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
)A list with the following components
\(T\) by \(k\) matrix of time series.
Matrix of (approximated) error terms.
Number of time series.
Maximum autoregressive lag order of the PhaseI VAR.
Matrix of estimated autoregressive coefficients of the Phase I VAR.
Vector of Phase I VAR intercepts.
Maximum autoregressive lag order of the VARMA.
Maximum moving average lag order of the VARMA.
Matrix of estimated autoregressive coefficients of the VARMA.
Matrix of estimated moving average coefficients of the VARMA.
Vector of VARMA intercepts.
names of time series
Phase I sparsity parameter grid
MSFE cross-validation scores for each value of the sparsity parameter in the considered grid
Phase I Optimal value of the sparsity parameter as selected by the time-series cross-validation procedure
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
Phase II sparsity parameter grid corresponding to Phi parameters
Phase II sparsity parameter grid corresponding to Theta parameters
Phase II Optimal value of the sparsity parameter (corresponding to Phi parameters) as selected by the time-series cross-validation procedure
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
Phase II Optimal value of the sparsity parameter (corresponding to Phi parameters) as selected by the time-series cross-validation procedure
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
Phase II MSFE cross-validation scores for each value in the two-dimensional sparsity grid
Forecast horizon h
A \(T\) by \(k\) matrix of time series. If k=1, a univariate autoregressive moving average model is estimated.
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.
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.
"HLag" (hierarchical sparse penalty) or "L1" (standard lasso penalty) penalization in PhaseI VAR.
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.
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.
Selection procedure for the first stage. Default is time series Cross-Validation. Alternatives are BIC, AIC, HQ
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.
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.
"HLag" (hierarchical sparse penalty) or "L1" (standard lasso penalty) penalization in the VARMA.
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.
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.
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.
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.
a small positive regularization parameter value corresponding to squared Frobenius penalty in VARMA. The default is zero.
selection procedure in the second stage. Default is "none"; Alternatives are cv, bic, aic, hq
Desired forecast horizon in time-series cross-validation procedure.
Proportion of observations used for model estimation in the time series cross-validation procedure. The remainder is used for forecast evaluation.
a small positive numeric value giving the tolerance for convergence in the proximal gradient algorithms.
Check whether data is standardised. Default is TRUE and is not recommended to be changed
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.
lagmatrix and directforecast
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
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