sparseVARMA {bigtime} R Documentation

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

### Description

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

### Usage

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 T by k matrix of time series. If k=1, a univariate autoregressive moving average model is estimated. U A T by k 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

### Value

A list with the following components

 Y T by k 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

### 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.

data(varma.example)