loglikelihood computes the log-likelihood of the specified GMAR, StMAR, or G-StMAR model.
Exists for convenience if one wants to for example plot profile log-likelihoods or employ other estimation algorithms.
Use minval to control what happens when the parameter vector is outside the parameter space.
loglikelihood(
data,
p,
M,
params,
model = c("GMAR", "StMAR", "G-StMAR"),
restricted = FALSE,
constraints = NULL,
conditional = TRUE,
parametrization = c("intercept", "mean"),
return_terms = FALSE,
minval = NA
)Returns the log-likelihood value or the terms described in return_terms.
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" \(\mu_{m} = \phi_{m,0}/(1-\sum\phi_{i,m})\)?
should the terms \(l_{t}: t=1,..,T\) in the log-likelihood function (see KMS 2015, eq.(13) or MPS 2018, eq.(15)) be returned instead of the log-likelihood value?
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.
fitGSMAR, GSMAR, quantile_residuals,
mixing_weights, calc_gradient
# GMAR model
params12 <- c(1.70, 0.85, 0.30, 4.12, 0.73, 1.98, 0.63)
loglikelihood(simudata, p=1, M=2, params=params12, model="GMAR")
# G-StMAR-model
params42gs <- c(0.04, 1.34, -0.59, 0.54, -0.36, 0.01, 0.06, 1.28, -0.36,
0.2, -0.15, 0.04, 0.19, 9.75)
loglikelihood(M10Y1Y, p=4, M=c(1, 1), params=params42gs, model="G-StMAR")
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