VAR Modelling: Estimation, Testing, and Prediction

Description

Estimation, Hypothesis Testing, Prediction in Stationary Vector Autoregressive Models

Details

The data set dat.rda is from Lutkepohl's book.

It is German Macrodata in log difference.

Bootstrap bias-correction and prediction intervals are also included.

Estimation and Forecasting based on Predictive Regression is also included.

Author(s)

Jae H. Kim

Maintainer: Jae H. Kim

Improved Augmented Regression Method for Predictive Regression

Description

Function for forecasting based on Imporved ARM

Usage

PR.Fore(x, y, M, h = 10)

Arguments

Details

Function for forecasting based on Imporved ARM

Value

Note

Kim J.H., 2014, Predictive Regression: Improved Augmented Regression Method, Journal of Empirical Finance 25, 13-15.

Author(s)

jae H. Kim

References

Kim J.H., 2014, Predictive Regression: Improved Augmented Regression Method, Journal of Empirical Finance 25, 13-15.

Examples

data(data1)
# Replicating Table 5 (excess return)
y=data1$ret.nyse.vw*100 -data1$tbill*100
x=cbind(log(data1$dy.nyse), data1$tbill*100); k=ncol(x) 
p=4
Rmat1=Rmatrix(p,k,type=1,index=1); Rmat=Rmat1$Rmat; rvec=Rmat1$rvec
M=PR.IARM(x,y,p,Rmat,rvec)
PRF=PR.Fore(x,y,M)

Improved Augmented Regression Method (IARM) for Predictive Regression

Description

Function for Improved ARM (IARM) estimation and testing

Usage

PR.IARM(x, y, p, Rmat = diag(k * p), rvec = matrix(0, nrow = k * p))

Arguments

Details

Kim J.H., 2014, Predictive Regression: Improved Augmented Regression Method, Journal of Empirical Finance, 26, 13-25.

Value

Note

Kim J.H., 2014, Predictive Regression: Improved Augmented Regression Method, Journal of Empirical Finance, 26, 13-25.

Author(s)

Jae H. Kim

References

Kim J.H., 2014, Predictive Regression: Improved Augmented Regression Method, Journal of Empirical Finance, 26, 13-25.

Examples

data(data1)
# Replicating Table 5 (excess return) of Kim (2014)
y=data1$ret.nyse.vw*100 -data1$tbill*100
x=cbind(log(data1$dy.nyse), data1$tbill*100); 

Rmat1=Rmatrix(p=1,k=2,type=1,index=0); Rmat=Rmat1$Rmat; rvec=Rmat1$rvec
M=PR.IARM(x,y,p=1,Rmat,rvec)

Improved Augmented Regression Method for Predictive Regression

Description

Function to select the order p by AIC or BIC

Usage

PR.order(x, y, pmax = 10)

Arguments

Details

Kim J.H., 2014, Predictive Regression: Improved Augmented Regression Method, Journal of Empirical Finance 25, 13-15.

Value

Note

Kim J.H., 2014, Predictive Regression: Improved Augmented Regression Method, Journal of Empirical Finance 25, 13-15.

Author(s)

Jae H. Kim

References

Kim J.H., 2014, Predictive Regression: Improved Augmented Regression Method, Journal of Empirical Finance 25, 13-15.

Examples

data(data1)
# Replicating Table 5 (excess return)
y=data1$ret.nyse.vw*100 -data1$tbill*100
x=cbind(log(data1$dy.nyse), data1$tbill*100); k=ncol(x) 

p=PR.order(x,y,pmax=10)$p.bic;  # AR(1) 

Improved Augmented Regression Method for Predictive Regression

Description

Function to generate restriction matrices

Usage

Rmatrix(p, k, type = 1, index = 0)

Arguments

Details

Function to generate restriction matrices

Value

Author(s)

Jae H. Kim

References

Kim J.H., 2014, Predictive Regression: Improved Augmented Regression Method, Journal of Empirical Finance 25, 13-15.

