forecast::Arima Models

Setup

First we need to generate a model. In this example, we won’t worry too much about whether or not the model is appropriate. Rather, we aim to illustrate that the corresponding equations are parsed correctly.

library(equatiomatic)
library(forecast)

# Build Arima (no regression)
simple_ts_mod <- Arima(simple_ts,
  order = c(1, 1, 1),
  seasonal = c(1, 0, 1),
  include.constant = TRUE
)

Extracting the equation

To extract the equation, we just call equatiomatic::extract_eq() on the resulting model object, equivalent to other model types. The model above outputs the following.

extract_eq(simple_ts_mod)

\[ (1 -\phi_{1}\operatorname{B} )\ (1 -\Phi_{1}\operatorname{B}^{\operatorname{4}} )\ (1 - \operatorname{B}) (y_{t} -\delta\operatorname{t}) = (1 +\theta_{1}\operatorname{B} )\ (1 +\Theta_{1}\operatorname{B}^{\operatorname{4}} )\ \varepsilon_{t} \]

Setup

# Build Exogenous Regressors
xregs <- as.matrix(data.frame(
  x1 = ts_reg_list$x1 + 5,
  x2 = ts_reg_list$x2 * 5
))

# Build Regression Model
ts_reg_mod <- Arima(ts(ts_reg_list$ts_rnorm, freq = 4),
  order = c(1, 1, 1),
  seasonal = c(1, 0, 1),
  xreg = xregs,
  include.constant = TRUE
)

Extracting the equation

Despite the extra complexity of the model, the code to pull the equation remains equivalent.

extract_eq(ts_reg_mod)

\[ \begin{alignat}{2} &\text{let}\quad &&y_{t} = \operatorname{y}_{\operatorname{0}} +\delta\operatorname{t} +\beta_{1}\operatorname{x1}_{\operatorname{t}} +\beta_{2}\operatorname{x2}_{\operatorname{t}} +\eta_{t} \\ &\text{where}\quad &&(1 -\phi_{1}\operatorname{B} )\ (1 -\Phi_{1}\operatorname{B}^{\operatorname{4}} )\ (1 - \operatorname{B}) \eta_{t} \\ & &&= (1 +\theta_{1}\operatorname{B} )\ (1 +\Theta_{1}\operatorname{B}^{\operatorname{4}} )\ \varepsilon_{t} \\ &\text{where}\quad &&\varepsilon_{t} \sim{WN(0, \sigma^{2})} \end{alignat} \]