loglikelihood_int computes the log-likelihood of the specified GMAR, StMAR, or G-StMAR model.
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
)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).
log-likelihood value of the specified model,
to_return=="mw":a size ((n_obs-p)xM) matrix containing the mixing weights: for m:th component in the m:th column.
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 \(\alpha_{m,T+1}\).
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.
to_return=="terms":a size ((n_obs-p)x1) numeric vector containing the terms \(l_{t}\).
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].
to_return=="regime_cmeans":a size ((n_obs-p)xM) matrix containing the regime specific conditional means.
to_return=="regime_cvars":a size ((n_obs-p)xM) matrix containing the regime specific conditional variances.
to_return=="total_cmeans":a size ((n_obs-p)x1) vector containing the total conditional means.
to_return=="total_cvars":a size ((n_obs-p)x1) vector containing the total conditional variances.
to_return=="qresiduals":a size ((n_obs-p)x1) vector containing the quantile residuals.
a numeric vector or class 'ts' object containing the data. NA values are not supported.
a positive integer specifying the autoregressive order of the model.
a positive integer specifying the number of mixture components.
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.
a real valued parameter vector specifying the model.
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).
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 \(2\).
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.
a logical argument stating whether the AR coefficients \(\phi_{m,1},...,\phi_{m,p}\) are restricted to be the same for all regimes.
specifies linear constraints imposed to each regime's autoregressive parameters separately.
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 \(\psi_{m}\)\(=(\psi_{m,1},...,\psi_{m,q_{m}})\).
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.
a logical argument specifying whether the conditional or exact log-likelihood function should be used.
is the model parametrized with the "intercepts" \(\phi_{m,0}\) or "means" \(\mu_{m} = \phi_{m,0}/(1-\sum\phi_{i,m})\)?
a logical argument. If TRUE, then loglikelihood returns minval if...
some component variance is not larger than zero,
some parametrized mixing weight \(\alpha_{1},...,\alpha_{M-1}\) 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 \(\nu_{m}\) is not larger than two.
Argument minval will be used only if boundaries==TRUE.
TRUE or FALSE specifying whether argument checks, such as stationarity checks, should be done.
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?
this will be returned when the parameter vector is outside the parameter space and boundaries==TRUE.
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(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.