Package 'BigVAR' reference manual

Title:	Dimension Reduction Methods for Multivariate Time Series
Description:	Estimates VAR and VARX models with Structured Penalties using the methods developed by Nicholson et al (2017)<doi:10.1016/j.ijforecast.2017.01.003> and Nicholson et al (2020) <doi:10.48550/arXiv.1412.5250>.
Authors:	Will Nicholson [cre, aut], David Matteson [aut], Jacob Bien [aut], Ines Wilms [aut]
Maintainer:	Will Nicholson <[email protected]>
License:	GPL (>= 2)
Version:	1.1.2
Built:	2024-10-31 06:55:12 UTC
Source:	CRAN

Generator for Simulated Multivariate Time Series

Description

Coefficient matrix for a stationary simulated multivariate time series

Details

Example generator matrix adapted from Table 3.2 of Gredenhoff and Karlsson (1997)

Author(s)

Will Nicholson

References

Gredenhoff, Mikael, and Sune Karlsson. "Lag-length selection in VAR-models using equal and unequal lag-length procedures." Computational Statistics 14.2 (1999): 171-187.

Dimension Reduction Methods for Multivariate Time Series.

Description

BigVAR implements the HLAG and VARX-L frameworks which allow for the estimation of vector autoregressions and vector autoregressions with exogenous variables using structured convex penalties. This package originated as a 2014 Google "Summer of Code" Project. The development version of this package is hosted on github: https://github.com/wbnicholson/BigVAR.

Details

To use the facilities of this package, starting with an $T \times k+m$ multivariate time series (in which T denotes the length of the series, k the number of endogenous or "model") and run constructModel to create an object of class BigVAR. cv.BigVAR creates an object of class BigVAR.results, which chooses an optimal penalty parameter based on minimizing h-step ahead forecasts on a specified cross-validation period over a grid of values as well as comparisons against AIC, BIC, unconditional mean, and a random walk. There are plot functions for both BigVAR (plot.BigVAR) and BigVAR.results (plot) as well as a predict function for BigVAR.results (predict).

Author(s)

Will Nicholson [email protected],

References

Lutkepohl "New Introduction to Multivariate Time Series", Banbura, Marta, Domenico Giannone, and Lucrezia Reichlin. 'Large Bayesian vector auto regressions.' Journal of Applied Econometrics 25.1 (2010): 71-92. Breheny P, Huang J (2011). “Coordinate descent algorithms for nonconvex penalized regression, with applications to biological feature selection.” Annals of Applied Statistics, 5(1), 232–253. Nicholson, William, I. Wilms, J. Bien, and D. S. Matteson. High dimensional forecasting via interpretable vector autoregression. Journal of Machine Learning Research, 21(166):1–52, 2020. William B. Nicholson, David S. Matteson, Jacob Bien,VARX-L: Structured regularization for large vector autoregressions with exogenous variables, International Journal of Forecasting, Volume 33, Issue 3, 2017, Pages 627-651, William B Nicholson, David S. Matteson, and Jacob Bien (2016), 'BigVAR: Tools for Modeling Sparse High-Dimensional Multivariate Time Series' arxiv:1702.07094

Examples

# Fit a Basic VAR-L(3,4) on simulated data
data(Y)
T1=floor(nrow(Y)/3)
T2=floor(2*nrow(Y)/3)
m1=constructModel(Y,p=4,struct="Basic",gran=c(50,10),verbose=FALSE,T1=T1,T2=T2,IC=FALSE)
plot(m1)
results=cv.BigVAR(m1)
plot(results)
predict(results,n.ahead=1)
# Fit a Basic VAR-L(3,4) on simulated data
data(Y)
T1=floor(nrow(Y)/3)
T2=floor(2*nrow(Y)/3)
m1=constructModel(Y,p=4,struct="Basic",gran=c(50,10),verbose=FALSE,T1=T1,T2=T2,IC=FALSE)
plot(m1)
results=cv.BigVAR(m1)
plot(results)
predict(results,n.ahead=1)

BigVAR Object Class

Description

An object class to be used with cv.BigVAR

Details

To construct an object of class BigVAR, use the function constructModel

Slots

Data: a $T \times k$ multivariate time series
model_data: processed time series and lag matrix
lagmax: Maximal lag order for modeled series
intercept: Indicator as to whether an intercept should be included
Structure: Penalty Structure
Relaxed: Indicator for relaxed VAR
Granularity: Granularity of penalty grid
horizon: Desired Forecast Horizon
crossval: Cross-Validation Procedure
Minnesota: Minnesota Prior Indicator
verbose: Indicator for Verbose output
dates: dates extracted from an xts object
ic: Indicator for including AIC and BIC benchmarks
VARX: VARX Model Specifications
VARXI: VARX Indicator
T1: Index of time series in which to start cross validation
T2: Index of times series in which to start forecast evaluation
ONESE: Indicator for 'One Standard Error Heuristic'
ownlambdas: Indicator for user-supplied lambdas
tf: Indicator for transfer function
alpha: Grid of candidate alpha values (applies only to Sparse VARX-L and Elastic Net models)
recursive: Indicator as to whether recursive multi-step forecasts are used (applies only to multiple horizon VAR models)
constvec: vector indicating variables to shrink toward a random walk instead of toward zero (valid only if Minnesota is TRUE)
tol: optimization tolerance
window.size: size of rolling window. If set to NULL an expanding window will be used.
separate_lambdas: indicator to use separate penalty parameter for each time series (default FALSE)
loss: Loss function to select penalty parameter (one of 'L1','L2','Huber').
delta: delta parameter for Huber loss (default 2.5)
gamma: gamma parameter for SCAD or MCP penalty (default 3)
rolling_oos: True or False: indicator to update the penalty parameter over the evaluation period (default False)
linear: indicator for linearly decrementing penalty grid (FALSE is log-linear).
refit_fraction: fraction of least squares refit to incorporate (default is 1).

