The ARIMA model is one of the most popular models for making short-term forecasts. The three groups of parameters described below are used to describe this model.
An autoregression process of the p-th order can be described as:
or:
where:
L. Lag operator: 
.
. Autoregression coefficients.
A model can also be regarded where some of the autoregression coefficients are assumed to be equal to zero. Under these conditions, the dependency of the current value yt only on yt-4 is described by the following model:
or:
A process of the moving average of the q-th order can be described as:
or
where:
{θ1; θ2; …; θq}. Moving average coefficients.
The process can also be modeled when some of the coefficients are zero:
The ARIMA model can also take into account seasonal autoregression. Quarterly frequency has the following model of autoregression of the p-th order that takes into account the first order seasonal autoregression:
where:
β. Coefficient of first order seasonal autoregression.
When p = 0, that is, if the non-seasonal autoregression is absent, the model containing seasonal autoregression is converted into a simple autoregression model of the following type:
Similarly to seasonal autoregression, the first order seasonal moving average is calculated for quarters following the formula below:
where:
ω. Coefficient of first order seasonal moving average.
Parameters of the described groups enable the user to create an ARMA model (autoregressive moving average model):
where:
L. Lag operator.
. p-th order autoregression coefficients.
{β1; β2; …}. Seasonal autoregression coefficients.
{θ1; θ2; …; θq}. Coefficients of q-th order moving average.
{ω1; ω2; …}. Seasonal moving average coefficients.
{P1; P2; …}. These are determined as seasonal period multiplied by the seasonal autoregression order.
{Q1; Q2; …}. These are determined as seasonal period multiplied by the seasonal autoregression order.
An ARIMA model (autoregressive integrated moving average model) is a generalization of ARMA model. It is generally viewed as a "cascade" of two models:
First, the source series is differentiated (if required).
Differentiated series is described using ARMA model.
The source series should be differentiated if it is non-stationary. For example, if the growth trend is observed. In this case the ARMA model should be applied to the series containing values of the increase of the source series.
A standard differentiation operator looks as follows:
where d - the order of differentiation.
Data seasonality can be accounted for when differentiating data. An example of differentiating quarterly data is given below:
where:
d. Non-seasonal differentiation order.
D. Seasonal differentiation order.
s. Seasonal period.
The user can apply only seasonal differentiation if he sets the non-seasonal differentiation order to zero:
Seasonal and non-seasonal parameters must satisfy the inequality:
C − (p + P + D + D·s) > 1
where:
C. Length of the source series.
p. Maximum autoregression order.
P. Maximum seasonal autoregression order.
d. Difference.
D. Seasonal difference.
s. Seasonal period.
If the source series contains missing data, the number of values used for calculation is reduced by n·(p + P + D + D·s), where n - the number of missing data in the source series.
ARIMA model assumes that ϵt is a sequence of independent uniformly distributed random values that have normal probability distribution with zero mean of distribution and some variance s2. If this assumption is not held true after parameters estimation, the selected model specification is incorrect.
ARMA model assumes that the output series is stationary. In this case all the roots of characteristic equations for autoregression and moving average processes must lie within a unit circle on a complex plane.
Roots of AR process are found by solving the equation:
Roots of MA process are found by solving the equation:
If the assumption that the roots lie within a unit circle is not confirmed after parameters estimation, the obtained process is non-stationary.
See also:
Library of Methods and Models | Regression Model with Autoregressive Residuals and Moving Average | ARIMA Model Coefficient Estimation | Modeling Container: The ARIMA Model | Time Series Analysis: ARIMA | IModelling.Arima | ISmLinearRegress