loglikelihood_int computes the log-likelihood value of the specified GMAR, StMAR or G-StMAR model for the given data.
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", "regime_cmeans", "regime_cvars",
"total_cmeans", "total_cvars"), minval)a numeric vector class 'ts' object containing the data. NA values are not supported.
a positive integer specifying the order of AR coefficients.
a positive integer specifying the number of mixture components.
a size (2x1) vector specifying the number of GMAR-type components M1 in the
first element and StMAR-type components M2 in the second. The total number of mixture components is M=M1+M2.
a real valued parameter vector specifying the model.
Size \((M(p+3)-1x1)\) vector \(\theta\)\(=\)(\(\upsilon_{1}\),...,\(\upsilon_{M}\), \(\alpha_{1},...,\alpha_{M-1}\)), where \(\upsilon_{m}\)\(=(\phi_{m,0},\)\(\phi_{m}\)\(, \sigma_{m}^2)\) and \(\phi_{m}\)=\((\phi_{m,1},...,\phi_{m,p}), m=1,...,M\).
Size \((M(p+4)-1x1)\) vector (\(\theta, \nu\))\(=\)(\(\upsilon_{1}\),...,\(\upsilon_{M}\), \(\alpha_{1},...,\alpha_{M-1}, \nu_{1},...,\nu_{M}\)).
Size \((M(p+3)+M2-1x1)\) vector (\(\theta, \nu\))\(=\)(\(\upsilon_{1}\),...,\(\upsilon_{M}\), \(\alpha_{1},...,\alpha_{M-1}, \nu_{M1+1},...,\nu_{M}\)).
Replace the vectors \(\phi_{m}\) with vectors \(\psi_{m}\) and provide a list of constraint matrices C that satisfy \(\phi_{m}\)\(=\)\(R_{m}\psi_{m}\) for all \(m=1,...,M\), where \(\psi_{m}\)\(=(\psi_{m,1},...,\psi_{m,q_{m}})\).
Size \((3M+p-1x1)\) vector \(\theta\)\(=(\phi_{1,0},...,\phi_{M,0},\)\(\phi\)\(, \sigma_{1}^2,...,\sigma_{M}^2,\alpha_{1},...,\alpha_{M-1})\), where \(\phi\)=\((\phi_{1},...,\phi_{M})\).
Size \((4M+p-1x1)\) vector (\(\theta, \nu\))\(=(\phi_{1,0},...,\phi_{M,0},\)\(\phi\)\(, \sigma_{1}^2,...,\sigma_{M}^2,\alpha_{1},...,\alpha_{M-1}, \nu_{1},...,\nu_{M})\).
Size \((3M+M2+p-1x1)\) vector (\(\theta, \nu\))\(=(\phi_{1,0},...,\phi_{M,0},\)\(\phi\)\(, \sigma_{1}^2,...,\sigma_{M}^2,\alpha_{1},...,\alpha_{M-1}, \nu_{M1+1},...,\nu_{M})\).
Replace the vector \(\phi\) with vector \(\psi\) and provide a constraint matrix \(C\) that satisfies \(\phi\)\(=\)\(R\psi\), where \(\psi\)\(=(\psi_{1},...,\psi_{q})\).
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 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 parameter \(\alpha\) is dropped, and in the case of StMAR or G-StMAR model
the degrees of freedom parameters \(\nu_{m}\) have to be larger than \(2\).
is "GMAR", "StMAR" or "G-StMAR" model considered? In 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 applied to the autoregressive parameters.
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 \(\psi\)\(=\psi_{1},...,\psi_{q}\).
Symbol \(\phi\) denotes an AR coefficient. Note that regardless of any constraints, the nominal order of AR coefficients is
alway 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...
any component variance is not larger than zero,
any 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 any degrees of freedom parameter \(\nu_{m}\) is not larger than two.
Argument minval will be used only if boundaries==TRUE.
an (optional) logical argument defining whether argument checks are made. If FALSE then no argument checks
such as stationary checks etc are made. The default is TRUE.
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)), regimewise conditional means, regimewise conditional variances,
total conditional means or total conditional variances?
Default is the log-likelihood value ("loglik").
this will be returned when the parameter vector is outside the parameter space.
Note that the first p observations are taken as the initial values so 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 m:th column.
to_return=="mw_tplus1":a size ((n_obs-p+1)xM) matrix containing the mixing weights: for m:th component in 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=="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.
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., 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. 2018. A mixture autoregressive model based on Student's t-distribution. arXiv:1805.04010 [econ.EM].
There are currently no published references for the G-StMAR model, but it's a straightforward generalization with theoretical properties similar to the GMAR and StMAR models.