Title:	Shrinkage Estimation Methods for Vector Autoregressive Models
Version:	0.3.3
Description:	Vector autoregressive (VAR) model is a fundamental and effective approach for multivariate time series analysis. Shrinkage estimation methods can be applied to high-dimensional VAR models with dimensionality greater than the number of observations, contrary to the standard ordinary least squares method. This package is an integrative package delivering nonparametric, parametric, and semiparametric methods in a unified and consistent manner, such as the multivariate ridge regression in Golub, Heath, and Wahba (1979) <doi:10.2307/1268518>, a James-Stein type nonparametric shrinkage method in Opgen-Rhein and Strimmer (2007) <doi:10.1186/1471-2105-8-S2-S3>, and Bayesian estimation methods using noninformative and informative priors in Lee, Choi, and S.-H. Kim (2016) <doi:10.1016/j.csda.2016.03.007> and Ni and Sun (2005) <doi:10.1198/073500104000000622>.
License:	GPL (≥ 3)
Depends:	R (≥ 3.5.0)
Imports:	vars (≥ 1.6.1), ars (≥ 0.6), corpcor (≥ 1.6.9), strucchange, stats, MASS, mvtnorm
Suggests:	knitr, rmarkdown, kableExtra
URL:	https://github.com/namgillee/VARshrink/
BugReports:	https://github.com/namgillee/VARshrink/issues/
Encoding:	UTF-8
RoxygenNote:	7.3.2
VignetteBuilder:	knitr
NeedsCompilation:	no
Packaged:	2026-01-10 00:25:55 UTC; namgil
Author:	Namgil Lee [aut, cre, cph], Heon Young Yang [ctb], Sung-Ho Kim [aut, cph]
Maintainer:	Namgil Lee <namgil.lee@kangwon.ac.kr>
Repository:	CRAN
Date/Publication:	2026-01-10 00:40:28 UTC

Coefficient matrices of the lagged endogenous variables

Description

Returns the estimated coefficient matrices of the lagged endogenous variables as a list of (K \times K) matrices.

Usage

Acoef_sh(x)

Arguments

Details

Consider VAR(p) model:

\mathbf{y}_t = \mathbf{A}_1 \mathbf{y}_{t-1} + ... + \mathbf{A}_p \mathbf{y}_{t-p} + \mathbf{C} \mathbf{d}_t + \mathbf{e}_t.

The function returns the (K \times K) matrices \mathbf{A}_1, ..., \mathbf{A}_p as a list object.

This function modifies vars::Acoef() for the class "varshrinkest", preventing redundant copying of data matrix objects.

Value

A list object with K-by-K VAR coefficient matrices \mathbf{A}_1, ..., \mathbf{A}_p.

See Also

Acoef

Examples

data(Canada, package = "vars")
y <- diff(Canada)
estim <- VARshrink(y, p = 2, type = "const", method = "ridge")
Acoef_sh(estim)

Coefficient matrix of an estimated VAR(p)

Description

Returns the estimated coefficients of a VAR(p) model as a matrix.

Usage

Bcoef_sh(x)

Arguments

Details

Consider VAR(p) model:

\mathbf{y}_t = \mathbf{A}_1 \mathbf{y}_{t-1} + ... + \mathbf{A}_p \mathbf{y}_{t-p} + \mathbf{C} \mathbf{d}_t + \mathbf{e}_t.

The function returns the concatenated matrix (\mathbf{A}_1, ..., \mathbf{A}_p, \mathbf{C}) as a matrix object.

This function modifies vars::Bcoef() for the class "varshrinkest", preventing redundant copying of data matrix objects.

Value

A matrix holding the estimated coefficients of a VAR.

See Also

Bcoef

Examples

data(Canada, package = "vars")
y <- diff(Canada)
estim <- VARshrink(y, p = 2, type = "const", method = "ridge")
Bcoef_sh(estim)

Coefficient matrices of the MA represention

Description

Returns the estimated coefficient matrices of the moving average representation of a stable VAR(p).

