uGMAR (version 3.2.0)

mixingWeights: Calculate mixing weights of GMAR, StMAR or G-StMAR model

Description

mixingWeights calculates the mixing weights of the specified GMAR, StMAR or G-StMAR model and returns them as a matrix.

Usage

mixingWeights(data, p, M, params, model = c("GMAR", "StMAR", "G-StMAR"),
  restricted = FALSE, constraints = NULL,
  parametrization = c("intercept", "mean"))

Arguments

data

a numeric vector class 'ts' object containing the data. NA values are not supported.

p

a positive integer specifying the order of AR coefficients.

M

For GMAR and StMAR models:

a positive integer specifying the number of mixture components.

For G-StMAR model:

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.

params

a real valued parameter vector specifying the model.

For non-restricted models:

For GMAR model:

Size $(M(p+3)-1x1)$ vector $\theta$$=$(${\upsilon }_1$,...,${\upsilon }_M$, ${\alpha }_1,...,{\alpha }_{M-1}$), where ${\upsilon }_m$$=({\varphi }_{m,0},$${\varphi }_m$$,{\sigma }_m^2)$ and ${\varphi }_m$=$({\varphi }_{m,1},...,{\varphi }_{m,p}),m=1,...,M$.

For StMAR model:

Size $(M(p+4)-1x1)$ vector ($\theta ,\nu$)$=$(${\upsilon }_1$,...,${\upsilon }_M$, ${\alpha }_1,...,{\alpha }_{M-1},{\nu }_1,...,{\nu }_M$).

For G-StMAR model:

Size $(M(p+3)+M2-1x1)$ vector ($\theta ,\nu$)$=$(${\upsilon }_1$,...,${\upsilon }_M$, ${\alpha }_1,...,{\alpha }_{M-1},{\nu }_{M1+1},...,{\nu }_M$).

With linear constraints:

Replace the vectors ${\varphi }_m$ with vectors ${\psi }_m$ and provide a list of constraint matrices C that satisfy ${\varphi }_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$$=({\varphi }_{1,0},...,{\varphi }_{M,0},$$\varphi$$,{\sigma }_1^2,...,{\sigma }_M^2,{\alpha }_1,...,{\alpha }_{M-1})$, where $\varphi$=$({\varphi }_1,...,{\varphi }_M)$.

For StMAR model:

Size $(4M+p-1x1)$ vector ($\theta ,\nu$)$=({\varphi }_{1,0},...,{\varphi }_{M,0},$$\varphi$$,{\sigma }_1^2,...,{\sigma }_M^2,{\alpha }_1,...,{\alpha }_{M-1},{\nu }_1,...,{\nu }_M)$.

For G-StMAR model:

Size $(3M+M2+p-1x1)$ vector ($\theta ,\nu$)$=({\varphi }_{1,0},...,{\varphi }_{M,0},$$\varphi$$,{\sigma }_1^2,...,{\sigma }_M^2,{\alpha }_1,...,{\alpha }_{M-1},{\nu }_{M1+1},...,{\nu }_M)$.

With linear constraints:

Replace the vector $\varphi$ with vector $\psi$ and provide a constraint matrix $C$ that satisfies $\varphi$$=$$R\psi$, where $\psi$$=({\psi }_1,...,{\psi }_q)$.

Symbol $\varphi$ 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 ${\varphi }_{m,0}$ with regimewise mean ${\mu }_m={\varphi }_{m,0}/(1-\sum {\varphi }_{i,m})$. 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 $2$.

model

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.

restricted

a logical argument stating whether the AR coefficients ${\varphi }_{m,1},...,{\varphi }_{m,p}$ are restricted to be the same for all regimes.

constraints

specifies linear constraints applied to the autoregressive parameters.

For non-restricted models:

a list of size $(px{q}_m)$ constraint matrices $C_m$ of full column rank satisfying ${\varphi }_m$$=$$C_m{\psi }_m$ for all $m=1,...,M$, where ${\varphi }_m$$=({\varphi }_{m,1},...,{\varphi }_{m,p})$ and ${\psi }_m$$=({\psi }_{m,1},...,{\psi }_{m,q_m})$.

For restricted models:

a size $(pxq)$ constraint matrix $C$ of full column rank satisfying $\varphi$$=$$C\psi$, where $\varphi$$=({\varphi }_1,...,{\varphi }_p)$ and $\psi$$={\psi }_1,...,{\psi }_q$.

Symbol $\varphi$ 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.

parametrization

is the model parametrized with the "intercepts" ${\varphi }_{m,0}$ or "means" ${\mu }_m={\varphi }_{m,0}/(1-\sum {\varphi }_{i,m})$?

Value

If to_return=="mw":

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

.

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., 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.

Examples

# NOT RUN {
# GMAR model
params12 <- c(0.18, 0.93, 0.01, 0.86, 0.68, 0.02, 0.88)
mixingWeights(logVIX, 1, 2, params12)

# Restricted GMAR model, outside parameter space
params12r <- c(0.21, 0.23, 0.92, 0.01, 0.02, 0.86)
mixingWeights(logVIX, 1, 2, params12r, restricted=TRUE)

# Non-mixture version of StMAR model, outside parameter space
params11t <- c(0.16, 0.93, 0.01, 3.01)
mixingWeights(logVIX, 1, 1, params11t, model="StMAR")

# G-StMAR model
params12gs <- c(0.86, 0.68, 0.02, 0.18, 0.93, 0.01, 0.11, 44.36)
mixingWeights(logVIX, 1, c(1, 1), params12gs, model="G-StMAR")

# Restricted G-StMAR model
params12gsr <- c(0.31, 0.33, 0.88, 0.01, 0.02, 0.77, 2.72)
mixingWeights(logVIX, 1, c(1, 1), params12gsr, model="G-StMAR", restricted=TRUE)

# GMAR model as a mixture of AR(2) and AR(1) models
constraints <- list(diag(1, ncol=2, nrow=2), as.matrix(c(1, 0)))
params22c <- c(0.61, 0.83, -0.06, 0.02, 0.21, 0.91, 0.01, 0.16)
mixingWeights(logVIX, 2, 2, params22c, constraints=constraints)

# Such StMAR(3,2) that the AR coefficients are restricted to be
# the same for both regimes and that the second AR coefficients are
# constrained to zero.
params32trc <- c(0.35, 0.33, 0.88, -0.02, 0.01, 0.01, 0.36, 4.53, 1000)
mixingWeights(logVIX, 3, 2, params32trc, model="StMAR", restricted=TRUE,
              constraints=matrix(c(1, 0, 0, 0, 0, 1), ncol=2))
# }