BigVAR Estimation

Description

Fit a BigVAR object with a structured penalty (VARX-L or HLAG).

Usage

BigVAR.est(object)
BigVAR.est(object)

Arguments

object

BigVAR object created from ConstructModel

Details

Fits HLAG or VARX-L model on a BigVAR object. Does not perform cross-validation. This method allows the user to construct their own penalty parameter selection procedure.

Value

An array of $k \times kp \times n$ or $k\times kp+ms \times n$ coefficient matrices; one for each of the n values of lambda.

Examples

data(Y)
Y=Y[1:100,]
#construct a Basic VAR-L
Model1=constructModel(Y,p=4,struct='Basic',gran=c(50,10))
BigVAR.est(Model1)
data(Y)
Y=Y[1:100,]
#construct a Basic VAR-L
Model1=constructModel(Y,p=4,struct='Basic',gran=c(50,10))
BigVAR.est(Model1)

Simple function to fit BigVAR model with fixed penalty parameter

Description

Simple function to fit BigVAR model with fixed penalty parameter

Usage

BigVAR.fit(
  Y,
  p,
  struct,
  lambda,
  alpha = NULL,
  VARX = list(),
  separate_lambdas = F,
  MN = F,
  C = as.double(NULL),
  intercept = TRUE,
  tf = F,
  tol = 1e-04,
  RVAR = F,
  refit_fraction = 1,
  beta = NULL,
  gamma = 3
)
BigVAR.fit(
  Y,
  p,
  struct,
  lambda,
  alpha = NULL,
  VARX = list(),
  separate_lambdas = F,
  MN = F,
  C = as.double(NULL),
  intercept = TRUE,
  tf = F,
  tol = 1e-04,
  RVAR = F,
  refit_fraction = 1,
  beta = NULL,
  gamma = 3
)

Arguments

`Y`	$T \times k$ multivariate time series or Y $T \times (k+m)$ endogenous and exogenous series, respectively
`p`	Predetermined maximal lag order (for modeled series)
`struct`	The choice of penalty structure (see details).
`lambda`	vector or matrix of penalty parameters.
`alpha`	grid of candidate parameters for the alpha in the Sparse Lag and Sparse Own/Other VARX-L
`VARX`	List containing VARX model specifications.
`separate_lambdas`	indicator for separate penalty parameters for each time series (default `FALSE`)
`MN`	Minnesota Prior Indicator
`C`	vector of coefficients to shrink toward a random walk (if `MN` is `TRUE`)
`intercept`	True or False: option to fit an intercept
`tf`	transfer function indicator (i.e. VARX in which p=0 & s>0) (default false)
`tol`	optimization tolerance (default 1e-4)
`RVAR`	True or False: option to refit based upon the support selected using the Relaxed-VAR procedure
`refit_fraction`	fraction of least squares refit to incorporate (default 1)
`beta`	optional $k\times (k\times p + m\times s +1)$ coefficient matrix to use as a 'warm start' (default `NULL`)
`gamma`	additional parameter for SCAD/MCP penalty (default 3) @details The choices for 'struct' are as follows 'Basic' (Basic VARX-L) 'BasicEN' (Basic Elastic Net VARX-L) 'Lag' (Lag Group VARX-L) 'SparseLag' (Lag Sparse Group VARX-L) 'OwnOther' (Own/Other Group VARX-L) 'SparseOO' (Own/Other Sparse Group VARX-L) 'EFX' (Endogenous First VARX-L) 'HLAGC' (Componentwise HLAG) 'HLAGOO' (Own/Other HLAG) 'HLAGELEM' (Elementwise HLAG) 'Tapered' (Lag weighted Lasso VAR) 'BGR' (Bayesian Ridge Regression (cf. Banbura et al)) 'MCP' (Minimax Concave Penalty (cf. Breheny and Huang)) 'SCAD' (Smoothly Clipped Absolute Deviation (cf. Breheny and Huang)) VARX specifications consist of a list with entry k denoting the series that are to be modeled and entry s to denote the maximal lag order for exogenous series. The argument alpha is ignored unless the structure choice is 'SparseLag' or 'Lag.' By default 'alpha' is set to `NULL` and will be initialized as 1/(k+1) in `cv.BigVAR` and `BigVAR.est`. Any user supplied values must be between 0 and 1.

