Fit Vector Autoregressive (VAR) Model Parameters using Lasso Regularization

This function estimates the parameters of a VAR model using the Lasso regularization method with cyclical coordinate descent. The Lasso method is used to estimate the autoregressive and cross-regression coefficients with sparsity.

Usage

FitVARLasso(YStd, XStd, lambda, max_iter, tol)

Arguments

YStd

Numeric matrix. Matrix of standardized dependent variables (Y).

XStd

Numeric matrix. Matrix of standardized predictors (X). XStd should not include a vector of ones in column one.

lambda

Numeric. Lasso hyperparameter. The regularization strength controlling the sparsity.

max_iter

Integer. The maximum number of iterations for the coordinate descent algorithm (e.g., max_iter = 10000).

tol

Numeric. Convergence tolerance. The algorithm stops when the change in coefficients between iterations is below this tolerance (e.g., tol = 1e-5).

Value

Matrix of estimated autoregressive and cross-regression coefficients.

Details

The FitVARLasso() function estimates the parameters of a Vector Autoregressive (VAR) model using the Lasso regularization method. Given the input matrices YStd and XStd, where YStd is the matrix of standardized dependent variables, and XStd is the matrix of standardized predictors, the function computes the autoregressive and cross-regression coefficients of the VAR model with sparsity induced by the Lasso regularization.

The steps involved in estimating the VAR model parameters using Lasso are as follows:

• Initialization: The function initializes the coefficient matrix beta with OLS estimates. The beta matrix will store the estimated autoregressive and cross-regression coefficients.

• Coordinate Descent Loop: The function performs the cyclical coordinate descent algorithm to estimate the coefficients iteratively. The loop iterates max_iter times, or until convergence is achieved. The outer loop iterates over the predictor variables (columns of XStd), while the inner loop iterates over the outcome variables (columns of YStd).

• Coefficient Update: For each predictor variable (column of XStd), the function iteratively updates the corresponding column of beta using the coordinate descent algorithm with L1 norm regularization (Lasso). The update involves calculating the soft-thresholded value c, which encourages sparsity in the coefficients. The algorithm continues until the change in coefficients between iterations is below the specified tolerance tol or when the maximum number of iterations is reached.

• Convergence Check: The function checks for convergence by comparing the current beta matrix with the previous iteration's beta_old. If the maximum absolute difference between beta and beta_old is below the tolerance tol, the algorithm is considered converged, and the loop exits.

See also

Other Fitting Autoregressive Model Functions: FitMLVARDynr(), FitMLVARMplus(), FitVARDynr(), FitVARLassoSearch(), FitVARMplus(), FitVAROLS(), LambdaSeq(), ModelVARP1Dynr(), ModelVARP2Dynr(), OrigScale(), PBootVARExoLasso(), PBootVARExoOLS(), PBootVARLasso(), PBootVAROLS(), RBootVARExoLasso(), RBootVARExoOLS(), RBootVARLasso(), RBootVAROLS(), SearchVARLasso(), StdMat()

Author

Ivan Jacob Agaloos Pesigan

Examples

YStd <- StdMat(dat_p2_yx$Y)
XStd <- StdMat(dat_p2_yx$X[, -1]) # remove the constant column
lambda <- 73.90722
FitVARLasso(
  YStd = YStd,
  XStd = XStd,
  lambda = lambda,
  max_iter = 10000,
  tol = 1e-5
)
#>           [,1]      [,2]      [,3]       [,4]      [,5]      [,6]
#> [1,] 0.3440823 0.0000000 0.0000000 0.08601767 0.0000000 0.0000000
#> [2,] 0.0000000 0.4580172 0.0000000 0.00000000 0.2130337 0.0000000
#> [3,] 0.0000000 0.0000000 0.5889874 0.00000000 0.0000000 0.2746474