ARX are auto-regressive models with exogenous inputs. The term exogenous variables should not be confused independent variables. Exogenous variables are determined outside of the process you are modeling. An exogenous variable can be a shift in the oil supply effecting prices or a change in consumer preferences for foreign manufactured products effecting price. Simply put, exogenous variable are independent of the process you are trying to model. Why are ARX model different then standard models? As an example let’s look at a model trying to predict the industrial output. In this model you may want to include lagged output, (the industrial capacity is carried over from one period to the next) and lagged interest rates (the past cost of money influences current contacts). Both lagged output and lagged interest rates are endogenous to the system. What effects output also affect the price of money (interest rates). In this model an exogenous variable would be an oil crisis or natural disaster. These events happened regardless of the values of output or interest rates.
How are ARX models different?
In a vector auto-regressive models (VARX) the distinction becomes clear. In a vector auto regression model (VAR) all the variables are assumed to be correlated with one another. To identify the model you make an assumption about how the variables are contemporaneously correlated with one another. EG. Interest rates effect money immediately but only lagged money effects interest rates today. With VARX model use estimate a system of correlated variables and exogenous variables. VARX allows outside shocks to be taken into consideration.
There are many variations of ARX models.
Non-linear auto-regressive models (NARX)
Additive nonlinear autoregressive exogenous
Vector auto-regressive models (VARX)