Title:	Shrinkage Estimation Methods for Vector Autoregressive Models
Version:	0.3.1
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 (≥ 1.5.3), ars (≥ 0.6), corpcor (≥ 1.6.9), strucchange, stats, MASS, mvtnorm
Suggests:	knitr, rmarkdown, rticles, kableExtra
URL:	https://github.com/namgillee/VARshrink/
BugReports:	https://github.com/namgillee/VARshrink/issues/
Encoding:	UTF-8
LazyData:	true
RoxygenNote:	6.1.1
VignetteBuilder:	knitr
NeedsCompilation:	no
Packaged:	2019-10-07 13:31:17 UTC; namgi
Author:	Namgil Lee [aut, cre], Heon Young Yang [ctb], Sung-Ho Kim [aut]
Maintainer:	Namgil Lee <namgil.lee@kangwon.ac.kr>
Repository:	CRAN
Date/Publication:	2019-10-09 15:10:03 UTC

Coefficient matrices of endogenous variables

Description

Returns the estimated coefficient matrices of the lagged endogenous variables of a VAR(p) model. This is a modification of vars::Acoef() for the class "varshrinkest".

Usage

Acoef_sh(x)

Arguments

Details

Consider VAR(p) model:

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

The function returns the K-by-K matrices A_1, ..., A_p as a list object.

Value

A list object with K-by-K VAR coefficient matrices A_1, A_2, ..., 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)

BQ function for class "varshrinkest"

Description

This is a modification of vars::BQ() for the class "varshrinkest".

Usage

BQ_sh(x)

Arguments

See Also

BQ

Examples

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

Coefficient matrix

Description

Returns the estimated coefficients of a VAR(p) model as a matrix. This is a modification of vars::Bcoef() for the class "varshrinkest".

Usage

Bcoef_sh(x)

Arguments

Details

Consider VAR(p) model:

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

The function returns the concatenated matrix (A_1, ..., A_p, C) as a matrix object.

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), of an SVAR as an array or a converted VECM to VAR. This is a modification of vars::Phi() for the class "varshrinkest".

Usage

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

Arguments

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 + ... + 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, ..., 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")

ARCH-LM test

Description

Performs univariate and multivariate ARCH-LM tests for a VAR. This is a modification of vars::arch.test() for the class "varshrinkest".

Usage

arch.test_sh(x, lags.single = 16, lags.multi = 5,
  multivariate.only = TRUE)

Arguments

See Also

arch.test

Examples

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

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

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 for class "varshrinkest"

Description

A modification of vars::causality() for the class "varshrinkest".

Usage

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

Arguments

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 + lambda0 * 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} + ... + A_p y_{t-p} + c + e_t,

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

Diagonal elements of each K-by-K matrix A_k 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, ..., 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)

Forecast Error Variance Decomposition

Description

Computes the forecast error variance decomposition of a VAR(p) for n.ahead steps. This is a modification of vars::fevd() for the class "varshrinkest".

Usage

## S3 method for class 'varshrinkest'
fevd(x, n.ahead = 10, ...)

Arguments

See Also

fevd

Impulse response function

Description

Computes the impulse response coefficients of a VAR(p) (or transformed VECM to VAR(p)) for n.ahead steps. This is a modification of vars::irf() for the class "varshrinkest".

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

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 ~ mvt(0, dof, 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 ~ mvt(0, dof, 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:

Y = X Psi + e.

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

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

Value

A list object with the components: 1) Psi - A list of estimated 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:

Y = X Psi + e, \quad e ~ mvt(0, dof, 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 the multivariate t-distribution with degree of freedom dof and scale matrix Sigma: e ~ mvt(0, dof, Sigma). The priors are informative priors: 1) a shrinkage prior for regression coefficients Psi, and 2) inverse Wishart prior for scale matrix Sigma, which can be either non-conjugate ("NCJ") or conjugate ("CJ") to the shrinkage prior for coefficients 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

Details

Acknowledgement: This code was contributed by Sung-Hoon Han & Dong-Han Lee @ Kangwon National University (2018.11.29.)

Examples

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

Normality, multivariate skewness and kurtosis test

Description

This function computes univariate and multivariate Jarque-Bera tests and multivariate skewness and kurtosis tests for the residuals of a VAR(p) or of a VECM in levels. This is a modification of vars::normality.test() for the class "varshrinkest".

Usage

normality.test_sh(x, multivariate.only = TRUE)

Arguments

See Also

normality.test

Examples

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

Predict method for objects of class varshrinkest

Description

Forecating a VAR object of class 'varshrinkest' with confidence bands. This is a modification of vars::predict.varest() for the class "varshrinkest".

Usage

## S3 method for class 'varshrinkest'
predict(object, ..., n.ahead = 10, ci = 0.95,
  dumvar = NULL)

Arguments

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 scale matrix Sigma and degree of freedom dof for multivariate t-distribution for residuals.

Restricted VAR

Description

This is a modification of vars::restrict() for the class "varshrinkest". Warning: THIS CODE IS NOT COMPLETE: this function may raise an error because it ignores shrinkage estimation.

Usage

restrict_sh(x, ...)

Arguments

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)-process

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

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 for VAR shrinkage estimate

Description

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

Usage

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

Arguments

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)

Stability function

Description

A variant of vars::stability(). Warning: this function has not been tested for small sample sizes yet.

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

See Also

stability

Examples

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

Summary method for class "shrinklm"

Description

Class "shrinklm" inherits the class "lm", and it extends the "lm" class to incorporate shrinkage estimates with effective number of parameter.

Usage

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

Arguments

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

Description

Extend summary.varest() to class 'varshrinest' to incorporate adapted methods for new classes: summary.shrinklm(), logLik.varshrinkest(), roots.varshrinkest().

Usage

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

Arguments

Details

Code is modified to avoid call to data matrices ($y, $datamat) and to use effective numbers of parameters of shrinkage estimates.

Output includes the scale matrix, Sigma, and degree-of-freedom, dof, for multivariate t-distribution for residuals.

Examples

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