Note

The specifications 'Basic', 'Lag,' 'SparseLag,' 'SparseOO,' and 'OwnOther' can accommodate both VAR and VARX models. EFX only applies to VARX models. 'HLAGC,' 'HLAGOO,' 'HLAGELEM,' and 'Tapered' can only be used with VAR models. Our implementation of the SCAD and MCP penalties is heavily influenced by the implementation in ncvreg.

References

Banbura, Marta, Domenico Giannone, and Lucrezia Reichlin. 'Large Bayesian vector auto regressions.' Journal of Applied Econometrics 25.1 (2010): 71-92. Breheny P, Huang J (2011). “Coordinate descent algorithms for nonconvex penalized regression, with applications to biological feature selection.” Annals of Applied Statistics, 5(1), 232–253. William B Nicholson, Jacob Bien, and David S Matteson. 'High Dimensional Forecasting via Interpretable Vector Autoregression.' arXiv preprint arXiv:1412.5250, 2016. William B. Nicholson, David S. Matteson, Jacob Bien,VARX-L: Structured regularization for large vector autoregressions with exogenous variables, International Journal of Forecasting, Volume 33, Issue 3, 2017, Pages 627-651, William B Nicholson, David S. Matteson, and Jacob Bien (2016), 'BigVAR: Tools for Modeling Sparse High-Dimensional Multivariate Time Series' arxiv:1702.07094

Examples

# VARX Example
# Fit a Basic VARX-L with k=2, m=1, s=2, p=4, lambda=1e-2
VARX=list()
VARX$k=2 # indicates that the first two series are modeled
VARX$s=2 # sets 2 as the maximal lag order for exogenous series
data(Y)
BigVAR.fit(Y,p=4,'Basic',lambda=1e-2,VARX=VARX)
# VARX Example
# Fit a Basic VARX-L with k=2, m=1, s=2, p=4, lambda=1e-2
VARX=list()
VARX$k=2 # indicates that the first two series are modeled
VARX$s=2 # sets 2 as the maximal lag order for exogenous series
data(Y)
BigVAR.fit(Y,p=4,'Basic',lambda=1e-2,VARX=VARX)

BigVAR.intermediate This class contains the in-sample results for cv.BigVAR

Description

It inherits the class BigVAR, and contains the results from rolling validation

Fields

ZFull: List containing full lag matrix and time series
InSampMSFE: In-sample MSFE from optimal value of lambda
LambdaGrid: Grid of candidate lambda values
index: Index order of optimal lambda value
OptimalLambda: Value of lambda that minimizes MSFE
Data: a $T \times k$ or $T\times k + m$ multivariate time Series
lagmax: Maximal lag order
Structure: Penalty structure
Relaxed: Indicator for relaxed VAR
Granularity: Granularity of penalty grid
horizon: Desired forecast horizon
crossval: Cross-Validation procedure
alpha: additional penalty parameter for Sparse Lag Group or Sparse Own/Other methods. Will contain either the heuristic choice of $1/(k+1)$ or the value selected by cross validation if the argument dual is set to TRUE
Minnesota: Minnesota Prior Indicator
verbose: verbose indicator
dual: indicator as to whether dual cross validation was conducted
contemp: indicator if contemporaneous exogenous predictors are used

Note

One can also access any object of class BigVAR from BigVAR.intermediate

Author(s)

Will Nicholson

BigVAR.results This class contains the results from cv.BigVAR.

Description

It inherits the class BigVAR, but contains substantially more information.

Fields

InSampMSFE: In-sample MSFE from optimal value of lambda
LambdaGrid: Grid of candidate lambda values
index: Rank of optimal lambda value
OptimalLambda: Value of lambda that minimizes MSFE
OOSMSFE: Average Out of sample MSFE of BigVAR model with optimal lambda
seoosfmsfe: Standard error of out of sample MSFE of BigVAR model with optimal lambda
MeanMSFE: Average out of sample MSFE of unconditional mean forecast
MeanSD: Standard error of out of sample MSFE of unconditional mean forecast
MeanPreds: predictions from conditional mean model
RWMSFE: Average out of sample MSFE of random walk forecast
RWPreds: Predictions from random walk model
RWSD: Standard error of out of sample MSFE of random walk forecast
AICMSFE: Average out of sample MSFE of AIC forecast
AICSD: Standard error of out of sample MSFE of AIC forecast
AICPreds: Predictions from AIC VAR/VARX model
AICpvec: Lag orders selected from AIC VAR model
AICpvec: Lag orders selected from AIC VARX model
BICMSFE: Average out of sample MSFE of BIC forecast
BICSD: Standard error of out of sample MSFE of BIC forecast
BICPreds: Predictions from BIC VAR/VARX model
BICpvec: Lag orders selected from BIC VAR model
BICpvec: Lag orders selected from BIC VARX model
betaPred: The final estimated $k\times kp+ms+1$ coefficient matrix, to be used for prediction
Zvals: The final lagged values of Y, to be used for prediction
fitted: fitted values obtained from betaPred
resids: residuals obtained from betaPred
Data: a $T \times k$ or $T\times k + m$ multivariate time Series
lagmax: Maximal lag order
Structure: Penalty structure
Relaxed: Indicator for relaxed VAR
Granularity: Granularity of penalty grid
horizon: Desired forecast horizon
crossval: Cross-Validation procedure
alpha: additional penalty parameter for Sparse Lag Group or Sparse Own/Other methods. Will contain either the heuristic choice of $1/(k+1)$ or the value selected by cross validation if the argument dual is set to TRUE
VARXI: VARX Indicator
Minnesota: Minnesota Prior Indicator
verbose: verbose indicator
dual: indicator as to whether dual cross validation was conducted
contemp: indicator if contemporaneous exogenous predictors are used
lagmatrix: matrix of lagged values used to compute residuals (of which Zvals is the final column)
betaArray: array of VAR/VARX coefficients from out of sample forecasts
sparse_count: average fraction of active coefficients in validation period
lambda_evolve_path: evolution of lambda over evaluation period

