# is_stationary: Check the stationary condition of specified GMAR, StMAR, or... In uGMAR: Estimate Univariate Gaussian and Student's t Mixture Autoregressive Models

## Description

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

## Usage

 ```1 2 3 4 5 6 7 8``` ```is_stationary( 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 θ=(υ_{1},...,υ_{M}, α_{1},...,α_{M-1},ν) where υ_{m}=(φ_{m,0},φ_{m},σ_{m}^2) φ_{m}=(φ_{m,1},...,φ_{m,p}), m=1,...,M ν=(ν_{M1+1},...,ν_{M}) M1 is the number of GMAR type regimes. In the GMAR model, M1=M and the parameter ν dropped. In the StMAR model, M1=0. If the model imposes linear constraints on the autoregressive parameters: Replace the vectors φ_{m} with the vectors ψ_{m} that satisfy φ_{m}=C_{m}ψ_{m} (see the argument `constraints`). For restricted models: Size (3M+M-M1+p-1x1) vector θ=(φ_{1,0},...,φ_{M,0},φ, σ_{1}^2,...,σ_{M}^2,α_{1},...,α_{M-1},ν), where φ=(φ_{1},...,φ_{p}) contains the AR coefficients, which are common for all regimes. If the model imposes linear constraints on the autoregressive parameters: Replace the vector φ with the vector ψ that satisfies φ=Cψ (see the argument `constraints`). Symbol φ denotes an AR coefficient, σ^2 a variance, α a mixing weight, and ν a degrees of freedom parameter. If `parametrization=="mean"`, just replace each intercept term φ_{m,0} with the regimewise mean μ_m = φ_{m,0}/(1-∑φ_{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 α are dropped, and in the case of StMAR or G-StMAR model, the degrees of freedom parameters ν 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 φ_{m,1},...,φ_{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 φ_{m}=C_{m}ψ_{m} for all m=1,...,M, where φ_{m}=(φ_{m,1},...,φ_{m,p}) and ψ_{m}=(ψ_{m,1},...,ψ_{m,q_{m}}). For restricted models:a size (pxq) constraint matrix C of full column rank satisfying φ=Cψ, where φ=(φ_{1},...,φ_{p}) and ψ=ψ_{1},...,ψ_{q}. The symbol φ 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.

## Value

Returns `TRUE` or `FALSE` accordingly.

## References

• Kalliovirta L., Meitz M. and Saikkonen P. 2015. Gaussian Mixture Autoregressive model for univariate time series. Journal of Time Series Analysis, 36, 247-266.

• Meitz M., Preve D., Saikkonen P. 2021. A mixture autoregressive model based on Student's t-distribution. Communications in Statistics - Theory and Methods, doi: 10.1080/03610926.2021.1916531

• Virolainen S. 2021. A mixture autoregressive model based on Gaussian and Student's t-distributions. Studies in Nonlinear Dynamics & Econometrics, doi: 10.1515/snde-2020-0060

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22``` ```# 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)) ```

uGMAR documentation built on Jan. 24, 2022, 5:10 p.m.