR(1), ARIMA(0,1,0) is I(1), and ARIMA(0,0,1) is MA(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 with p=p'−d, 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 autoregressive polynomial with d unit roots. (For this reason, every ARIMA model with d>0 is not wide sense stationary.)

The above can be generalized as follows.

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

This defines an ARIMA(p,d,q) process with drift δ/(1−Σφ_i).

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; this factor has the effect of re-expressing the data as changes from s periods ago. Another example is the factor $\left( 1 -\sqrt{3} L + L^2 \right)$ , which includes a (non-stationary) seasonality of period 2. 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

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 forecasts can be made for the process $Y_t$ , using a generalization of the method of autoregressive forecasting.

Examples

Some well-known special cases arise naturally. For example:

An ARIMA(0,1,0) model (or I(1) model) is given by $X_t = X_{t-1} + \varepsilon_t$ — which is simply a random walk.
An ARIMA(0,1,0) with a constant, given by $X_t = c + X_{t-1} + \varepsilon_t$ — which is a random walk with drift.
An ARIMA(0,2,2) model is given by $X_t = X_{t-1} + X_{t-2} +(\alpha + \beta - 2)\varepsilon_{t-1} + (1-\alpha)\varepsilon_{t-2} + \varepsilon_{t}$ — which is equivalent to Holt's linear method with additive errors.

Variations and extensions

A number of variations on the ARIMA model are commonly employed. 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; in that case, it is generally 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 allowed to have 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.

EViews

EViews has extensive ARIMA and SARIMA capabilities.

Julia

Julia contains an ARIMA implementation in the TimeModels package^[4]

Mathematica

Mathematica includes ARIMAProcess function.

MATLAB

MATLAB's Econometrics Toolbox includes ARIMA models and regression with ARIMA errors

Python

In Python, the "statsmodels" package includes models for time series analysis - univariate time series analysis: AR, ARIMA - vector autoregressive models, VAR and structural VAR - descriptive statistics and process models for time series analysis.

R Language

In R, the standard stats package includes an arima function, which 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 CRAN task view on Time Series is the reference with many more links.

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.

Ruby

In Ruby, the "statsample-timeseries" gem is used for time series analysis, including ARIMA models and Kalman Filtering.

SAS

SAS includes extensive ARIMA processing in its Econometric and Time Series Analysis system: SAS/ETS.

SPSS

IBM SPSS includes ARIMA modeling in its Statistics and Modeler statistical packages. The default Expert Modeler feature evaluates a range of seasonal and non-seasonal autoregressive (p), integrated (d), and moving average (q) settings and seven exponential smoothing models. The Expert Modeler can also transform the target time-series data into its square root or natural log. The user also has the option to restrict the Expert Modeler to ARIMA models, or to manually enter ARIMA nonseasonal and seasonal p, d, and q settings without Expert Modeler. Automatic outlier detection is available for seven types of outliers, and the detected outliers will be accommodated in the time-series model if this feature is selected.

The APO-FCS package^[5] in SAP ERP from SAP allows creation and fitting of ARIMA models using the Box-Jenkins methodology.

SQL

SQL Server Analysis Services from Microsoft includes ARIMA as a Data Mining algorithm.

Stata

Stata includes ARIMA modelling (using its arima command) as of Stata 9.

References

↑ For further information on Stationarity and Differencing see https://www.otexts.org/fpp/8/1
↑ "Notation for ARIMA Models". Time Series Forecasting System. SAS Institute. Retrieved 19 May 2015.
↑ Hyndman, Rob J; Athanasopoulos, George. "8.9 Seasonal ARIMA models". Forecasting: principles and practice. oTexts. Retrieved 19 May 2015.
↑ TimeModels.jl www.github.com
↑ "Box Jenkins model". SAP. Retrieved 8 March 2013.

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