# loglikelihood_int: Compute the log-likelihood of GMAR, StMAR, or G-StMAR model In uGMAR: Estimate Univariate Gaussian and Student's t Mixture Autoregressive Models

## Description

`loglikelihood_int` computes the log-likelihood of the specified GMAR, StMAR, or G-StMAR model.

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17``` ```loglikelihood_int( data, p, M, params, model = c("GMAR", "StMAR", "G-StMAR"), restricted = FALSE, constraints = NULL, conditional = TRUE, parametrization = c("intercept", "mean"), boundaries = TRUE, checks = TRUE, to_return = c("loglik", "mw", "mw_tplus1", "loglik_and_mw", "terms", "term_densities", "regime_cmeans", "regime_cvars", "total_cmeans", "total_cvars", "qresiduals"), minval ) ```

## Arguments

 `data` a numeric vector or class `'ts'` object containing the data. `NA` values are not supported. `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. `conditional` a logical argument specifying whether the conditional or exact log-likelihood function should be used. `parametrization` is the model parametrized with the "intercepts" φ_{m,0} or "means" μ_{m} = φ_{m,0}/(1-∑φ_{i,m})? `boundaries` a logical argument. If `TRUE`, then `loglikelihood` returns `minval` if... some component variance is not larger than zero, some parametrized mixing weight α_{1},...,α_{M-1} is not larger than zero, sum of the parametrized mixing weights is not smaller than one, if the model is not stationary, or if `model=="StMAR"` or `model=="G-StMAR"` and some degrees of freedom parameter ν_{m} is not larger than two. Argument `minval` will be used only if `boundaries==TRUE`. `checks` `TRUE` or `FALSE` specifying whether argument checks, such as stationarity checks, should be done. `to_return` should the returned object be the log-likelihood value, mixing weights, mixing weights including value for alpha_{m,T+1}, a list containing log-likelihood value and mixing weights, the terms l_{t}: t=1,..,T in the log-likelihood function (see KMS 2015, eq.(13)), the densities in the terms, regimewise conditional means, regimewise conditional variances, total conditional means, total conditional variances, or quantile residuals? `minval` this will be returned when the parameter vector is outside the parameter space and `boundaries==TRUE`.

## Value

Note that the first p observations are taken as the initial values so the mixing weights and conditional moments start from the p+1:th observation (interpreted as t=1).

By default:

log-likelihood value of the specified model,

If `to_return=="mw"`:

a size ((n_obs-p)xM) matrix containing the mixing weights: for m:th component in the m:th column.

If `to_return=="mw_tplus1"`:

a size ((n_obs-p+1)xM) matrix containing the mixing weights: for m:th component in the m:th column. The last row is for α_{m,T+1}.

If `to_return=="loglik_and_mw"`:

a list of two elements. The first element contains the log-likelihood value and the second element contains the mixing weights.

If `to_return=="terms"`:

a size ((n_obs-p)x1) numeric vector containing the terms l_{t}.

If `to_return=="term_densities"`:

a size ((n_obs-p)xM) matrix containing the conditional densities that summed over in the terms l_{t}, as `[t, m]`.

If `to_return=="regime_cmeans"`:

a size ((n_obs-p)xM) matrix containing the regime specific conditional means.

If `to_return=="regime_cvars"`:

a size ((n_obs-p)xM) matrix containing the regime specific conditional variances.

If `to_return=="total_cmeans"`:

a size ((n_obs-p)x1) vector containing the total conditional means.

If `to_return=="total_cvars"`:

a size ((n_obs-p)x1) vector containing the total conditional variances.

If `to_return=="qresiduals"`:

a size ((n_obs-p)x1) vector containing the quantile residuals.

## References

• Galbraith, R., Galbraith, J. 1974. On the inverses of some patterned matrices arising in the theory of stationary time series. Journal of Applied Probability 11, 63-71.

• Kalliovirta L. (2012) Misspecification tests based on quantile residuals. The Econometrics Journal, 15, 358-393.

• 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

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