Note

One can also access any object of class BigVAR from BigVAR.results

Author(s)

Will Nicholson

Default coef method BigVAR-results, returns the last coefficient matrix from the evaluation period

Description

Default coef method BigVAR-results, returns the last coefficient matrix from the evaluation period

Usage

## S4 method for signature 'BigVAR.results'
coef(object)
## S4 method for signature 'BigVAR.results'
coef(object)

Arguments

object

BigVAR.results object created from cv.BigVAR

Details

displays formatted coefficient matrix

Construct an object of class BigVAR

Description

Construct an object of class BigVAR

Usage

constructModel(
  Y,
  p,
  struct,
  gran,
  h = 1,
  cv = "Rolling",
  verbose = TRUE,
  IC = TRUE,
  VARX = list(),
  T1 = floor(nrow(Y)/3),
  T2 = floor(2 * nrow(Y)/3),
  ONESE = FALSE,
  ownlambdas = FALSE,
  recursive = FALSE,
  dates = as.character(NULL),
  window.size = 0,
  separate_lambdas = FALSE,
  linear = FALSE,
  loss = "L2",
  rolling_oos = FALSE,
  model.controls = list()
)
constructModel(
  Y,
  p,
  struct,
  gran,
  h = 1,
  cv = "Rolling",
  verbose = TRUE,
  IC = TRUE,
  VARX = list(),
  T1 = floor(nrow(Y)/3),
  T2 = floor(2 * nrow(Y)/3),
  ONESE = FALSE,
  ownlambdas = FALSE,
  recursive = FALSE,
  dates = as.character(NULL),
  window.size = 0,
  separate_lambdas = FALSE,
  linear = FALSE,
  loss = "L2",
  rolling_oos = FALSE,
  model.controls = list()
)

Arguments

`Y`	$T \times k$ multivariate time series or Y $T \times (k+m)$ endogenous and exogenous series, respectively.
`p`	Predetermined maximal lag order (for modeled series).
`struct`	The choice of penalty structure (see details).
`gran`	vector of penalty parameter specifications.
`h`	Desired forecast horizon.
`cv`	Cross-validation approach, either 'Rolling' for rolling cross-validation or 'LOO' for leave-one-out cross-validation. 'None' for use with BigVAR.fit.
`verbose`	Verbose output while estimating.
`IC`	True or False: whether to include AIC and BIC benchmarks.
`VARX`	List containing VARX model specifications.
`T1`	Index of time series in which to start cross validation.
`T2`	Index of times series in which to start forecast evaluation.
`ONESE`	True or False: whether to use the 'One Standard Error Heuristic.'
`ownlambdas`	True or False: Indicator for user-supplied penalty parameters.
`recursive`	True or False: Indicator as to whether iterative multi-step predictions are desired in the VAR context if the forecast horizon is greater than 1.
`dates`	optional vector of dates corresponding to $Y$ .
`window.size`	size of rolling window. If set to 0 an expanding window will be used.
`separate_lambdas`	indicator for separate penalty parameters for each time series (default `FALSE`).
`linear`	indicator for linearly decrementing penalty grid (FALSE is log-linear; default `FALSE`).
`loss`	Loss function to select penalty parameter (one of 'L1','L2','Huber')
`rolling_oos`	True or False: indicator to update the penalty parameter over the evaluation period (default `False`)
`model.controls`	named list of control parameters for BigVAR model estimation (see details).

Details

The choices for 'struct' are as follows

'Basic' (Basic VARX-L)
'BasicEN' (Elastic Net VARX-L)
'Lag' (Lag Group VARX-L)
'SparseLag' (Lag Sparse Group VARX-L)
'OwnOther' (Own/Other Group VARX-L)
'SparseOO' (Own/Other Sparse Group VARX-L)
'EFX' (Endogenous First VARX-L)
'HLAGC' (Componentwise HLAG)
'HLAGOO' (Own/Other HLAG)
'HLAGELEM' (Elementwise HLAG)
'Tapered' (Lag weighted Lasso VAR)
'BGR' (Bayesian Ridge Regression (cf. Banbura et al))
'MCP' (Minimax Concave Penalty (cf. Breheny and Huang))
'SCAD' (Smoothly Clipped Absolute Deviation Penalty (cf. Breheny and Huang))

The first number in the vector 'gran' specifies how deep to construct the penalty grid and the second specifies how many penalty parameters to use If ownlambas is set to TRUE, gran should contain the user-supplied penalty parameters.