Usage

## S3 method for class 'varshrinkest'
Phi(x, nstep = 10, ...)

Arguments

Details

This is a modification of vars::Phi.varest() for the class "varshrinkest", preventing redundant copying of data matrix objects.

Value

An array with dimension K \times K \times (nstep + 1) holding the estimated coefficients of the moving average representation. The first slice of the array is the starting value, i.e., \mathbf{\Phi}_0.

See Also

Phi

Shrinkage estimation of VAR parameters

Description

Shrinkage estimation methods for high-dimensional VAR models. Consider VAR(p) model:

y_t = A_1 y_{t-1} + \cdots + A_p y_{t-p} + C d_t + e_t,

where y_t is K-dimensional time series, d_t is deterministic regressors, e_t is a noise process, and A_1, \ldots, A_p, and C are coefficient matrices. Exogenous variables can be included additionally as regressors.

Usage

VARshrink(
  y,
  p = 1,
  type = c("const", "trend", "both", "none"),
  season = NULL,
  exogen = NULL,
  method = c("ridge", "ns", "fbayes", "sbayes", "kcv"),
  lambda = NULL,
  lambda_var = NULL,
  dof = Inf,
  ...
)

Arguments

Details

Shrinkage estimation methods can estimate the coefficients even when the dimensionality K is larger than the number of observations.

Value

An object of class "varshrinkest" with the components: varresult, datamat, y, type, p, K, obs, totobs, restrictions, method, lambda, call. The class "varshrinkest" inherits the class "varest" in the package vars.

Examples

data(Canada, package = "vars")
y <- diff(Canada)
VARshrink(y, p = 2, type = "const", method = "ridge")

Sum of squared errors (SSE) between coefficients of two VARs

Description

Compute sum of squared errors of coefficients of lagged endogenous variables (Acoef) of two VAR models.

Usage

calcSSE_Acoef(Acoef1, Acoef2)

Arguments

Details

Consider VAR(p) model:

y_t = A_1 y_{t-1} + ... + A_p y_{t-p} + C d_t + e_t.

The SSE of two VAR(p) models is expressed as

sum_{k=1}^p sum_{i=1}^K sum_{j=1}^K ( (A_k)_{ij} - (A_k')_{ij} )^2.

Value

An SSE value.

Examples

data(Canada, package = "vars")
y <- diff(Canada)
estim1 <- VARshrink(y, p = 2, type = "const", method = "fbayes")
Acoef1 <- Acoef_sh(estim1)
estim2 <- VARshrink(y, p = 2, type = "const", method = "ridge")
Acoef2 <- Acoef_sh(estim2)
calcSSE_Acoef(Acoef1, Acoef2)

Causality Analysis

Description

Computes the test statistics for Granger- and Instantaneous causality for a VAR(p).

Usage

causality_sh(x, cause = NULL, vcov. = NULL, boot = FALSE, boot.runs = 100)

Arguments

Details

This function runs vars::causality() for an object of class "varshrinkest".

Value

A list of class attribute "htest" with the following elements: Granger, Instant.

See Also

causality

Examples

data(Canada, package = "vars")
y <- diff(Canada)
estim <- VARshrink(y, p = 2, type = "const", method = "ridge")
causality_sh(estim, cause = "e")

Convert format for VAR coefficients from Psi to varresult

Description

Convert a matrix of VAR coefficients estimated by a shrinkage method into a list of "shrinklm" object, where the class "shrinklm" inherits the class "lm".

Usage

convPsi2varresult(
  Psi,
  Y,
  X,
  lambda0,
  type = c("const", "trend", "both", "none"),
  ybar = NULL,
  xbar = NULL,
  Q_values = NULL,
  callstr = ""
)

Arguments

Details

Consider VAR(p) model:

y_t = A_1 y_{t-1} + ... + A_p y_{t-p} + C d_t + e_t.

