quantile_residuals {uGMAR}

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

Description

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

Usage

quantile_residuals(
  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:

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).

For restricted models:

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.

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 imposed to each regime's autoregressive parameters separately.

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}.

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.

parametrization

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

Details

Numerical integration is employed if the quantile residuals cannot be obtained analytically with the hypergeometric function using the package 'gsl'.

Value

Returns a (Tx1) numeric vector containing the quantile residuals of the specified GMAR, StMAR or G-StMAR model. Note that there are no quantile residuals for the first p observations as they are 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(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.

Examples

# GMAR model
params12 <- c(1.70, 0.85, 0.30, 4.12, 0.73, 1.98, 0.63)
quantile_residuals(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)
quantile_residuals(M10Y1Y, p=4, M=c(1, 1), params=params42gs, model="G-StMAR")