Seasonal Autoregressive Integrated Moving Average exogeneous (SARIMAX) Model

In principle, an SARIMAX model is a linear regression model that uses a SARIMA-type process (i.e. ) This model is useful in cases we suspect that residuals may exhibit a seasonal trend or pattern.


$$w_t = y_t - \beta_1 x_{1,t}-\beta_2 x_{2,t} - \cdots - \beta_b x_{b,t}$$ $$(1-\sum_{i=1}^p {\phi_i L^i})(1-\sum_{j=1}^P {\Phi_j L^{j \times s}})(1-L)^d (1-L^s)^D w_t -\eta = (1+\sum_{i=1}^q {\theta_i L^i})(1+\sum_{j=1}^Q {\Theta_j L^{j \times s}}) a_t$$ $$a_t \sim \textrm{i.i.d} \sim \Phi(0,\sigma^2)$$

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 non-seasonal AR component.
  • $P$ is the order of the seasonal AR component.
  • $q$ is the order of the non-seasonal MA component.
  • $Q$ is the order of the seasonal MA component.
  • $s$ is the seasonal length.
  • $D$ is the seasonal integration order of the time series.
  • $\eta$ is a constant in the SARIMA model
  • $a_t$ 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. $\Phi(0,\sigma^2)$)

Re-ordering the terms in the equation above and assuming the differenced (both seasonal and non-seasonal) results in a stationary time series ($z_t$) yields the following:
$$z_t = (1-L)^d(1-L^s)^D w_t$$
$$\mu = E[z_t] = \frac{\eta}{(1-\phi_1-\phi_2-\cdots-\phi_p)(1-\Phi_1-\Phi_2-\cdots-\Phi_P)}$$ $$(1-\sum_{i=1}^p {\phi_i L^i})(1-\sum_{j=1}^P {\Phi_j L^{j \times s}})(1-L)^d (1-L^s)^D (w_t -\mu) = (1+\sum_{i=1}^q {\theta_i L^i})(1+\sum_{j=1}^Q {\Theta_j L^{j \times s}}) a_t$$

 

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.

 

References

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