| Title: | All-Purpose Toolkit for Analyzing Multivariate Time Series (MTS) and Estimating Multivariate Volatility Models |
|---|---|
| Description: | Multivariate Time Series (MTS) is a general package for analyzing multivariate linear time series and estimating multivariate volatility models. It also handles factor models, constrained factor models, asymptotic principal component analysis commonly used in finance and econometrics, and principal volatility component analysis. (a) For the multivariate linear time series analysis, the package performs model specification, estimation, model checking, and prediction for many widely used models, including vector AR models, vector MA models, vector ARMA models, seasonal vector ARMA models, VAR models with exogenous variables, multivariate regression models with time series errors, augmented VAR models, and Error-correction VAR models for co-integrated time series. For model specification, the package performs structural specification to overcome the difficulties of identifiability of VARMA models. The methods used for structural specification include Kronecker indices and Scalar Component Models. (b) For multivariate volatility modeling, the MTS package handles several commonly used models, including multivariate exponentially weighted moving-average volatility, Cholesky decomposition volatility models, dynamic conditional correlation (DCC) models, copula-based volatility models, and low-dimensional BEKK models. The package also considers multiple tests for conditional heteroscedasticity, including rank-based statistics. (c) Finally, the MTS package also performs forecasting using diffusion index , transfer function analysis, Bayesian estimation of VAR models, and multivariate time series analysis with missing values.Users can also use the package to simulate VARMA models, to compute impulse response functions of a fitted VARMA model, and to calculate theoretical cross-covariance matrices of a given VARMA model. |
| Authors: | Ruey S. Tsay [aut, cre], David Wood [aut], Jon Lachmann [ctb] |
| Maintainer: | Ruey S. Tsay <[email protected]> |
| License: | Artistic License 2.0 |
| Version: | 1.2.1 |
| Built: | 2024-11-10 04:14:24 UTC |
| Source: | https://github.com/cran/MTS |
Multivariate Time Series (MTS) is a general package for analyzing multivariate linear time series and estimating multivariate volatility models. It also handles factor models, constrained factor models, asymptotic principal component analysis commonly used in finance and econometrics, and principal volatility component analysis. (a) For the multivariate linear time series analysis, the package performs model specification, estimation, model checking, and prediction for many widely used models, including vector AR models, vector MA models, vector ARMA models, seasonal vector ARMA models, VAR models with exogenous variables, multivariate regression models with time series errors, augmented VAR models, and Error-correction VAR models for co-integrated time series. For model specification, the package performs structural specification to overcome the difficulties of identifiability of VARMA models. The methods used for structural specification include Kronecker indices and Scalar Component Models. (b) For multivariate volatility modeling, the MTS package handles several commonly used models, including multivariate exponentially weighted moving-average volatility, Cholesky decomposition volatility models, dynamic conditional correlation (DCC) models, copula-based volatility models, and low-dimensional BEKK models. The package also considers multiple tests for conditional heteroscedasticity, including rank-based statistics. (c) Finally, the MTS package also performs forecasting using diffusion index, transfer function analysis, Bayesian estimation of VAR models, and multivariate time series analysis with missing values.Users can also use the package to simulate VARMA models, to compute impulse response functions of a fitted VARMA model, and to calculate theoretical cross-covariance matrices of a given VARMA model.
| Package: | MTS |
| Type: | Package |
| License: | Artistic License 2.0 |
Ruey S. Tsay and David Wood
Perform asymptotic PCA for a data set. Typically for cases in which the number of variables is greater than the number of data points.
apca(da, m)apca(da, m)
da |
A T-by-k data set matrix, where T is the sample size and k is the dimension |
m |
The number of common factors |
Perform the PCA analysis of interchanging the roles of variables and observations.
sdev |
Square root of the eigenvalues |
factors |
The common factors |
loadings |
The loading matrix |
Ruey S. Tsay
Tsay (2014, Chapter 6). Multivariate Time Series Analysis with R and Financial Applications. John Wiley. Hoboken, NJ.
rtn=matrix(rnorm(1200),12,100) sp100=apca(rtn,3)rtn=matrix(rnorm(1200),12,100) sp100=apca(rtn,3)
Perform tests to check the conditional heteroscedasticity in a time series. The Ljung-Box statistics of squared series and a rank-based Ljung-Box test are used.
archTest(rt, lag = 10)archTest(rt, lag = 10)
rt |
A scalar time series. If rt is a matrix, only the first column is used. |
lag |
The number of lags of ACF used in the Ljung-Box statistics. The default is 10. |
The Ljung-Box statistics based on the squared series are computed first. The rank series of the squared time series is than used to test the conditional heteroscedasticity.
