Autoregressive Moving Average with Exogenous Inputs (ARMAX) Model

In principle, an ARMAX model is a linear regression model that uses an ARMA-type process (i.e. $w_t$) to model residuals:.

$$y_t = \alpha_o + \beta_1 x_{1,t} + \beta_2 x_{2,t} + \cdots + \beta_b x_{b,t} + w_t$$

$$(1-\phi_1 L - \phi_2L^2-\cdots-\phi_pL^p)(y_t-\alpha_o -\beta_1 x_{1,t} - \beta_2 x_{2,t} - \cdots - \beta_b x_{b,t})=(1+ \theta_1 L + \theta_2 L^2 + \cdots + \theta_q L^q)a_t$$

$$(1-\phi_1 L - \phi_2 L^2 - \cdots - \phi_p L^p)w_t=(1+\theta_1 L+ \theta_2 L^2 + \cdots + \theta_q L^q ) a_t$$

$$a_t \sim$ i.i.d $\sim \Phi (0,\sigma^2)$$

Where:

  • $L$ is the lag (aka back-shift) operator.
  • $y_t$ is the observed output at time t.
  • $x_{k,t}$ is the k-th exogenous input variable at time t.
  • $\beta_k$ is the coefficient value for the k-th exogenous (explanatory) input variable.
  • $b$ is the number of exogenous input variables.
  • $w_t$ is the auto-correlated regression residuals.
  • $p$ is the order of the last lagged variables.
  • $q$ is the order of the last lagged innovation or shock.
  • $a_t$ is the innovation, shock or error term at time t.
  • $a_t$ time series observations are independent and identically distributed (i.e. i.i.d) and follow a Gaussian distribution (i.e. $\Phi(0,\sigma^2)$)

Assuming $y_t$ and all exogenous input variables are stationary, then taking the expectation from both sides, we can express $\alpha_o$ as follows:

$\alpha_o = \mu - \sum_{i=1}^b {\beta_i E[x_i] }= \mu - \sum_{i=1}^b {\beta_i \bar{x_i} }
\bar x_k$ is the long-run average of the i-th exogenous input variable.

In the event that $y_t$ is not stationary, then one must verify that: (a) one or more variables in $\{x_1,x_2,\cdots,x_b\}$ is not stationary and (b) the time series variables in $\{y, x_1,x_2,\cdots,x_b\}$ are cointegrated, so there is at least one linear combination of those variables that yields a stationary process (i.e. ARMA).

notes

  1. The variance of the shocks is constant or time-invariant.
  2. The order of an AR component process is solely determined by the order of the last lagged auto-regressive variable with a non-zero coefficient (i.e. $w_{t-p}$).
  3. The order of an MA component process is solely determined by the order of the last moving average variable with a non-zero coefficient (i.e. $a_{t-q}$).
  4. In principle, you can have fewer parameters than the orders of the model.

    Example: Consider the following ARMA(12,2) process:

    $$(1-\phi_1 L -\phi_{12} L^{12} )(y_t - \mu) = (1+\theta L^2)a_t$$

Files Examples

 

References

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