get_test_Omega function

Generate the covariance matrix Omega for quantile residual tests

get_test_Omega generates the covariance matrix Omega used in the quantile residual tests.

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

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 $=$Cpsi (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 $=$Cpsi , 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" $\mu_{m} = \phi_{m,0}/(1-\sum\phi_{i,m})$?

• g: a function specifying the transformation.

• dim_g: output dimension of the transformation g.

Returns

Returns size (dim_gxdim_g) covariance matrix Omega.

Details

This function is used for quantile residuals tests in quantile_residual_tests.

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.

See Also

quantile_residual_tests