VARX specifications consist of a named list with entry k denoting the series that are to be modeled and entry s to denote the maximal lag order for exogenous series.

As the capabilities of BigVAR have expanded, we have decided to consolidate parameters in the list model.controls. These parameters include:

'alpha:' grid of candidate parameters for the alpha in the Basic Elastic Net, Sparse Lag, Sparse Own/Other VARX-L.
'C:' vector of coefficients to shrink toward a random walk (if MN is TRUE).
'delta:' parameter for Huber loss (default 2.5)
'intercept:' option to fit an intercept, default TRUE
'loss:' Loss function to select penalty parameter (one of 'L1','L2','Huber')
'MN:' Minnesota Prior Indicator, default FALSE
'RVAR:' option to refit based upon the support selected using the Relaxed-VAR procedure (default FALSE).
'refit_fraction:' If RVAR is TRUE, proportional tradeoff between least squares fit and penalized fit (default 1).
'tol:' optimization tolerance (default 1e-4)

The argument alpha is ignored unless the structure choice is 'SparseLag' or 'Lag.' By default 'alpha' is set to NULL and will be initialized as 1/(k+1) in cv.BigVAR and BigVAR.est. Any user supplied values must be between 0 and 1.

Note

The specifications 'Basic','BasicEN', 'Lag,' 'SparseLag,' 'SparseOO','OwnOther', 'MCP', and 'SCAD.' can accommodate both VAR and VARX models. EFX only applies to VARX models. 'HLAGC,' 'HLAGOO,' 'HLAGELEM,' and 'Tapered' can only be used with VAR models. Our implementation of the SCAD and MCP penalties is heavily influenced by the package ncvreg.

References

Banbura, Marta, Domenico Giannone, and Lucrezia Reichlin. 'Large Bayesian vector auto regressions.' Journal of Applied Econometrics 25.1 (2010): 71-92. Breheny P, Huang J (2011). “Coordinate descent algorithms for nonconvex penalized regression, with applications to biological feature selection.” Annals of Applied Statistics, 5(1), 232–253. Nicholson, William, I. Wilms, J. Bien, and D. S. Matteson. High dimensional forecasting via interpretable vector autoregression. Journal of Machine Learning Research, 21(166):1–52, 2020. William B. Nicholson, David S. Matteson, Jacob Bien,VARX-L: Structured regularization for large vector autoregressions with exogenous variables, International Journal of Forecasting, Volume 33, Issue 3, 2017, Pages 627-651, William B Nicholson, David S. Matteson, and Jacob Bien (2016), 'BigVAR: Tools for Modeling Sparse High-Dimensional Multivariate Time Series' arxiv:1702.07094

Examples

# VARX Example
# Create a Basic VARX-L with k=2, m=1, s=2, p=4
VARX=list()
VARX$k=2 # indicates that the first two series are modeled
VARX$s=2 # sets 2 as the maximal lag order for exogenous series
data(Y)
T1=floor(nrow(Y)/3)
T2=floor(2*nrow(Y)/3)
Model1=constructModel(Y,p=4,struct='Basic',gran=c(50,10),verbose=FALSE,VARX=VARX,T1=T1,T2=T2)
# VARX Example
# Create a Basic VARX-L with k=2, m=1, s=2, p=4
VARX=list()
VARX$k=2 # indicates that the first two series are modeled
VARX$s=2 # sets 2 as the maximal lag order for exogenous series
data(Y)
T1=floor(nrow(Y)/3)
T2=floor(2*nrow(Y)/3)
Model1=constructModel(Y,p=4,struct='Basic',gran=c(50,10),verbose=FALSE,VARX=VARX,T1=T1,T2=T2)

Cross Validation for BigVAR

Description

Cross Validation for BigVAR

Usage

cv.BigVAR(object)
cv.BigVAR(object)

Arguments

object

BigVAR object created from ConstructModel

Details

The main function of the BigVAR package. Performs cross validation to select penalty parameters over a training sample (as the minimizer of in-sample MSFE), then evaluates them over a test set. Compares against sample mean, random walk, AIC, and BIC benchmarks. Creates an object of class BigVAR.results

Value

An object of class BigVAR.results.

