loglikelihood_int: Compute the log-likelihood of Gaussian or Student's t Mixture Autoregressive model

Description

FOR INTERNAL USE. loglikelihood_int computes the log-likelihood value of the specified GMAR or StMAR model for the given data.

Usage

loglikelihood_int(data, p, M, params, StMAR = FALSE, restricted = FALSE,
  constraints = FALSE, R, conditional = TRUE, boundaries = FALSE,
  checks = TRUE, returnTerms = FALSE, epsilon, minval)

Arguments

data

a numeric vector or column matrix 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 or regimes.

params

a real valued parameter vector specifying the model.

For non-restricted models:

For GMAR 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 $\phi_{m}$=$(\phi_{m,1},...,\phi_{m,p}), m=1,...,M$.
For StMAR model:: Size $(M(p+4)-1x1)$ vector ($\theta, \nu$)$=$($\upsilon_{1}$,...,$\upsilon_{M}$, $\alpha_{1},...,\alpha_{M-1}, \nu_{1},...,\nu_{M}$).
With linear constraints:: Replace the vectors $\phi_{m}$ with vectors $\psi_{m}$ and provide a list of constraint matrices R that satisfy $\phi_{m}$$=$$R_{m}\psi_{m}$ for all $m=1,...,M$, where $\psi_{m}$$=(\psi_{m,1},...,\psi_{m,q_{m}})$.

For restricted models:

For GMAR model:: Size $(3M+p-1x1)$ vector $\theta$$=(\phi_{1,0},...,\phi_{M,0},$$\phi$$, \sigma_{1}^2,...,\sigma_{M}^2,\alpha_{1},...,\alpha_{M-1})$, where $\phi$=$(\phi_{1},...,\phi_{M})$.
For StMAR model:: Size $(4M+p-1x1)$ vector ($\theta, \nu$)$=(\phi_{1,0},...,\phi_{M,0},$$\phi$$, \sigma_{1}^2,...,\sigma_{M}^2,\alpha_{1},...,\alpha_{M-1}, \nu_{1},...,\nu_{M})$.
With linear constraints:: Replace the vector $\phi$ with vector $\psi$ and provide a constraint matrix $R$ that satisfies $\phi$$=$$R\psi$, where $\psi$$=(\psi_{1},...,\psi_{q})$.

Symbol $\phi$ denotes an AR coefficient, $\sigma^2$ a variance, $\alpha$ a mixing weight and $v$ a degrees of freedom parameter. Note that in the case M=1 the parameter $\alpha$ is dropped, and in the case of StMAR model the degrees of freedom parameters $\nu_{m}$ have to be larger than $2$.

StMAR

an (optional) logical argument stating whether StMAR model should be considered instead of GMAR model. Default is FALSE.

restricted

an (optional) logical argument stating whether the AR coefficients $\phi_{m,1},...,\phi_{m,p}$ are restricted to be the same for all regimes. Default is FALSE.

constraints

an (optional) logical argument stating whether general linear constraints should be applied to the model. Default is FALSE.

Specifies the linear constraints.

For non-restricted models:: a list of size $(pxq_{m})$ constraint matrices $R_{m}$ of full column rank satisfying $\phi_{m}$$=$$R_{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}})$.
For restricted models:: a size $(pxq)$ constraint matrix $R$ of full column rank satisfying $\phi$$=$$R\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. This argument is ignored if constraints==FALSE.

conditional

an (optional) logical argument specifying wether the conditional or exact log-likehood function should be used. Default is TRUE.

boundaries

an (optional) 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 StMAR=TRUE and any degrees of freedom parameter $v_{m}$ is not larger than two or is larger than 342-p.

Default is FALSE.

checks

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.

returnTerms

set TRUE if the terms $l_{t}: t=1,..,T$ in the log-likelihood function (see KMS 2015, eq.(13)) should not be summed to calculate the log-likelihood value, but to be returned as a numeric vector. Default is FALSE.

epsilon

an (optional) negative real number specifying the logarithm of the smallest positive non-zero number that will be handled without external packages. Too small value may lead to a failure or biased results and too large value will make the code run significantly slower. Default is round(log(.Machine$double.xmin)+10) and should not be adjusted too much.

minval

a negative real number defining the log-likelihood value that will be returned with boundaries==TRUE when the parameter is out of the boundaries. Ignored if boundaries==FALSE.

Value

By default:: log-likelihood value of the specified GMAR or StMAR model,
If returnTerms==TRUE:: size $Tx1$ numeric vector containing the terms $l_{t}$.

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.
Lutkepohl H. New Introduction to Multiple Time Series Analysis, Springer, 2005.
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.
References regarding the StMAR model and general linear constraints will be updated after they are published.