ARIMA model
Introduction
The ARIMA model, standing for AutoRegressive Integrated Moving Average, is a popular forecasting method used in statistics and econometrics. This model is particularly useful for time series data that shows a clear trend or seasonal patterns. ARIMA is an acronym that stands for AutoRegressive Integrated Moving Average, reflecting the key components of the model.
Overview
ARIMA models 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 one or more times to eliminate the non-stationarity Non-stationary time series.
The AR part of ARIMA indicates that the evolving variable of interest is regressed on its own lagged (i.e., prior) values. The MA part indicates that the regression error is actually a linear combination of error terms whose values occurred contemporaneously and at various times in the past. The I (for "integrated") indicates that the data values have been replaced with the difference between their values and the previous values (and this differencing process may have been performed more than once). The purpose of each of these features is to make the model fit the data as well as possible.
Mathematical Formulation
The mathematical structure of an ARIMA model is a linear equation where the predictors consist of lags of the dependent variable and/or lags of the forecast errors. That is, the predictors are quantities that we don't know if we're forecasting one or more periods ahead.
The general form of the ARIMA model is (p, d, q) where 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.
ARIMA Model Components
Autoregressive (AR) Component
The autoregressive part of ARIMA indicates that the evolving variable of interest is regressed on its own lagged (i.e., prior) values. The autoregressive component (p) refers to the use of past values in the regression equation for the series Y. The term autoregression indicates that it is a regression of the variable against itself.
Integrated (I) Component
The integrated part of ARIMA indicates that the data values have been replaced with the difference between their values and the previous values. This differencing can help stabilize the mean of a time series by removing changes in the level of a time series, and so eliminating trend and seasonality.
Moving Average (MA) Component
The moving average part of ARIMA indicates that the regression error is actually a linear combination of error terms whose values occurred contemporaneously and at various times in the past. The moving average component (q) represents the error of the model as a combination of previous error terms. The order of the MA process refers to the number of lagged forecast errors in the prediction equation.
Application of ARIMA Models
ARIMA models are widely used for forecasting in finance, economics, and operations research. They are used to predict future values based on past values, such as predicting stock prices or economic indicators. They are also used in signal processing for signal detection and estimation.
Limitations of ARIMA Models
While ARIMA models are powerful, they do have some limitations. They assume a constant variance over time (homoscedasticity), and they assume that the relationships between variables are linear. If these assumptions are not met, the model may not provide a good fit to the data.