The Q-statistic and its p-value. Also, the rank-based Q statistic and its p-value.
Ruey Tsay
MarchTest
rt=rnorm(200) archTest(rt)rt=rnorm(200) archTest(rt)
Perform out-of-sample prediction of a given ARIMA model and compute the summary statistics
backtest(m1, rt, orig, h = 1, xre = NULL, fixed = NULL, inc.mean = TRUE, reest = 1, method = c("CSS-ML"))backtest(m1, rt, orig, h = 1, xre = NULL, fixed = NULL, inc.mean = TRUE, reest = 1, method = c("CSS-ML"))
m1 |
An output of the arima command for scalar time series |
rt |
The time series under consideration |
orig |
The starting forecast origin. It should be less than the length of the underlying time series |
h |
The forecast horizon. For a given h, it computes 1-step to h-step ahead forecasts |
inc.mean |
A logical switch. It is true if mean vector is estimated. |
fixed |
A vector of the length of the number of coefficients of the ARIMA model. It is used in R for parameter constraint. |
xre |
A matrix containing the exogeneous variables used in the ARIMA model |
reest |
A control variable used to re-fit the model in prediction. The program will re-estimate the model for every new reest observations. The default is 1. That is, re-estimate the model with every new data point. |
method |
Estimation method in the ARIMA model |
Perform estimation-prediction-reestimation in the forecasting subsample, and to compuate the summary statistics
origion |
Forecast origin |
error |
forecast errors |
forecasts |
forecasts |
rmse |
Root mean squared forecast errors |
mabso |
Mean absolute forecast errors |
reest |
Return the reest value |
Ruey S. Tsay
Tsay (2010). Analysis of Financial Time Series, 3rd. John Wiley. Hoboken, NJ.
Estimation of a BEKK(1,1) Model for a k-dimensional time series. Only k = 2 or 3 is available
BEKK11(rt, include.mean = T, cond.dist = "normal", ini.estimates = NULL)BEKK11(rt, include.mean = T, cond.dist = "normal", ini.estimates = NULL)
rt |
A T-by-k data matrix of k-dimensional asset returns |
include.mean |
A logical switch to include a constant vector in the mean equation. Default is with a constant vector. |
cond.dist |
Conditional innovation distribution. Only Gaussian innovations are used in the current version. |
ini.estimates |
Optional initial estimates. |
estimates |
Parameter estimates |
HessianMtx |
Hessian matrix of the estimates |
Sigma.t |
The multivariate volatilities, each row contains k-by-k elements of the volatility matrix Sigma(t) |
Ruey S. Tsay
Tsay (2014, Chapter 7)
#data("mts-examples",package="MTS") #da=ibmspko #rtn=log(da[,2:3]+1) #m1=BEKK11(rtn)#data("mts-examples",package="MTS") #da=ibmspko #rtn=log(da[,2:3]+1) #m1=BEKK11(rtn)
Perform back-test of transfer function model with 2 input variable. For a specified tfm2 model and a given forecast origin, the command iterated between estimation and 1-step ahead prediction starting at the forecast origin until the (T-1)th observation, where T is the sample size.
