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

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


  • \( L \) 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:

\[ z_t = (1-L)^d(1-L^s)^D w_t\] \[ \mu = E[z_t] = \frac{\eta}{(1-\phi_1-\phi_2-\cdots-\phi_p)(1-\Phi_1-\Phi_2-\cdots-\Phi_P)}\] \[ (1-\sum_{i=1}^p {\phi_i L^i})(1-\sum_{j=1}^P {\Phi_j L^{j \times s}})(1-L)^d (1-L^s)^D (w_t -\mu) = (1+\sum_{i=1}^q {\theta_i L^i})(1+\sum_{j=1}^Q {\Theta_j L^{j \times s}}) a_t\]
  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.
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* Hamilton, J .D.; Time Series Analysis , Princeton University Press (1994), ISBN 0-691-04289-6
* Tsay, Ruey S.; Analysis of Financial Time Series John Wiley & SONS. (2005), ISBN 0-471-690740
* D. S.G. Pollock; Handbook of Time Series Analysis, Signal Processing, and Dynamics; Academic Press; Har/Cdr edition(Nov 17, 1999), ISBN: 125609906
* Box, Jenkins and Reisel; Time Series Analysis: Forecasting and Control; John Wiley & SONS.; 4th edition(Jun 30, 2008), ISBN: 470272848