Examples

Rmat1=Rmatrix(p=1,k=1,type=2,index=1); Rmat=Rmat1$Rmat; rvec=Rmat1$rvec

Bootstrap Prediction Intervals for VAR(p) Model

Description

No Bias-correction is given

Usage

VAR.BPR(x, p, h, nboot = 500, type = "const", alpha = 0.95)

Arguments

Details

Bootstrap Prediction Intervals for VAR(p) Model

Value

Note

No Bias-correction is given

Author(s)

Jae H. Kim

References

Kim, J. H. (2001). Bootstrap-after-bootstrap prediction intervals for autoregressive models, Journal of Business & Economic Statistics, 19, 117-128.

Examples

data(dat)
VAR.BPR(dat,p=2,h=10,nboot=200,type="const",alpha=0.95)
# nboot is set to a low number for fast execution in the example
# In actual implementation, use higher number such as nboot=1000

Bootstrap-after-Bootstrap Prediction Intervals for VAR(p) Model

Description

Bias-correction given with stationarity Correction

Usage

VAR.BaBPR(x, p, h, nboot = 500, nb = 200, type = "const", alpha = 0.95)

Arguments

Details

Bias-correction given with stationarity Correction

Value

Note

Bias-correction given with stationarity Correction

Author(s)

Jae H. Kim

References

Kim, J. H. (2001). Bootstrap-after-bootstrap prediction intervals for autoregressive models, Journal of Business & Economic Statistics, 19, 117-128.

Examples

data(dat)
VAR.BaBPR(dat,p=2,h=10,nboot=200,nb=100,type="const",alpha=0.95)
# nboot and nb are set to low numbers for fast execution in the example
# In actual implementation, use higher numbers such as nboot=1000, nb=200

Bootstrapping VAR(p) model: bias-correction based on the bootstrap

Description

The function returns bias-corrected parmater estimators and Bias estimators based on the bootstrap

Usage

VAR.Boot(x, p, nb = 200, type = "const")

Arguments

Details

Kilian's (1998) stationarity-correction is used for bias-correction

Value

Author(s)

Jae H. Kim

References

Kilian, L. (1998). Small sample confidence intervals for impulse response functions, The Review of Economics and Statistics, 80, 218 - 230.

Examples

data(dat)
VAR.Boot(dat,p=2,nb=200,type="const")

VAR Forecasting

Description

Generate point forecasts and prediction intervals

Usage

VAR.FOR(x, p, h, type = "const", alpha = 0.95)

Arguments

Details

Prediction Intervals are based on normal approximation

Value

Note

See Chapter 3 of Lutkepohl (2005)

Author(s)

Jae H. Kim

References

Lutkepohl, H. 2005, New Introduction to Multiple Time Series Analysis, Springer

Examples

#replicating Section 3.5.3 of Lutkepohl (2005)
data(dat)
VAR.FOR(dat,p=2,h=10,type="const",alpha=0.95)

VAR Forecasting

Description

Generate point forecasts using the estimated VAR coefficient matrix

Usage

VAR.Fore(x, b, p, h, type = "const")

Arguments

Details

Generate point forecasts using the estimated VAR coefficient matrix

Value

Note

See Chapter 3 of Lutkepohl (2005)

Author(s)

Jae H. Kim

References

Lutkepohl, H. 2005, New Introduction to Multiple Time Series Analysis, Springer

Examples

#replicating Section 3.5.3 of Lutkepohl (2005)
data(dat)
b=VAR.est(dat,p=2,type="const")$coef
VAR.Fore(dat,b,p=2,h=10,type="const")

The Likelihood Ratio test for parameter restrictions

Description

Likelihood Ratio test for zero parameter restrictions based on system VAR estimation

Bootstrap option is available: iid bootstrap or wild bootstrap

Bootstrap is conducted under the null hypothesis using estimated GLS estimation: see Kim (2014)

Usage

VAR.LR(x, p, restrict0, restrict1, type = "const",bootstrap=0,nb=500)

Arguments

Details

Restriction matrix is of m by 3 matrix where m is the number of restrictions. A typical row of this matrix (k,i,j), which means that (i,j) element of Ak matrix is set to 0. Ak is a VAR coefficient matrix (k = 1,....p).