Examples

data(Y)
# Fit a Basic VARX-L with rolling cross validation 
Model1=constructModel(Y,p=4,struct='Basic',gran=c(50,10), verbose=FALSE)
results=cv.BigVAR(Model1)
data(Y)
# Fit a Basic VARX-L with rolling cross validation 
Model1=constructModel(Y,p=4,struct='Basic',gran=c(50,10), verbose=FALSE)
results=cv.BigVAR(Model1)

Simulate a VAR

Description

Simulate a VAR

Usage

MultVarSim(k, A1, p, Sigma, T)
MultVarSim(k, A1, p, Sigma, T)

Arguments

`k`	Number of Series
`A1`	Either a $k \times k$ coefficient matrix or a $kp \times kp$ matrix created using `VarptoVar1MC`.
`p`	Maximum Lag Order
`Sigma`	Residual Covariance Matrix of dimension $k\times k$
`T`	Number of simulations

Value

Returns a $T \times k$ of realizations from a VAR.

References

Lutkepohl, 'A New Introduction to Multiple Time Series Analysis'

Examples

k=3;p=6
B=matrix(0,nrow=k,ncol=p*k)
A1<- matrix(c(.4,-.02,.01,-.02,.3,.02,.01,.04,.3),ncol=3,nrow=3)
A2 <- matrix(c(.2,0,0,0,.3,0,0,0,.13),ncol=3,nrow=3)
B[,1:k]=A1
B[,(4*k+1):(5*k)]=A2
A <- VarptoVar1MC(B,p,k)
Y <-MultVarSim(k,A,p,.1*diag(k),100)
k=3;p=6
B=matrix(0,nrow=k,ncol=p*k)
A1<- matrix(c(.4,-.02,.01,-.02,.3,.02,.01,.04,.3),ncol=3,nrow=3)
A2 <- matrix(c(.2,0,0,0,.3,0,0,0,.13),ncol=3,nrow=3)
B[,1:k]=A1
B[,(4*k+1):(5*k)]=A2
A <- VarptoVar1MC(B,p,k)
Y <-MultVarSim(k,A,p,.1*diag(k),100)

Plot an object of class BigVAR.results

Description

Plot an object of class BigVAR.results

Usage

## S4 method for signature 'BigVAR.results'
plot(x, y = NULL, ...)
## S4 method for signature 'BigVAR.results'
plot(x, y = NULL, ...)

Arguments

`x`	BigVAR.results object created from `cv.BigVAR`
`y`	NULL
`...`	additional arguments

Details

Plots the in sample MSFE of all values of lambda with the optimal value highlighted.

Plot a BigVAR object

Description

Plot a BigVAR object

Usage

## S4 method for signature 'BigVAR'
plot(x, y = NULL, ...)
## S4 method for signature 'BigVAR'
plot(x, y = NULL, ...)

Arguments

`x`	BigVAR object created from `ConstructModel`
`y`	NULL
`...`	additional plot arguments

Details

Uses plot.zoo to plot each indivdual series of Y on a single plot

Value

NA, side effect is graph

Forecast using a BigVAR.results object

Description

Forecast using a BigVAR.results object

Usage

predict(object,...)
predict(object,...)

Arguments

`object`	BigVAR.results object from `cv.BigVAR`
`...`	additional arguments affecting the predictions produced (e.g. `n.ahead`, `confint`)

Details

Provides n.ahead step forecasts using the model produced by cv.BigVAR. If confint is set to TRUE, a 95 percent confidence interval will also be returned.

Examples

data(Y)
Y=Y[1:100,]
Model1=constructModel(Y,p=4,struct='Basic',gran=c(50,10),verbose=FALSE)
results=cv.BigVAR(Model1)
predict(results,n.ahead=1)
data(Y)
Y=Y[1:100,]
Model1=constructModel(Y,p=4,struct='Basic',gran=c(50,10),verbose=FALSE)
results=cv.BigVAR(Model1)
predict(results,n.ahead=1)

One-step ahead predictions for VARX models

Description

One-step ahead predictions for VARX models

Usage

PredictVARX(VARXRes)
PredictVARX(VARXRes)

Arguments

VARXRes

the results from VARXFit

Value

Returns a vector consisting of the out-of-sample forecasts for the provided VARXFit model.

Examples

data(Y)
# fit a VAR_3(3)
mod <- VARXFit(Y,3,NULL,NULL)
pred <-PredictVARX(mod)

data(Y)
# fit a VAR_3(3)
mod <- VARXFit(Y,3,NULL,NULL)
pred <-PredictVARX(mod)

Default show method for an object of class BigVAR.results

Description

Default show method for an object of class BigVAR.results

Usage

## S4 method for signature 'BigVAR.results'
show(object)
## S4 method for signature 'BigVAR.results'
show(object)

Arguments

object

BigVAR.results object created from cv.BigVAR

Details

prints forecast results and additional diagnostic information as well as comparisons with mean, random walk, and AIC, and BIC benchmarks

Default show method for an object of class BigVAR

Description

Default show method for an object of class BigVAR

Usage

## S4 method for signature 'BigVAR'
show(object)
## S4 method for signature 'BigVAR'
show(object)

Arguments

object

BigVAR object created from ConstructModel

Value

Displays the following information about the BigVAR object:

Prints the first 5 rows of Y
Penalty Structure
Relaxed Least Squares Indicator
Maximum lag order
VARX Specifications (if applicable)
Start, end of cross validation period

Sparsity Plot of a BigVAR.results object

Description

Sparsity Plot of a BigVAR.results object

Usage

SparsityPlot.BigVAR.results(object)
SparsityPlot.BigVAR.results(object)

Arguments

object

BigVAR.results object

Details

Uses levelplot from the lattice package to plot the magnitude of each coefficient in the last coefficient estimated by cv.BigVAR.