Btfm2(y,x,x2=NULL,wt=NULL,ct=NULL,orderN=c(1,0,0),orderS=c(0,0,0),sea=12, order1=c(0,1,0),order2=c(0,-1,0),orig=(length(y)-1))Btfm2(y,x,x2=NULL,wt=NULL,ct=NULL,orderN=c(1,0,0),orderS=c(0,0,0),sea=12, order1=c(0,1,0),order2=c(0,-1,0),orig=(length(y)-1))
y |
Data vector of dependent variable |
x |
Data vector of the first input (or independent) variable |
x2 |
Data vector of the second input variable if any |
ct |
Data vector of a given deterministic variable such as time trend, if any |
wt |
Data vector of co-integrated series between input and output variables if any |
orderN |
Order (p,d,q) of the regular ARMA part of the disturbance component |
orderS |
Order (P,D,Q) of the seasonal ARMA part of the disturbance component |
sea |
Seasonalityt, default is 12 for monthly data |
order1 |
Order (r,s,b) of the transfer function model of the first input variable, where r and s are the degrees of denominator and numerator polynomials and b is the delay |
order2 |
Order (r2,s2,b2) of the transfer function model of the second input variable, where 2r and s2 are the degrees of denominator and numerator polynomials and b2 is the delay |
orig |
Forecast origin with default being T-1, where T is the sample size |
Perform out-of-sample 1-step ahead prediction to evaluate a fitted tfm2 model
ferror |
1-step ahead forecast errors, starting at the given forecast origin |
mse |
out-of-sample mean squared forecast errors |
rmse |
root mean squared forecast errors |
mae |
out-of-sample mean absolute forecast errors |
nobf |
The number of 1-step ahead forecast errors computed |
rAR |
Regular AR coefficients |
Ruey S. Tsay
Box, G. E. P., Jenkins, G. M., and Reinsel, G. C. (1994). Time Series Analysis: Forecasting and Control, 3rd edition, Prentice Hall, Englewood Cliffs, NJ.
tfm2
Estimate a VAR(p) model using Bayesian approach, including the use of Minnesota prior
BVAR(z,p=1,C,V0,n0=5,Phi0=NULL,include.mean=T)BVAR(z,p=1,C,V0,n0=5,Phi0=NULL,include.mean=T)
z |
A matrix of vector time series, each column represents a series. |
p |
The AR order. Default is p=1. |
C |
The precision matrix of the coefficient matrix. With constant, the dimension of C is (kp+1)-by-(kp+1). The covariance matrix of the prior for the parameter vec(Beta) is Kronecker(Sigma_a,C-inverse). |
V0 |
A k-by-k covariance matrix to be used as prior for the Sigma_a matrix |
n0 |
The degrees of freedom used for prior of the Sigma_a matrix, the covariance matrix of the innovations. Default is n0=5. |
Phi0 |
The prior mean for the parameters. Default is set to NULL, implying that the prior means are zero. |
include.mean |
A logical switch controls the constant term in the VAR model. Default is to include the constant term. |
for a given prior, the program provide the posterior estimates of a VAR(p) model.
est |
Posterior means of the parameters |
Sigma |
Residual covariance matrix |
Ruey S. Tsay
Tsay (2014, Chapter 2).
data("mts-examples",package="MTS") z=log(qgdp[,3:5]) zt=diffM(z)*100 C=0.1*diag(rep(1,7)) V0=diag(rep(1,3)) BVAR(zt,p=2,C,V0)data("mts-examples",package="MTS") z=log(qgdp[,3:5]) zt=diffM(z)*100 C=0.1*diag(rep(1,7)) V0=diag(rep(1,3)) BVAR(zt,p=2,C,V0)
Computes sample cross-correlation matrices of a multivariate time series, including simplified ccm matrix and p-value plot of Ljung-Box statistics.
ccm(x, lags = 12, level = FALSE, output = T)ccm(x, lags = 12, level = FALSE, output = T)
x |
A matrix of vector time series, each column represents a series. |
lags |
The number of lags of CCM to be computed. Default is 12. |
level |
A logical switch. When level=T, numerical values of CCM is printed. Default is no printing of CCM. |
output |
A logical switch. If ouput=F, no output is given. Default is with output. |
The p-value of Ljung-Box statistics does not include any adjustment in degrees of freedom.
ccm |
Sample cross-correlation matrices |
pvalue |
p-values for each lag of CCM being a zero matrix |
Ruey S. Tsay
Tsay (2014, Chapter 1). Multivariate Time Series Analysis with R and Financial Applications. John Wiley. Hoboken, NJ.