The bootstrap test is conducted using the GLS estimation under the parameter restrictions implied by the null hypothesis: see Kim (2014) for details.

Kim (2014) found that the bootstrap based on OLS can show inferior small sample properties.

There are two versions of the bootstrap: the first is based on the iid resampling and the second based on wild bootstrapping.

The Wild bootstrap is conducted with Mammen's two-point distribution.

Value

Note

See Chapter 4 of Lutkepohl (2005)

Author(s)

Jae H. Kim

References

Lutkepohl, H. 2005, New Introduction to Multiple Time Series Analysis, Springer

Kim, J.H. 2014, Testing for parameter restrictions in a stationary VAR model: a bootstrap alternative. Economic Modelling, 41, 267-273.

Examples

data(dat)
#replicating Table 4.4 of Lutkepohl (2005)
restrict1="full";
restrict0 = rbind(c(4,1,1), c(4,1,2), c(4,1,3), c(4,2,1),
c(4,2,2),c(4,2,3),c(4,3,1),c(4,3,2),c(4,3,3))
VAR.LR(dat,p=4,restrict0,restrict1,type="const")

Bias-correction for VAR parameter estimators based on Pope's formula

Description

The function returns bias-corrected parmater estimators and Bias estimators based on Pope's asymptotic formula

Usage

VAR.Pope(x, p, type = "const")

Arguments

Details

Kilian's (1998) stationarity-correction is used for bias-correction

Value

Author(s)

Jae H. Kim

References

Kim, J. H. 2004, Bias-corrected bootstrap prediction regions for Vector Autoregression, Journal of FOrecasting 23, 141-154.

Kilian, L. (1998). Small sample confidence intervals for impulse response functions, The Review of Economics and Statistics, 80, 218 - 230.

Nicholls DF, Pope AL. 1988, Bias in estimation of multivariate autoregression. Australian Journal of Statistics, 30A, 296-309.

Pope AL. 1990. Biases of estimators in multivariate non-Gaussian autoregression, Journal of Time Series Analysis 11, 249-258.

Examples

data(dat)
VAR.Pope(dat,p=2,type="const")

VAR parameter estimation with parameter restrictions

Description

Estimation of VAR with 0 restrictions on parameters

Usage

VAR.Rest(x, p, restrict, type = "const", method = "gls")

Arguments

Details

Restriction matrix is of m by 3 matrix where m is the number of restrictions. A typical row of this matrix (k,i,j), which means that (i,j) element of Ak matrix is set to 0. Ak is a VAR coefficient matrix (k = 1,....p).

Value

Note

See Chapter 5 of Lutkepohl

Author(s)

Jae H. Kim

References

Lutkepohl, H. 2005, New Introduction to Multiple Time Series Analysis, Springer

Examples

data(dat) 
#replicating Section 5.2.10 of Lutkepohl (2005)
restrict = rbind( c(1,1,2),c(1,1,3),c(1,2,1),c(1,2,2), c(1,3,1),
c(2,1,1), c(2,1,2),c(2,1,3), c(2,2,2), c(2,2,3),c(2,3,1), c(2,3,3),
c(3,1,1), c(3,1,2), c(3,1,3), c(3,2,1), c(3,2,2), c(3,2,3), c(3,3,1),c(3,3,3),
c(4,1,2), c(4,1,3), c(4,2,1), c(4,2,2), c(4,2,3), c(4,3,1),c(4,3,2),c(4,3,3))
M= VAR.Rest(dat,p=4,restrict,type="const",method="gls")
print(M$coef)
print(M$tstat)