Value

NA, side effect is graph

Examples

data(Y)
Y <- Y[1:100,]
Model1 <- constructModel(Y,p=4,struct='Basic',gran=c(50,10),verbose=FALSE)
SparsityPlot.BigVAR.results(cv.BigVAR(Model1))
data(Y)
Y <- Y[1:100,]
Model1 <- constructModel(Y,p=4,struct='Basic',gran=c(50,10),verbose=FALSE)
SparsityPlot.BigVAR.results(cv.BigVAR(Model1))

Converts a VAR coefficient matrix of order p to multiple companion form

Description

Converts a VAR coefficient matrix of order p to multiple companion form

Usage

VarptoVar1MC(B, p, k)
VarptoVar1MC(B, p, k)

Arguments

`B`	a $k \times kp$ coefficient matrix
`p`	Lag order
`k`	Number of Series

Value

Returns a $kp \times kp$ coefficient matrix representing all coefficient matrices contained in Ai as a VAR(1).

References

See page 15 of Lutkepohl, 'A New Introduction to Multiple Time Series Analysis'

Examples

k=3;p=6
B=matrix(0,nrow=k,ncol=p*k)
A1<- matrix(c(.4,-.02,.01,-.02,.3,.02,.01,.04,.3),ncol=3,nrow=3)
A2 <- matrix(c(.2,0,0,0,.3,0,0,0,.13),ncol=3,nrow=3)
B[,1:k]=A1
B[,(4*k+1):(5*k)]=A2
A <- VarptoVar1MC(B,p,k)
k=3;p=6
B=matrix(0,nrow=k,ncol=p*k)
A1<- matrix(c(.4,-.02,.01,-.02,.3,.02,.01,.04,.3),ncol=3,nrow=3)
A2 <- matrix(c(.2,0,0,0,.3,0,0,0,.13),ncol=3,nrow=3)
B[,1:k]=A1
B[,(4*k+1):(5*k)]=A2
A <- VarptoVar1MC(B,p,k)

Fit a VAR or VARX model by least squares

Description

Fit a VAR or VARX model by least squares

Usage

VARXFit(Y, p, IC, VARX = NULL)
VARXFit(Y, p, IC, VARX = NULL)

Arguments

`Y`	a $t \times k$ multivariate time series
`p`	maximum lag order
`IC`	Information criterion indicator, if set to `NULL`, it will fit a least squares VAR(X) of orders p and s. Otherwise, if set to 'AIC' or 'BIC' it return the model with lag orders that minimize the given IC.
`VARX`	a list of VARX specifications (as in `constructModel` (or NULL )

Details

This function uses a modified form of the least squares technique proposed by Neumaier and Schneider (2001). It fits a least squares VAR or VARX via a QR decomposition that does not require explicit matrix inversion. This results in improved computational performance as well as numerical stability over the conventional least squares approach.

Value

Returns a list with four entries:

'Bhat'Estimated $k\times kp+ms$ coefficient matrix
'SigmaUEstimated $k\times k$ residual covariance matrix
'phat'Selected lag order for VAR component
'shat'Selected lag order for VARX component
'Y'multivariate time series retained for prediction purposes
'Y'number of endogenous (modeled) time series

References

Neumaier, Arnold, and Tapio Schneider. 'Estimation of parameters and eigenmodes of multivariate autoregressive models.' ACM Transactions on Mathematical Software (TOMS) 27.1 (2001): 27-57.

Examples

data(Y)
# fit a VAR_3(3)
mod <- VARXFit(Y,3,NULL,NULL)
# fit a VAR_3 with p= 6 and lag selected according to AIC
modAIC <- VARXFit(Y,6,'AIC',NULL)
# Fit a VARX_{2,1} with p=6, s=4 and lags selected by BIC
modXBIC <- VARXFit(Y,6,'BIC',list(k=1,s=4))

data(Y)
# fit a VAR_3(3)
mod <- VARXFit(Y,3,NULL,NULL)
# fit a VAR_3 with p= 6 and lag selected according to AIC
modAIC <- VARXFit(Y,6,'AIC',NULL)
# Fit a VARX_{2,1} with p=6, s=4 and lags selected by BIC
modXBIC <- VARXFit(Y,6,'BIC',list(k=1,s=4))

Evaluate forecasts from a VAR or VARX with lag orders selected by AIC/BIC

Description

Evaluate forecasts from a VAR or VARX with lag orders selected by AIC/BIC

Usage

VARXForecastEval(
  Y,
  X,
  p,
  s,
  T1,
  T2,
  IC,
  h,
  iterated = FALSE,
  loss = "L2",
  delta = 2.5
)
VARXForecastEval(
  Y,
  X,
  p,
  s,
  T1,
  T2,
  IC,
  h,
  iterated = FALSE,
  loss = "L2",
  delta = 2.5
)

