The only difference between ARIMA and ARIMAX is the addition of an exogenous (external) variable. The ARIMA model works on a single time series data (univariate) whereas ARIMAX uses multiple variables to include the external feature.
Equation of ARIMAX
Δr(t)= c + φ Δr(t-1) + θ ε(t-1) + ε(t) + β x
- Δr(t)= r(t)-r(t-1) , difference in consecutive period.
- ε(t),ε(t-1) = current error term and one period ago.
- c = baseline constant factor.
- φ = value coefficient, what part of the last period value is relevant in explaining the current value.
- θ = error coefficient, what part of the last period value is relevant in explaining the current error value.
- β = coefficient for the external variable.
- x = value of the external variable.
Equation of ARIMA
The equation of the ARIMA model is the same as ARIMAX except the external factor.
Δr(t)= c + φ Δr(t-1) + θ ε(t-1) + ε(t)