xt=matrix(rnorm(1500),500,3) ccm(xt) ccm(xt,lag=20)xt=matrix(rnorm(1500),500,3) ccm(xt) ccm(xt,lag=20)
Compute the principal volatility components based on the residuals of a VAR(p) model.
comVol(rtn, m = 10, p = 1, stand = FALSE)comVol(rtn, m = 10, p = 1, stand = FALSE)
rtn |
A T-by-k data matrix of k-dimensional asset returns |
m |
The number of lags used to compute generalized cross-Kurtosis matrix |
p |
VAR order for the mean equation |
stand |
A logical switch to standardize the returns |
Perform a VAR(p) fit, if any. Then, use the residual series to perform principal volatility component analysis. The ARCH test statistics are also computed for the sample principal components
residuals |
The residuals of a VAR(p) fit |
values |
Eigenvalues of the principal volatility component analysis |
vectors |
Eigenvectors of the principal volatility component analysis |
M |
The transformation matrix |
Ruey S. Tsay and Y.B. Hu
Tsay (2014, Chapter 7)
data("mts-examples",package="MTS") zt=diffM(log(qgdp[,3:5])) m1=comVol(zt,p=2) names(m1)data("mts-examples",package="MTS") zt=diffM(log(qgdp[,3:5])) m1=comVol(zt,p=2) names(m1)
For a given dependent variable and an input variable, the program computes the Corner table for specifying the order (r,s,d) of a transfer function
Corner(y,x,Nrow=11,Ncol=7)Corner(y,x,Nrow=11,Ncol=7)
y |
A pre-whitened dependent (or output) variable |
x |
A pre-whitened independent (or input) variable. It should be a white noise series |
Nrow |
The number of rows of the Corner table. Default is 11. |
Ncol |
The number of columns of the Corner table. Default is 7. |
For the pair of pre-whitened output and input variables, the program compute the Corner table and its simplified version for specifying the order of a transfer function.
corner |
The Corner table |
Ruey S. Tsay
Fits a DCC model using either multivariate Gaussian or multivariate Student-t innovations. Two types of DCC models are available. The first type is proposed by Engle and the other is by Tse and Tsui. Both models appear in the Journal of Business and Economic Statistics, 2002.
dccFit(rt, type = "TseTsui", theta = c(0.90, 0.02), ub = c(0.95, 0.049999), lb = c(0.4,0.00001), cond.dist = "std", df = 7, m = 0)dccFit(rt, type = "TseTsui", theta = c(0.90, 0.02), ub = c(0.95, 0.049999), lb = c(0.4,0.00001), cond.dist = "std", df = 7, m = 0)
rt |
The T-by-k data matrix of k-dimensional standardized asset returns. Typically, they are the standardized residuals of the command dccPre. |
type |
A logical switch to specify the type of DCC model. Type="TseTsui" for Tse and Tsui's DCC model. Type = "Engle" for Engle's DCC model. Default is Tse-Tsui model. |
theta |
The initial parameter values for theta1 and theta2 |
ub |
Upper bound of parameters |
lb |
Lower bound of parameters |
cond.dist |
Conditional innovation distribution with std for multivariate Student-t innovations. |
df |
degrees of freedom of the multivariate Student-t innovations. |
m |
For Tse and Tsui method only, m denotes the number of returns used in local correlation matrix estimation |
estimates |
Parameter estimates |
Hessian |
Hessian matrix of the estimates |
rho.t |
Time-varying correlation matrices. Each row contains elements of a cross-correlation matrix. |
Ruey S. Tsay
Tsay (2014, Chapter 7). Multivariate Time Series Analysis with R and Financial Applications. John Wiley. Hoboken, NJ.
dccPre
This program fits marginal GARCH models to each component of a vector return series and returns the standardized return series for further analysis. The garchFit command of fGarch package is used.
