sparseVARMA function

Sparse Estimation of the Vector AutoRegressive Moving Average (VARMA) Model

Sparse Estimation of the Vector AutoRegressive Moving Average (VARMA) Model

sparseVARMA(
  Y,
  U = NULL,
  VARp = NULL,
  VARpen = "HLag",
  VARlseq = NULL,
  VARgran = NULL,
  VARselection = c("cv", "bic", "aic", "hq"),
  VARMAp = NULL,
  VARMAq = NULL,
  VARMApen = "HLag",
  VARMAlPhiseq = NULL,
  VARMAPhigran = NULL,
  VARMAlThetaseq = NULL,
  VARMAThetagran = NULL,
  VARMAalpha = 0,
  VARMAselection = c("none", "cv", "bic", "aic", "hq"),
  h = 1,
  cvcut = 0.9,
  eps = 10^-3,
  check_std = TRUE
)

Arguments

  • Y: A TT by kk matrix of time series. If k=1, a univariate autoregressive moving average model is estimated.
  • U: A TT by kk matrix of (approximated) error terms. Typical usage is to have the program estimate a high-order VAR model (Phase I) to get approximated error terms U.
  • VARp: User-specified maximum autoregressive lag order of the PhaseI VAR. Typical usage is to have the program compute its own maximum lag order based on the time series length.
  • VARpen: "HLag" (hierarchical sparse penalty) or "L1" (standard lasso penalty) penalization in PhaseI VAR.
  • VARlseq: User-specified grid of values for regularization parameter in the PhaseI VAR. Typical usage is to have the program compute its own grid. Supplying a grid of values overrides this. WARNING: use with care.
  • VARgran: User-specified vector of granularity specifications for the penalty parameter grid of the PhaseI VAR: First element specifies how deep the grid should be constructed. Second element specifies how many values the grid should contain.
  • VARselection: Selection procedure for the first stage. Default is time series Cross-Validation. Alternatives are BIC, AIC, HQ
  • VARMAp: User-specified maximum autoregressive lag order of the VARMA. Typical usage is to have the program compute its own maximum lag order based on the time series length.
  • VARMAq: User-specified maximum moving average lag order of the VARMA. Typical usage is to have the program compute its own maximum lag order based on the time series length.
  • VARMApen: "HLag" (hierarchical sparse penalty) or "L1" (standard lasso penalty) penalization in the VARMA.
  • VARMAlPhiseq: User-specified grid of values for regularization parameter corresponding to the autoregressive coefficients in the VARMA. Typical usage is to have the program compute its own grid. Supplying a grid of values overrides this. WARNING: use with care.
  • VARMAPhigran: User-specified vector of granularity specifications for the penalty parameter grid corresponding to the autoregressive coefficients in the VARMA: First element specifies how deep the grid should be constructed. Second element specifies how many values the grid should contain.
  • VARMAlThetaseq: User-specified grid of values for regularization parameter corresponding to the moving average coefficients in the VARMA. Typical usage is to have the program compute its own grid. Supplying a grid of values overrides this. WARNING: use with care.
  • VARMAThetagran: User-specified vector of granularity specifications for the penalty parameter grid corresponding to the moving average coefficients in the VARMA: First element specifies how deep the grid should be constructed. Second element specifies how many values the grid should contain.
  • VARMAalpha: a small positive regularization parameter value corresponding to squared Frobenius penalty in VARMA. The default is zero.
  • VARMAselection: selection procedure in the second stage. Default is "none"; Alternatives are cv, bic, aic, hq
  • h: Desired forecast horizon in time-series cross-validation procedure.
  • cvcut: Proportion of observations used for model estimation in the time series cross-validation procedure. The remainder is used for forecast evaluation.
  • eps: a small positive numeric value giving the tolerance for convergence in the proximal gradient algorithms.
  • check_std: Check whether data is standardised. Default is TRUE and is not recommended to be changed

Returns

A list with the following components - Y: TT by kk matrix of time series.

  • U: Matrix of (approximated) error terms.

  • k: Number of time series.

  • VARp: Maximum autoregressive lag order of the PhaseI VAR.

  • VARPhihat: Matrix of estimated autoregressive coefficients of the Phase I VAR.

  • VARphi0hat: Vector of Phase I VAR intercepts.

  • VARMAp: Maximum autoregressive lag order of the VARMA.

  • VARMAq: Maximum moving average lag order of the VARMA.

  • Phihat: Matrix of estimated autoregressive coefficients of the VARMA.

  • Thetahat: Matrix of estimated moving average coefficients of the VARMA.

  • phi0hat: Vector of VARMA intercepts.

  • series_names: names of time series

  • PhaseI_lambas: Phase I sparsity parameter grid

  • PhaseI_MSFEcv: MSFE cross-validation scores for each value of the sparsity parameter in the considered grid

  • PhaseI_lambda_opt: Phase I Optimal value of the sparsity parameter as selected by the time-series cross-validation procedure

  • PhaseI_lambda_SEopt: Phase I Optimal value of the sparsity parameter as selected by the time-series cross-validation procedure and after applying the one-standard-error rule

  • PhaseII_lambdaPhi: Phase II sparsity parameter grid corresponding to Phi parameters

  • PhaseII_lambdaTheta: Phase II sparsity parameter grid corresponding to Theta parameters

  • PhaseII_lambdaPhi_opt: Phase II Optimal value of the sparsity parameter (corresponding to Phi parameters) as selected by the time-series cross-validation procedure

  • PhaseII_lambdaPhi_SEopt: Phase II Optimal value of the sparsity parameter (corresponding to Theta parameters) as selected by the time-series cross-validation procedure and after applying the one-standard-error rule

  • PhaseII_lambdaTheta_opt: Phase II Optimal value of the sparsity parameter (corresponding to Phi parameters) as selected by the time-series cross-validation procedure

  • PhaseII_lambdaTheta_SEopt: Phase II Optimal value of the sparsity parameter (corresponding to Theta parameters) as selected by the time-series cross-validation procedure and after applying the one-standard-error rule

  • PhaseII_MSFEcv: Phase II MSFE cross-validation scores for each value in the two-dimensional sparsity grid

  • h: Forecast horizon h

Examples

data(varma.example)
VARMAfit <- sparseVARMA(Y = scale(Y.varma)) # sparse VARMA
y <- matrix(Y.varma[,1], ncol=1)
ARMAfit <- sparseVARMA(Y=scale(y)) # sparse ARMA

References

Wilms Ines, Sumanta Basu, Bien Jacob and Matteson David S. (2021), “Sparse Identification and Estimation of Large-Scale Vector AutoRegressive Moving Averages”, Journal of the American Statistical Association, doi: 10.1080/01621459.2021.1942013.

See Also

lagmatrix and directforecast

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