Arguments

`Y`	a $T \times k$ multivariate time series
`X`	a $T \times m$ multivariate time series of unmodeled exogenous variables
`p`	maximum lag order for endogenous series
`s`	maximum lag order for exogenous series
`T1`	start of forecast evaluation period.
`T2`	end of forecast evaluation period
`IC`	specifies whether to select lag order according to 'AIC' or 'BIC'
`h`	desired forecast horizon
`iterated`	indicator as to whether to use iterated or direct multistep forecasts (if applicable, VAR context only)
`loss`	loss function (default 'L2', one of 'L1','L2','Huber')
`delta`	delta for Huber loss function (default 2.5)

Details

This function evaluates the one-step ahead forecasts of a VAR or VARX fit by least squares over an evaluation period. At every point in time, lag orders for the endogenous and exogenous series are selected according to AIC or BIC. This function is run automatically when cv.BigVAR is called unless ic is set to FALSE in constructModel.

Value

Returns the one-step ahead MSFE as well as the forecasts over the evaluation period and lag order selected.

References

Neumaier, Arnold, and Tapio Schneider. 'Estimation of parameters and eigenmodes of multivariate autoregressive models.' ACM Transactions on Mathematical Software (TOMS) 27.1 (2001): 27-57.

Examples

data(Y)

# Evaluate the performance of a VAR with lags selected by BIC.
p <- 4
T1 <- floor(nrow(Y))/3
T2 <- floor(2*nrow(Y))/3
# Matrix of zeros for X
X <- matrix(0,nrow=nrow(Y),ncol=ncol(Y))
BICMSFE <- VARXForecastEval(Y,X,p,0,T1,T2,'BIC',1)

data(Y)

# Evaluate the performance of a VAR with lags selected by BIC.
p <- 4
T1 <- floor(nrow(Y))/3
T2 <- floor(2*nrow(Y))/3
# Matrix of zeros for X
X <- matrix(0,nrow=nrow(Y),ncol=ncol(Y))
BICMSFE <- VARXForecastEval(Y,X,p,0,T1,T2,'BIC',1)

Construct a VAR or VARX lag matrix

Description

Construct a VAR or VARX lag matrix

Usage

VARXLagCons(Y, X = NULL, p, s = 0, oos = FALSE, contemp = FALSE)
VARXLagCons(Y, X = NULL, p, s = 0, oos = FALSE, contemp = FALSE)

Arguments

`Y`	a $T \times k$ matrix of endogenous (modeled) series
`X`	a $T \times m$ matrix of exogenous (unmodeled) series (default NULL)
`p`	Endogenous Lag order
`s`	exogenous lag order (default zero)
`oos`	indicator as to whether the data should be constructed for out of sample prediction (i.e. last available entries of Y as final lags default FALSE)
`contemp`	indicator as to whether to use contemporaneous exogenous predictors (for example, if exogenous series become available before exogenous default FALSE).

Details

This function is not required unless you which to design your own cross validation routine.

Value

list with two entries:

'Z' $kp+ms+1\times T-max(p,s)$ VARX lag matrix
'Y'adjusted $k\times T-max(p,s)$ endogenous series

References

See page 15 of Lutkepohl, 'A New Introduction to Multiple Time Series Analysis

Examples

data(Y)
# construct VAR lag matrix with p=4
ZZ<-VARXLagCons(Y,X=NULL,p=4,s=0)
data(Y)
# construct VAR lag matrix with p=4
ZZ<-VARXLagCons(Y,X=NULL,p=4,s=0)

Simulated Multivariate Time Series

Description

Realization of a simulated multivariate time series

Details

$100 \times 3$ multivariate time series distributed according to the generator matrix A.

Author(s)

Will Nicholson

Package 'BigVAR'

Help Index

Generator for Simulated Multivariate Time Series

Description

Details

Author(s)

References

Dimension Reduction Methods for Multivariate Time Series.

Description

Details

Author(s)

References

See Also

Examples

BigVAR Object Class

Description

Details

Slots

See Also

BigVAR Estimation

Description

Usage

Arguments

Details

Value

See Also

Examples

Simple function to fit BigVAR model with fixed penalty parameter

Description

Usage

Arguments

Note

References

See Also

Examples

BigVAR.intermediate This class contains the in-sample results for cv.BigVAR

Description

Fields

Note

Author(s)

BigVAR.results This class contains the results from cv.BigVAR.

Description

Fields

Note

Author(s)

Default coef method BigVAR-results, returns the last coefficient matrix from the evaluation period

Description

Usage

Arguments

Details

Construct an object of class BigVAR

Description

Usage

Arguments

Details

Note

References

See Also

Examples

Cross Validation for BigVAR

Description

Usage

Arguments

Details

Value

See Also

Examples

Simulate a VAR

Description

Usage

Arguments

Value

References

See Also

Examples

Plot an object of class BigVAR.results

Description

Usage

Arguments

Details