uGMAR (version 3.5.2)

is_stationary: Check the stationary condition of specified GMAR, StMAR, or G-StMAR model.

Description

is_stationary checks the stationarity condition of the specified GMAR, StMAR, or G-StMAR model.

Usage

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

Value

Returns TRUE or FALSE accordingly.

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 \(\phi\)\(=\)\(C\psi\) (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 \(2\).

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.

restricted

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

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 \(\phi\)\(=\)\(C\psi\), 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.

Details

This function falsely returns FALSE for stationary models when the parameter is extremely close to the boundary of the stationarity region.

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.

Examples

# GMAR model
params22 <- c(0.4, 0.39, 0.6, 0.3, 0.4, 0.1, 0.6, 0.3, 0.8)
is_stationary(p=2, M=2, params=params22)

# StMAR model
params12t <- c(-0.3, 1, 0.9, 0.1, 0.8, 0.6, 0.7, 10, 12)
is_stationary(p=1, M=2, params=params12t, model="StMAR")

# G-StMAR model
params12gs <- c(1, 0.1, 1, 2, 0.2, 2, 0.8, 20)
is_stationary(p=1, M=c(1, 1), params=params12gs, model="G-StMAR")

# Restricted GMAR model
params13r <- c(0.1, 0.2, 0.3, -0.99, 0.1, 0.2, 0.3, 0.5, 0.3)
is_stationary(p=1, M=3, params=params13r, restricted=TRUE)

# Such StMAR(3, 2) that the AR coefficients are restricted to be the
# same for both regimes and that the second AR coefficients are
# constrained to zero.
params32trc <- c(1, 2, 0.8, -0.3, 1, 2, 0.7, 11, 12)
is_stationary(p=3, M=2, params=params32trc, model="StMAR", restricted=TRUE,
              constraints=matrix(c(1, 0, 0, 0, 0, 1), ncol=2))