dccPre(rtn, include.mean = T, p = 0, cond.dist = "norm")dccPre(rtn, include.mean = T, p = 0, cond.dist = "norm")
rtn |
A T-by-k data matrix of k-dimensional asset returns |
include.mean |
A logical switch to include a mean vector. Deafult is to include the mean. |
p |
VAR order for the mean equation |
cond.dist |
The conditional distribution of the innovations. Default is Gaussian. |
The program uses fGarch package to estimate univariate GARCH model for each residual series after a VAR(p) fitting, if any.
marVol |
A matrix of the volatility series for each return series |
sresi |
Standardized residual series |
est |
Parameter estimates for each marginal volatility model |
se.est |
Standard errors for parameter estimates of marginal volatility models |
fGarch package is used
Ruey S. Tsay
Tsay (2014, Chapter 7). Multivariate Time Series Analysis with R and Financial Applications. John Wiley. Hoboken, NJ.
dccFit
Performs the difference operation of a vector time series
diffM(zt, d = 1)diffM(zt, d = 1)
zt |
A vector time series (T by k, with sample size T and dimension k) |
d |
Order of differencing. Default is d=1. |
When d = 1, the command is equivalent to apply(zt,2,diff)
The differenced time series
Ruey S Tsay
data("mts-examples",package="MTS") zt=log(qgdp[,3:5]) xt=diffM(zt)data("mts-examples",package="MTS") zt=log(qgdp[,3:5]) xt=diffM(zt)
Compute the extended cross-correlation matrices and the associated two-way table of p-values of multivariate Ljung-Box statistics of a vector time series.
Eccm(zt, maxp = 5, maxq = 6, include.mean = FALSE, rev = TRUE)Eccm(zt, maxp = 5, maxq = 6, include.mean = FALSE, rev = TRUE)
zt |
Data matrix (T-by-k) of a vector time series, where T is the sample size and k is the dimension. |
maxp |
Maximum AR order entertained. Default is 5. |
maxq |
Maximum MA order entertained. Default is 6. |
include.mean |
A logical switch controlling the mean vector in estimation. Default assumes zero mean. |
rev |
A logical switch to control the cross-correlation matrices used to compute the multivariate Ljung-Box statistics. Traditional way is to compute test statistics from lag-1 to lag-m. If rev = TRUE, then the test statistics are compute from lag-(m-1) to lag-m, from lag-(m-2) to lag-m, etc. |
pEccm |
A two-way table of the p-values of extended cross-correlation matrices |
vEccm |
The sample extended cross-correlation matrices |
ARcoef |
AR coefficient matrices of iterated VAR fitting |
Ruey S. Tsay
Tsay (2014, Chapter 3). Multivariate Time Series Analysis with R and Financial Applications. John Wiley. Hoboken, NJ.
zt=matrix(rnorm(900),300,3) m1=Eccm(zt)zt=matrix(rnorm(900),300,3) m1=Eccm(zt)
Performs estimation of an Error-Correction VAR(p) model using the Quasi Maximum Likelihood Method.
ECMvar(x, p, ibeta, include.const = FALSE, fixed = NULL, alpha = NULL, se.alpha = NULL, se.beta = NULL, phip = NULL, se.phip = NULL)ECMvar(x, p, ibeta, include.const = FALSE, fixed = NULL, alpha = NULL, se.alpha = NULL, se.beta = NULL, phip = NULL, se.phip = NULL)
x |
A T-by-k data matrix of a k-dimensional co-integrated VAR process |
p |
VAR order |
ibeta |
Initial estimate of the co-integrating matrix. The number of columns of ibeta is the number of co-integrating series |
include.const |
A logical switch to include a constant term in the model. The default is no constant |
fixed |
A logical matrix to set zero parameter constraints. |
alpha |
Initial estimate of alpha, if any |
