uGMAR (version 3.2.4)

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 or class 'ts' object containing the data. NA values are not supported.

p

a positive integer specifying the autoregressive order of the model.

M

For GMAR and StMAR models:

a positive integer specifying the number of mixture components.

For G-StMAR models:

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.

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 (\(\theta, \nu\))\(=\)(\(\upsilon_{1}\),...,\(\upsilon_{M}\), \(\alpha_{1},...,\alpha_{M-1}, \nu_{1},...,\nu_{M}\)).

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 \(\phi_{m}\) 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 \(\psi_{m}\)\(=(\psi_{m,1},...,\psi_{m,q_{m}})\).

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 (\(\theta, \nu\))\(=(\phi_{1,0},...,\phi_{M,0},\)\(\phi\)\(, \sigma_{1}^2,...,\sigma_{M}^2,\alpha_{1},...,\alpha_{M-1}, \nu_{1},...,\nu_{M})\).

For G-StMAR model:

Size \((3M+M2+p-1x1)\) vector (\(\theta, \nu\))\(=(\phi_{1,0},...,\phi_{M,0},\)\(\phi\)\(, \sigma_{1}^2,...,\sigma_{M}^2,\alpha_{1},...,\alpha_{M-1}, \nu_{M1+1},...,\nu_{M})\).

With linear constraints:

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 \(2\).

model

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.

restricted

a logical argument stating whether the AR coefficients \(\phi_{m,1},...,\phi_{m,p}\) 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 \((pxq_{m})\) 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 \(\psi_{m}\)\(=(\psi_{m,1},...,\psi_{m,q_{m}})\).

For restricted models:

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 autoregressive order is always p for all regimes. Ignore or set to NULL if applying linear constraints is not desired.

parametrization

is the model parametrized with the "intercepts" \(\phi_{m,0}\) or "means" \(\mu_m = \phi_{m,0}/(1-\sum\phi_{i,m})\)?

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 \(\alpha_{m,T+1}\)

.

Details

The first p observations are taken to be the initial values.

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. (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, 247-266.

• Meitz M., Preve D., Saikkonen P. 2018. A mixture autoregressive model based on Student's t-distribution. arXiv:1805.04010 [econ.EM].

• Virolainen S. 2020. A mixture autoregressive model based on Gaussian and Student's t-distribution. arXiv:2003.05221 [econ.EM].

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))
# }