It can be written in the matrix form:

Y = X \Psi + E,

where \Psi is a concatenated M-by-K matrix, \Psi = (A_1, ..., A_p, C)^T. It can be written in the multiple linear regression form of a VAR(p) model:

y_j = X \psi_j + e_j, \quad j=1,...,K,

where y_j, \psi_j, and e_j are the j-th column vectors of Y, \Psi, and E, respectively. This function converts \Psi into a list of "shrinklm" objects, where each "shrinklm" object contains the length-M vector \psi_j as coefficients.

Considering that each coefficient vector \psi_j is estimated by a shrinkage method, the effective number of parameters, k_{eff}, is computed as:

k_{eff} = Trace(X (X^T X + \lambda_0 * I)^{-1} X^T).

Then, the degree of freedom of residuals is computed as:

df.residual = N - k_{eff},

where N is the number of rows of data matrices Y and X.

Value

A list object with objects of class c("shrinklm", "lm"). Each "shrinklm" object has components: coefficients, residuals, fitted.values, rank, df.residual, lambda0, call, terms, svd

Create coefficients of a VAR model

Description

Randomly create sparse lower-triangular matrices for VAR coefficients of lagged endogenous variables, and set a constant vector.

Usage

createVARCoefs_ltriangular(
  p = 1,
  K = 5,
  diag_val = 1/p,
  num_nonzero = 0,
  const_vector = NULL,
  range_min = 0.2,
  range_max = 1/p
)

Arguments

Details

Consider VAR(p) model:

y_t = A_1 y_{t-1} + \cdots + A_p y_{t-p} + c + e_t,

with the constant deterministic variable (d_t = 1). The function creates the coefficient matrices A_1, \ldots, A_p and constant vector c.

Diagonal elements of each K-by-K matrix A_i are all equal to diag_val, and off-diagonal elements are all zero except for a few randomly selected nonzero elements. Nonzero off-diagonal elements are selected from lower-triangular parts of A_i and the values are drawn from a uniform distribution over [-range_max, -range_min] U [range_min, range_max].

Value

A list object with components $A and $c. $A is a list of K-by-K matrices A_1, \ldots, A_p, and $c is a constant vector of length K.

Examples

p <- 1; K <- 20;
const_vector <- c(rep(0.2, 5), rep(0.7, 15))
createVARCoefs_ltriangular(p = p, K = K, diag_val = 0.6,
num_nonzero = K, const_vector = const_vector, range_max = 1)

Impulse response function

Description

Computes the impulse response coefficients of a VAR(p) (or transformed VECM to VAR(p)) for n.ahead steps.

Usage

## S3 method for class 'varshrinkest'
irf(
  x,
  impulse = NULL,
  response = NULL,
  n.ahead = 10,
  ortho = TRUE,
  cumulative = FALSE,
  boot = TRUE,
  ci = 0.95,
  runs = 100,
  seed = NULL,
  ...
)

Arguments

Details

This function runs vars::irf() for an object of class "varshrinkest".

Value

An object of class "varirf", computed by vars::irf().

See Also

irf

K-fold Cross Validation for Selection of Shrinkage Parameters of Semiparametric Bayesian Shrinkage Estimator for Multivariate Regression

Description

Estimate regression coefficients and scale matrix for noise by using semiparametric Bayesian shrinkage estimator, whose shrinkage parameters are selected by k-fold cross validation (KCV).

Usage

lm_ShVAR_KCV(
  Y,
  X,
  dof = Inf,
  lambda = NULL,
  lambda_var = NULL,
  prior_type = c("NCJ", "CJ"),
  num_folds = 5,
  m0 = ncol(Y)
)

Arguments

Details

The shrinkage parameters, lambda and lambda_var, for the semiparametric Bayesian shrinkage estimator are selected by KCV. See help(lm_semi_Bayes_PCV) for details about semiparametric Bayesian estimator.