Wald test for parameter restrictions

Description

Wald test for zero parameter restrictions based on system VAR estimation

Bootstrap option is available: iid bootstrap or wild bootstrap

Bootstrap is conducted under the null hypothesis using estimated GLS estimation: see Kim (2014)

Usage

VAR.Wald(x, p, restrict, type = "const",bootstrap=0,nb=500)

Arguments

Details

Restriction matrix is of m by 3 matrix where m is the number of restrictions. A typical row of this matrix (k,i,j), which means that (i,j) element of Ak matrix is set to 0. Ak is a VAR coefficient matrix (k = 1,....p). Under H1, the model is full VAR.

The bootstrap test is conducted using the GLS estimation under the parameter restrictions implied by the null hypothesis: see Kim (2014) for details.

Kim (2014) found that the bootstrap based on OLS can show inferior small sample properties.

There are two versions of the bootstrap: the first is based on the iid resampling and the second based on wild bootstrapping.

The Wild bootstrap is conducted with Mammen's two-point distribution.

Value

Note

See Chapter 3 of Lutkepohl

Author(s)

Jae H. Kim

References

Lutkepohl, H. 2005, New Introduction to Multiple Time Series Analysis, Springer.

Kim, J.H. 2014, Testing for parameter restrictions in a stationary VAR model: a bootstrap alternative. Economic Modelling, 41, 267-273.

Examples

data(dat)
#replicating Section 3.6.2 of Lutkepohl (2005)
restrict = rbind( c(1,1,2),c(1,1,3), c(2,1,2),c(2,1,3))
VAR.Wald(dat,p=2,restrict,type="const")

Estimation of unrestricted VAR(p) model parameters

Description

This function returns least-squares estimation results for VAR(p) model

Usage

VAR.est(x, p, type = "const")

Arguments

Details

VAR estimation

Value

Note

See Chapter 3 of Lutkepohl (2005)

Author(s)

Jae H. Kim

References

Lutkepohl, H. 2005, New Introduction to Multiple Time Series Analysis, Springer

Examples

#replicating Section 3.2.3 of of Lutkepohl (2005)
data(dat)
M=VAR.est(dat,p=2,type="const")
print(M$coef)
print(M$tratio)

Orthogonalized impluse response functions from an estimated VAR(p) model

Description

This function returns Orthogonalized impluse response functions

Usage

VAR.irf(b, p, sigu, h=10,graphs=FALSE)

Arguments

Details

VAR impulse response functions

Value

Note

See Lutkepohl (2005) for details

Author(s)

Jae H. Kim

References

Lutkepohl, H. 2005, New Introduction to Multiple Time Series Analysis, Springer

Examples

#replicating Table 3.4 and Figure 3.11 Lutkepohl (2005)
data(dat)
M=VAR.est(dat,p=2,type="const")
b=M$coef; sigu=M$sigu
VAR.irf(b,p=2,sigu,graphs=TRUE)

Order Selection for VAR models

Description

AIC, HQ, or SC can be used

Usage

VAR.select(x, type = "const", ic = "aic", pmax)

Arguments

Details

Order Section Criterion

Value

Note

See Chapter 4 of Lutkepohl

Author(s)

JAe H. Kim

References

Lutkepohl, H. 2005, New Introduction to Multiple Time Series Analysis, Springer

Examples

data(dat)
#replicating Section 4.3.1 of Lutkepohl (2005)
VAR.select(dat,pmax=4,ic="aic")

German investment income consumption in log difference

Description

Lutkepohl's data

Usage

data(dat)

References

Lutkepohl, H. 2005, New Introduction to Multiple Time Series Analysis, Springer

Examples

data(dat)

stock return data used in Kim (2014)

Description

stock return data used in Kim (2014)

Usage

data(data1)

References

Kim, J.H. 2014, Testing for parameter restrictions in a stationary VAR model: a bootstrap alternative. Economic Modelling, 41, 267-273.

Examples

data(data1)