How to add knots to auto.arima() regressors?

Question

I am using forecast package in R by Rob Hyndman. I am using the auto.arima() function and I have various regressors.

One of my regressors is temperature values, including both negative and positive values. I only want the temperature regressor to be effective when it is below 0 and I want the other values to be ignored.

Similarly I have another regressor (Production Prognosis) which has values that are greater than 0 Ranging from 200 to 5000. I only want to use this regressor for its values below 1000.

What is the best way to do this?

Answer 1

Given the ARMAX model below, where $\phi_p(B)$ and $\theta_q(B)$ are backshift operators with lag orders $p$ and $q$:

$$\phi_p(B) y_t=\beta \cdot x_t+\theta_q(B)\epsilon_t$$

What you want is to incorporate an indicator variable telling if $x_t$ is contained in an interval $\tau$. Therefore, you actually end with the following model, which is simply allowing different slopes per category

$$\phi_p(B) y_t= \beta_{(x_t \in \tau)} \cdot I_{(x_t \in \tau)} \cdot x_t+\beta_{(x_t \not\in \tau)} \cdot I_{(x_t \not\in \tau)}\cdot x_t+\theta_q(B)\epsilon_t$$

As you said you don't want to use the exogenous variable if it's outside the interval, $\beta_{(x_t \not\in \tau)}=0$, so $$\phi_p(B) y_t= \beta_{(x_t \in \tau)} \cdot I_{(x_t \in \tau)} \cdot x_t+\theta_q(B)\epsilon_t$$

See $x_t^*=I_{(x_t \in \tau)} \cdot x_t$ is simply using $x_t$ if $(x_t \in \tau)$, and using $0$ when $(x_t \not\in \tau)$, resulting in a simple ARMAX like $$\phi_p(B) y_t= \beta_{(x_t \in \tau)} \cdot x_t^*+\theta_q(B)\epsilon_t$$