References

N. Lee, H. Choi, and S.-H. Kim (2016). Bayes shrinkage estimation for high-dimensional VAR models with scale mixture of normal distributions for noise. Computational Statistics & Data Analysis 101, 250-276. doi: 10.1016/j.csda.2016.03.007

Full Bayesian Shrinkage Estimation Method for Multivariate Regression

Description

Estimate regression coefficients and scale matrix for noise by using Gibbs MCMC algorithm. The function assumes 1) multivariate t-distribution for noise as a sampling distribution, and 2) noninformative priors for regression coefficients and scale matrix for noise.

Usage

lm_full_Bayes_SR(Y, X, dof = Inf, burnincycle = 1000, mcmccycle = 2000)

Arguments

Details

Consider the multivariate regression:

Y = X \Psi + e, \quad e \sim MVT(0, \nu, \Sigma).

\Psi is a M-by-K matrix of regression coefficients and \Sigma is a K-by-K scale matrix for multivariate t-distribution for noise.

Sampling distribution for noise e is multivariate t-distribution with degree of freedom dof and scale matrix \Sigma: e \sim MVT(0, \nu, \Sigma). The priors are noninformative priors: 1) the shrinkage prior for regression coefficients \Psi, and 2) the reference prior for scale matrix \Sigma.

The function implements Gibbs MCMC algorithm for estimating regression coefficients Psi and scale matrix Sigma.

Value

A list object with estimated parameters: Psi, Sigma, dof, delta (delta is the reciprocal of lambda), and lambda. Additional components are se.param (standard error of the parameters) and LINEXVARmodel (estimates under LINEX loss).

References

S. Ni and D. Sun (2005). Bayesian estimates for vector autoregressive models. Journal of Business & Economic Statistics 23(1), 105-117.

Multivariate Ridge Regression

Description

Estimate regression coefficients by using ridge regression.

Usage

lm_multiv_ridge(Y, X, lambda = 0, do_scale = FALSE)

Arguments

Details

Consider the multivariate regression:

\mathbf{Y} = \mathbf{X \Psi} + \mathbf{e}.

\mathbf{\Psi} is a M-by-K matrix of regression coefficients. The ridge regression estimate for the coefficients is

\mathbf{\Psi} = (\mathbf{X'X} + \lambda \mathbf{I})^{-1} \mathbf{X'Y}.

Value

A list object with the components: 1) Psi - A list of estimated \mathbf{\Psi} matrices, 2) lambda - A vector of \lambda values, 3) GCV - A vector of GCV values

References

G. H. Golub, M. Heath, G. Wahba (1979). Generalized cross-validation as a method for choosing a good ridge parameter. Technometrics 21(2), 215-223. doi: 10.2307/1268518

Semiparametric Bayesian Shrinkage Estimation Method for Multivariate Regression

Description

Estimate regression coefficients and scale matrix for noise by using a parameterized cross validation (PCV). The function assumes 1) multivariate t-distribution for noise as a sampling distribution, and 2) informative priors for regression coefficients and scale matrix for noise.

Usage

lm_semi_Bayes_PCV(
  Y,
  X,
  dof = Inf,
  lambda = NULL,
  lambda_var = NULL,
  prior_type = c("NCJ", "CJ"),
  num_folds = 5,
  m0 = ncol(Y)
)

Arguments

Details

Consider the multivariate regression:

\mathbf{Y} = \mathbf{X} \mathbf{\Psi} + \mathbf{e}, \quad \mathbf{e} \sim MVT(0, \nu, \mathbf{\Sigma}).

\mathbf{\Psi} is a (M \times K) matrix of regression coefficients and \mathbf{\Sigma} is a (K \times K) scale matrix for multivariate t-distribution for noise.

Sampling distribution for noise \mathbf{e} is the multivariate t-distribution with the degrees-of-freedom \nu and scale matrix \mathbf{\Sigma}: \mathbf{e} \sim MVT(0, \nu, \mathbf{\Sigma}). The priors are informative priors: 1) a shrinkage prior for regression coefficients \mathbf{Psi}, and 2) inverse Wishart prior for scale matrix \mathbf{\Sigma}, which can be either non-conjugate ("NCJ") or conjugate ("CJ") to the shrinkage prior for coefficients \mathbf{\Psi}.

