is_stationary_int function

Check the stationarity and identification conditions of specified GMAR, StMAR, or G-StMAR model.

is_stationary_int checks the stationarity condition and is_identifiable checks the identification condition of the specified GMAR, StMAR, or G-StMAR model.

is_stationary_int(p, M, params, restricted = FALSE)

is_identifiable(
  p,
  M,
  params,
  model = c("GMAR", "StMAR", "G-StMAR"),
  restricted = FALSE,
  constraints = NULL
)

Arguments

• p: a positive integer specifying the autoregressive order of the model.

• M: - For GMAR and StMAR models:: a positive integer specifying the number of mixture components.

  • For G-StMAR models:: a size (2x1) integer vector specifying the number of GMAR type components M1 in the first element and StMAR type components M2 in the second element. The total number of mixture components is M=M1+M2.

• params: a real valued parameter vector specifying the model.

  • For non-restricted models:: Size $(M(p+3)+M-M1-1x1)$ vector theta $=$(upsilon_{1} $,...,$upsilon_{M} , $\alpha_{1},...,\alpha_{M-1},$nu ) where

      * upsilon_{m} $=(\phi_{m,0},$phi_{m} $,$$\sigma_{m}^2)$
      * phi_{m} $=(\phi_{m,1},...,\phi_{m,p}), m=1,...,M$
      * nu $=(\nu_{M1+1},...,\nu_{M})$
      * $M1$ is the number of GMAR type regimes.
     
     In the GMAR model, $M1=M$ and the parameter nu dropped. In the StMAR model, $M1=0$.
     
     If the model imposes linear constraints on the autoregressive parameters: Replace the vectors phi_{m} with the vectors psi_{m} that satisfy phi_{m} $=$C_{m}psi_{m} (see the argument `constraints`).
    

  • For restricted models:: Size $(3M+M-M1+p-1x1)$ vector theta $=(\phi_{1,0},...,\phi_{M,0},$phi $,$

      $\sigma_{1}^2,...,\sigma_{M}^2,$$\alpha_{1},...,\alpha_{M-1},$nu ), where phi =$(\phi_{1},...,\phi_{p})$
     
     contains the AR coefficients, which are common for all regimes.
     
     If the model imposes linear constraints on the autoregressive parameters: Replace the vector phi with the vector psi that satisfies phi $=$Cpsi (see the argument `constraints`).
    

  Symbol $\phi$ denotes an AR coefficient, $\sigma^2$ a variance, $\alpha$ a mixing weight, and $\nu$ a degrees of freedom parameter. If parametrization=="mean", just replace each intercept term $\phi_{m,0}$ with the regimewise mean $\mu_m = \phi_{m,0}/(1-\sum\phi_{i,m})$. In the G-StMAR model, the first M1 components are GMAR type

  and the rest M2 components are StMAR type. Note that in the case M=1 , the mixing weight parameters $\alpha$ are dropped, and in the case of StMAR or G-StMAR model, the degrees of freedom parameters $\nu$ have to be larger than $2$.

• restricted: a logical argument stating whether the AR coefficients $\phi_{m,1},...,\phi_{m,p}$ are restricted to be the same for all regimes.

• model: is "GMAR", "StMAR", or "G-StMAR" model considered? In the G-StMAR model, the first M1 components are GMAR type and the rest M2 components are StMAR type.

• constraints: specifies linear constraints imposed to each regime's autoregressive parameters separately.

  • For non-restricted models:: a list of size $(pxq_{m})$ constraint matrices C_{m} of full column rank satisfying phi_{m} $=$C_{m}psi_{m} for all $m=1,...,M$, where phi_{m} $=(\phi_{m,1},...,\phi_{m,p})$ and psi_{m} $=(\psi_{m,1},...,\psi_{m,q_{m}})$.
  • For restricted models:: a size $(pxq)$ constraint matrix C of full column rank satisfying phi $=$Cpsi , where phi $=(\phi_{1},...,\phi_{p})$ and psi $=\psi_{1},...,\psi_{q}$.

  The symbol $\phi$ denotes an AR coefficient. Note that regardless of any constraints, the autoregressive order is always p for all regimes. Ignore or set to NULL if applying linear constraints is not desired.

Returns

Returns TRUE or FALSE accordingly.

Details

is_stationary_int does not support models imposing linear constraints. In order to use it for a model imposing linear constraints, one needs to expand the constraints first to obtain an unconstrained parameter vector.

Note that is_stationary_int returns FALSE for stationary parameter vectors if they are extremely close to the boundary of the stationarity region.

is_identifiable checks that the regimes are sorted according to the mixing weight parameters and that there are no duplicate regimes.

Warning

These functions don't have any argument checks!

References

• Kalliovirta L., Meitz M. and Saikkonen P. 2015. Gaussian Mixture Autoregressive model for univariate time series. Journal of Time Series Analysis, 36 (2), 247-266.
• Meitz M., Preve D., Saikkonen P. 2023. A mixture autoregressive model based on Student's t-distribution. Communications in Statistics - Theory and Methods, 52 (2), 499-515.
• Virolainen S. 2022. A mixture autoregressive model based on Gaussian and Student's t-distributions. Studies in Nonlinear Dynamics & Econometrics, 26 (4) 559-580.