Dynamic Regression Models 📂Statistical Analysis

Dynamic Regression Models

Model

Dynamic regression models are, simply put, models that combine ARIMA models with regression models.

Description

It is also called ARIMAX $ARIMAX$, which means adding independent variables other than ARIMA $X$. In programming, especially as explained below in the practice section, in R, $X$ is emphasized with xreg.

In fact, at this point, formulas are more comfortable than words. Let’s say the time series data to be analyzed as the dependent variable is $\left\{ y_{t} \right\}$, and there is another time series data $\left\{ x_{t} \right\}$ as the independent variable to explain this data.

If $x_{t}$ explains $y_{t}$ well, it is possible to perform time series regression analysis on its own, and it would be better if you could also do time series analysis on parts that are not explained by that alone. For example, if $\left\{ y_{t} \right\}$ itself follows the ARMA model $ARMA(1,1)$, the following formula can be considered. $$ \begin{align} y_{t} = y_{t-1} + e_{t} - \theta e_{t-1} + \beta x_{t} \end{align} $$ It might just appear to be a combination of formulas, and indeed it is. Understanding both regression analysis and time series analysis makes dynamic regression models an easy concept. The reason why the term dynamic is used in the name is that it is easier to understand going from regression models to time series analysis models. In the realm of science and engineering, dynamic means that the state of a system can be represented by its previous state. From that perspective, formula $(1)$ is considering previous data while performing regression analysis.

Practice

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In R, such dynamic regression models are implemented as an option of the arima() function. If you can use dynamic regression models, you can almost consider that you have understood time series analysis based on the ARIMA model, and it would be good to refer to the following cheat sheet:

(1) x: The time series data $\left\{ y_{t} \right\}$ to be analyzed goes in.
(2) order: If it is the ARIMA model $ARIMA(p,d,q)$, you put it in as order=c(p,d,q).
(3) seasonal: If it is the seasonal ARIMA model $ARIMA(p,d,q) \times (P,D,Q)_{s}$, you put it in as seasonal=list(order=c(P,D,Q), period=s). Note that it must be entered as a list.
(4) xreg: The time series data $\left\{ x_{t} \right\}$ that will be the independent variable goes in. In the case of additive outliers, dummy data is also entered in this way for processing.
(5) io: Innovative outliers go in, and if data numbers 38 and 96 are problematic, you put it in as io=c(38,96).