se.alpha |
Initial estimate of the standard error of alpha, if any |
se.beta |
Initial estimate of the standard error of beta, if any |
phip |
Initial estimate of the VAR coefficients, if any |
se.phip |
Initial estimate of the standard error of the VAR coefficients, if any |
data |
The vector time series |
ncoint |
The number of co-integrating series |
arorder |
VAR order |
include.const |
Logical switch to include constant |
alpha, se.alpha
|
Estimates and their standard errors of the alpha matrix |
beta, se.beta
|
Estimates and their standard errors of the beta matrix |
aic, bic
|
Information criteria of the fitted model |
residuals |
The residual series |
Sigma |
Residual covariance matrix |
Phip, se.Phip
|
Estimates and their standard errors of VAR coefficients |
Ruey S. Tsay
Tsay (2014, Chapter 5)
ECMvar1
phi=matrix(c(0.5,-0.25,-1.0,0.5),2,2); theta=matrix(c(0.2,-0.1,-0.4,0.2),2,2) Sig=diag(2) mm=VARMAsim(300,arlags=c(1),malags=c(1),phi=phi,theta=theta,sigma=Sig) zt=mm$series[,c(2,1)] beta=matrix(c(1,0.5),2,1) m1=ECMvar(zt,3,ibeta=beta) names(m1)phi=matrix(c(0.5,-0.25,-1.0,0.5),2,2); theta=matrix(c(0.2,-0.1,-0.4,0.2),2,2) Sig=diag(2) mm=VARMAsim(300,arlags=c(1),malags=c(1),phi=phi,theta=theta,sigma=Sig) zt=mm$series[,c(2,1)] beta=matrix(c(1,0.5),2,1) m1=ECMvar(zt,3,ibeta=beta) names(m1)
Perform least-squares estimation of an ECM VAR(p) model with known co-integrating processes
ECMvar1(x, p, wt, include.const = FALSE, fixed = NULL, output = TRUE)ECMvar1(x, p, wt, include.const = FALSE, fixed = NULL, output = TRUE)
x |
A T-by-k data matrix of a k-dimensional co-integrated VAR process |
p |
VAR order |
wt |
A T-by-m data matrix of m-dimensional co-integrated process |
include.const |
A logical switch to include a constant term. Default is no constant. |
fixed |
A logical matrix to set zero parameter constraints |
output |
A logical switch to control output |
data |
The vector time series |
wt |
The co-integrated series |
arorder |
VAR order |
include.const |
Logical switch to include constant |
coef |
Parameter estimates |
aic, bic
|
Information criteria of the fitted model |
residuals |
The residual series |
Sigma |
Residual covariance matrix |
Ruey S. Tsay
Tsay (2014, Chapter 5). Multivariate Time Series Analysis with R and Financial Applications. John Wiley. Hoboken, NJ.
ECMvar
phi=matrix(c(0.5,-0.25,-1.0,0.5),2,2); theta=matrix(c(0.2,-0.1,-0.4,0.2),2,2) Sig=diag(2) mm=VARMAsim(300,arlags=c(1),malags=c(1),phi=phi,theta=theta,sigma=Sig) zt=mm$series wt=0.5*zt[,1]+zt[,2] m1=ECMvar1(zt,3,wt) names(m1)phi=matrix(c(0.5,-0.25,-1.0,0.5),2,2); theta=matrix(c(0.2,-0.1,-0.4,0.2),2,2) Sig=diag(2) mm=VARMAsim(300,arlags=c(1),malags=c(1),phi=phi,theta=theta,sigma=Sig) zt=mm$series wt=0.5*zt[,1]+zt[,2] m1=ECMvar1(zt,3,wt) names(m1)
Use exponentially weighted moving-average method to compute the volatility matrix
EWMAvol(rtn, lambda = 0.96)EWMAvol(rtn, lambda = 0.96)
rtn |
A T-by-k data matrix of k-dimensional asset returns, assuming the mean is zero |
lambda |
Smoothing parameter. The default is 0.96. If lambda is negative, then the multivariate Gaussian likelihood is used to estimate the smoothing parameter. |
Sigma.t |
The volatility matrix with each row representing a volatility matrix |
return |
The data |
lambda |
The smoothing parameter lambda used |
Ruey S. Tsay
Tsay (2014, Chapter 7). Multivariate Time Series Analysis with R and Financial Applications. John Wiley. Hoboken, NJ.
