statespace_arima

ARMA and ARIMA components (DK §3.4, §3.6.2, §5.6.2-5.6.4).

The stationary part \(y^*_t = \Delta^d \Delta_s^D y_t\) follows the ARMA model \(\phi(L)\Phi(L^s) y^*_t = \theta(L)\Theta(L^s) \zeta_t\), put in DK’s form (DK eqs. 3.19-3.20): with \(r = \max(p + sP, q + sQ + 1)\),

\[\begin{split}T = \begin{bmatrix} \phi^* & I \\ & 0 \end{bmatrix}, \quad R = (1, \theta^*_1, \dots, \theta^*_{r-1})', \quad Z = e_1',\end{split}\]

where \(\phi^*, \theta^*\) are the coefficients of the multiplied polynomials. For d > 0 the state is extended by the cumulative differences (DK §3.4), and for D = 1 by the s lags of \(\Delta^d y\). The differenced states are diffuse and the ARMA block stationary (DK §5.6.3). The layout equals statsmodels’ SARIMAX, so the states can be compared one by one.

A constant or regressors, as in regression with ARMA errors (DK §3.6.2, §5.6.4), are added as a regression_t component.

arima_t

type, extends(component_t) :: arima_t
  integer :: ar = 0, d = 0, ma = 0            ! p, d, q
  integer :: sar = 0, sd = 0, sma = 0         ! P, D, Q (D at most 1)
  integer :: s = 0                            ! seasonal period
  integer :: series = 1
  logical :: enforce_stationarity = .true.
  logical :: enforce_invertibility = .true.
end type

Parameters: AR (p), seasonal AR (P), MA (q), seasonal MA (Q), then the variance of \(\zeta_t\), named ar.L1, ..., ar.S.L12, ..., ma.L1, ..., sigma2. AR polynomials are kept stationary and MA polynomials invertible by Monahan’s (1984) transform, as in statsmodels; the variance uses \(\exp(2\psi)\).

Example

An ARMA(1, 1) with a constant (DK §5.6.4):

type(component_holder_t) :: comps(2)
comps(1)%c = regression_t(x=reshape(spread(1.0_dp, 1, n), [n, 1]))
comps(2)%c = arima_t(ar=1, ma=1)
model = structural_model(y, comps, info)