uGMAR (version 3.0.1)

quantileResiduals: Compute quantile residuals of GMAR, StMAR or G-StMAR model

Description

quantileResiduals computes the quantile residuals of the specified GMAR, StMAR or G-StMAR model.

Usage

quantileResiduals(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 \(\upsilon_{m}\)\(=(\phi_{m,0},\)\(\phi_{m}\)\(, \sigma_{m}^2)\) 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 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 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" \(\phi_{m,0}\) or "means" \(\mu_m = \phi_{m,0}/(1-\sum\phi_{i,m})\)?

Value

Returns a \((Tx1)\) numeric vector containing the quantile residuals associated with the specified GMAR, StMAR or G-StMAR model.

Suggested packages

Install the suggested package "gsl" for faster evaluation of the quantile residuals of StMAR and G-StMAR models.

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.

• Lutkepohl H. 2005. New Introduction to Multiple Time Series Analysis. Springer.

• 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 G-StMAR model, but it's a straight forward generalization with theoretical properties similar to GMAR and StMAR models.

Examples

# NOT RUN {
# GMAR model
params12 <- c(1.12, 0.9, 0.29, 4.53, 0.7, 3.22, 0.84)
qr12 <- quantileResiduals(VIX, 1, 2, params12)

# Restricted GMAR model
params12r <- c(1.4, 1.8, 0.88, 0.29, 3.18, 0.84)
qr12r <- quantileResiduals(VIX, 1, 2, params12r, restricted=TRUE)

# StMAR model
params12t <- c(1.1, 0.9, 0.3, 4.5, 0.7, 3.2, 0.8, 5, 8)
qr12t <- quantileResiduals(VIX, 1, 2, params12t, model="StMAR")

# Non-mixture version of StMAR model
params11t <- c(0.76, 0.93, 1.35, 2.4)
qr11t <- quantileResiduals(VIX, 1, 1, params11t, model="StMAR")

# G-StMAR model
params12gs <- c(1.5, 0.8, 1.5, 2.9, 0.8, 1.1, 0.6, 3)
qr12gs <- quantileResiduals(VIX, 1, c(1, 1), params12gs, model="G-StMAR")

# Restricted G-StMAR model
params13gsr <- c(1.3, 1, 1.4, 0.8, 0.4, 2, 0.2, 0.25, 0.15, 20)
qr12gsr <- quantileResiduals(VIX, 1, c(2, 1), params13gsr, 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(1.2, 0.85, 0.04, 0.3, 3.3, 0.77, 2.8, 0.77)
qr22c <- quantileResiduals(VIX, 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(2.2, 1.8, 0.88, -0.03, 2.4, 0.27, 0.40, 3.9, 1000)
qr32trc <- quantileResiduals(VIX, 3, 2, params32trc, model="StMAR",
 restricted=TRUE, constraints=matrix(c(1, 0, 0, 0, 0, 1), ncol=2))
# }