The function implements parameterized cross validation (PCV) for selecting a shrinkage parameter lambda for estimating regression coefficients (0 < lambda <= 1). In addition, the function uses a Stein-type shrinkage method for selecting a shrinkage parameter lambda_var for estimating variances of time series variables.

References

N. Lee, H. Choi, and S.-H. Kim (2016). Bayes shrinkage estimation for high-dimensional VAR models with scale mixture of normal distributions for noise. Computational Statistics & Data Analysis 101, 250-276. doi: 10.1016/j.csda.2016.03.007

Log-likelihood method for class "varshrinkest"

Description

Returns the log-likelihood of a VAR model estimated by VARshrink(). It extends vars::logLik.varest() to incorporate 1) multivariate t-distribution for residuals, 2) scale matrix Sigma provided by shrinkage methods, and 3) effective number of parameters provided by shrinkage methods.

Usage

## S3 method for class 'varshrinkest'
logLik(object, ...)

Arguments

Value

log-likelihood of the fitted VAR model, with the class attribute "logLik" and the attributes df and nobs.

Examples

data(Canada, package = "vars")
y <- diff(Canada)
estim <- VARshrink(y, p = 2, type = "const", method = "ridge")
logLik(estim)

Print method for class "varshrinkest"

Description

print method for an object of class "varshrinkest"

Usage

## S3 method for class 'varshrinkest'
print(x, digits = max(3, getOption("digits") - 3), ...)

Arguments

Examples

data(Canada, package = "vars")
y <- diff(Canada)
estim <- VARshrink(y, p = 2, type = "const", method = "ridge")
print(estim)

Print method for class "varshsum"

Description

print method for an object obtained by summary.varshrinkest().

Usage

## S3 method for class 'varshsum'
print(
  x,
  digits = max(3, getOption("digits") - 3),
  signif.stars = getOption("show.signif.stars"),
  ...
)

Arguments

Details

This function extends print.varsum() for VAR models estimated by shrinkage methods. The output includes the covariance matrix Sigma and the degrees-of-freedom dof for the multivariate t-distribution for the residuals.

Restricted VAR

Description

Estimation of a VAR by imposing zero restrictions manually or by significance.

Usage

restrict_sh(x, ...)

Arguments

Details

This is a modification of vars::restrict() for the class "varshrinkest". Given an estimated VAR object of class "varest" or "varshrinkest", a restricted VAR is obtained by choosing method "ser" or "manual". Note: this function fits a restricted VAR using ordinary least squares rather than a shrinkage method.

Value

An object of class "varest", a VAR model fitted using ordinary least squares. It contains an additional element restrictions, which is a zero-one matrix indicating which variables were fixed to zero.

See Also

restrict

Examples

data(Canada, package = "vars")
y <- diff(Canada)
estim <- VARshrink(y, p = 2, type = "const", method = "ridge")
restrict_sh(estim)

Eigenvalues of the companion coefficient matrix of a VAR(p)

Description

This is a variant of vars::roots() for an object of class "varshrinkest", VAR parameters estimated by VARshrink().

Usage

roots_sh(x, modulus = TRUE)

Arguments

Value

A vector with the eigenvalues of the companion matrix, or their modulus (default).

See Also

roots

Examples

data(Canada, package = "vars")
y <- diff(Canada)
estim <- VARshrink(y, p = 2, type = "const", method = "ridge")
roots_sh(estim)

Test for serially correlated errors

Description

Computes the multivariate Portmanteau- and Breusch-Godfrey test for serially correlated erros.

Usage

serial.test_sh(
  x,
  lags.pt = 16,
  lags.bg = 5,
  type = c("PT.asymptotic", "PT.adjusted", "BG", "ES")
)

Arguments

Details

An extension of vars::serial.test() to the class "varshrinkest".

Value

An object of class "varcheck" computed by vars::serial.test().

See Also

serial.test

Examples

data(Canada, package = "vars")
y <- diff(Canada)
estim <- VARshrink(y, p = 2, type = "const", method = "ridge")
serial.test_sh(estim)

Semiparametric Bayesian Shrinkage Estimator for Multivariate Regression

Description

Compute the semiparametric Bayesian shrinkage estimator of Psi and Sigma for a given shrinkage parameter lambda. The function is a private function for lm_semi_Bayes_PCV() and lm_ShVAR_KCV().

Usage

shrinkVARcoef(
  Y,
  X,
  lambda,
  dof = Inf,
  prior_type = "NCJ",
  TolDRes = 1e-04,
  m0 = ncol(Y)
)

Arguments

References

N. Lee, H. Choi, and S.-H. Kim (2016). Bayes shrinkage estimation for high-dimensional VAR models with scale mixture of normal distributions for noise. Computational Statistics & Data Analysis 101, 250-276. doi: 10.1016/j.csda.2016.03.007

Generate multivariate time series data using the given VAR model

Description

Generate a multivariate time series data set using the given VAR model.

Usage

simVARmodel(numT, model, burnin = 0)

Arguments

Details

First, it creates (p+burnin+numT x K) data, then it remove the first (p+burnin) vectors. Finally, it returns (numT x K) data.

Value

A numT-by-K matrix

Examples

myCoef <- list(A = list(matrix(c(0.5, 0, 0, 0.5), 2, 2)), c = c(0.2, 0.7))
myModel <- list(Coef = myCoef, Sigma = diag(0.1^2, 2), dof = Inf)
simVARmodel(numT = 100, model = myModel, burnin = 10)

Structural stability of a VAR(p)

Description

Computes empirical fluctuation processes for VAR estimates. Utilizes strucchange::efp() for the VAR estimates of each time series variable.

Usage

stability_sh(
  x,
  type = c("OLS-CUSUM", "Rec-CUSUM", "Rec-MOSUM", "OLS-MOSUM", "RE", "ME", "Score-CUSUM",
    "Score-MOSUM", "fluctuation"),
  h = 0.15,
  dynamic = FALSE,
  rescale = TRUE,
  ...
)

Arguments

Details

A variant of vars::stability() for an object of class "varshrinkest".

Value

A list with class attribute "varstabil" which contains the following elements: stability, endog, K. The stability is a list of strucchange::efp() outputs.

See Also

stability

Examples

data(Canada, package = "vars")
y <- diff(Canada)
estim <- VARshrink(y, p = 2, type = "const", method = "ridge")
stabil <- stability_sh(estim)
plot(stabil)

Summarizing shrinkage estimates of an AR model

Description

summary method for an object of class "shrinklm".

Usage

## S3 method for class 'shrinklm'
summary(object, correlation = FALSE, symbolic.cor = FALSE, ...)

Arguments

Details

The class "shrinklm" inherits from the class "lm", and extends it to incorporate shrinkage estimates with the effective number of parameters.

Value

Returns a list with class attribute "summary.lm", which contains summary statistics of the fitted linear model given in object.

Summary method for an object of class "varshrinkest", VAR parameters estimated by VARshrink()

Description

Extend summary.varest() to the class "varshrinkest" to incorporate adapted methods for the new classes: summary.shrinklm(), logLik.varshrinkest(), roots_sh().

Usage

## S3 method for class 'varshrinkest'
summary(object, equations = NULL, ...)

Arguments

Details

The code has been modified to eliminate direct calls to the data matrices ($y, $datamat) and to use the effective numbers of parameters obtained from the shrinkage estimates. The output additionally includes the covariance matrix Sigma and the degrees-of-freedom dof of the multivariate t-distribution for the residuals.

Value

Returns a list with class attribute "varshsum" and "varsum" which contains the following elements: names, logLik, obs, roots, type, call, varresult, covres, corres, Sigma, dof.

Examples

data(Canada, package = "vars")
y <- diff(Canada)
estim <- VARshrink(y, p = 2, type = "const", method = "ridge")
summary(estim)