data("mts-examples",package="MTS") rtn=log(ibmspko[,2:4]+1) m1=EWMAvol(rtn)data("mts-examples",package="MTS") rtn=log(ibmspko[,2:4]+1) m1=EWMAvol(rtn)
Computes the forecast error variance decomposition of a VARMA model
FEVdec(Phi, Theta, Sig, lag = 4)FEVdec(Phi, Theta, Sig, lag = 4)
Phi |
VAR coefficient matrices in the form Phi=[Phi1, Phi2, ..., Phip], a k-by-kp matrix. |
Theta |
VMA coefficient matrices in form form Theta=[Theta1, Theta2, ..., Thetaq], a k-by-kq matrix. |
Sig |
The residual covariance matrix Sigma, a k-by-k positive definite matrix. |
lag |
The number of lags of forecast errors variance to be computed. Default is 4. |
Use the psi-weight matrices to compute the forecast error covariance and use Cholesky decomposition to perform the decomposition
irf |
Impulse response matrices |
orthirf |
Orthogonal impulse response matrices |
Omega |
Forecast error variance matrices |
OmegaR |
Forecast error variance decomposition |
Ruey S. Tsay
Tsay (2014, Chapter 3)
p1=matrix(c(0.2,-0.6,0.3,1.1),2,2) theta1=matrix(c(-0.5,0,0,-0.6),2,2) Sig=matrix(c(3,1,1,1),2,2) m1=FEVdec(p1,theta1,Sig) names(m1)p1=matrix(c(0.2,-0.6,0.3,1.1),2,2) theta1=matrix(c(-0.5,0,0,-0.6),2,2) Sig=matrix(c(3,1,1,1),2,2) m1=FEVdec(p1,theta1,Sig) names(m1)
Performs Granger causality test using a vector autoregressive model
GrangerTest(X,p=1,include.mean=T,locInput=c(1))GrangerTest(X,p=1,include.mean=T,locInput=c(1))
X |
a T-by-p data matrix with T denoting sample size and p the number of variables |
p |
vector AR order. |
include.mean |
Indicator for including a constant in the model. Default is TRUE. |
locInput |
Locators for the input variables in the data matrix. Default is the first column being the input variable. Multiple inputs are allowed. |
Perform VAR(p) and constrained VAR(p) estimations to test the Granger causality. It uses likelihood ratio and asymptotic chi-square.
data |
Original data matrix |
cnst |
logical variable to include a constant in the model |
order |
order of VAR model used |
coef |
Coefficient estimates |
constraints |
Implied constraints of Granger causality |
aic, bic, hq
|
values of information criteria |
residuals |
residual vector |
secoef |
standard errors of coefficient estimates |
Sigma |
Residual covariance matrix |
Phi |
Matrix of VAR coefficients |
Ph0 |
constant vector |
omega |
Estimates of constrained coefficients |
covomega |
covariance matrix of constrained parameters |
locInput |
Locator vector for input variables |
Ruey S. Tsay
Tsay (2014, Chapter 2)
Performs factor model analysis with a given constrained matrix
hfactor(X, H, r)hfactor(X, H, r)
X |
A T-by-k data matrix of an observed k-dimensional time series |
H |
The constrained matrix with each column representing a constraint |
r |
The number of common factor |
Results of the traditional PCA and constrained factor models are given
Ruey S. Tsay
Tsay (2014, Chapter 6). Tsai and Tsay (2010, JASA)
data("mts-examples",package="MTS") rtn=log(tenstocks[,2:11]+1) # compute log returns h1=c(1,1,1,1,rep(0,6)) # specify the constraints h2=c(0,0,0,0,1,1,1,0,0,0) h3=c(rep(0,7),1,1,1) H=cbind(h1,h2,h3) m1=hfactor(rtn,H,3)data("mts-examples",package="MTS") rtn=log(tenstocks[,2:11]+1) # compute log returns h1=c(1,1,1,1,rep(0,6)) # specify the constraints h2=c(0,0,0,0,1,1,1,0,0,0) h3=c(rep(0,7),1,1,1) H=cbind(h1,h2,h3) m1=hfactor(rtn,H,3)
Monthly simple returns of the stocks of International Business Machines (IBM) and Coca Cola (KO) and the S&P Composite index (SP). The sample period is from January 1961 to December 2011. The original data were from the Center for Research in Security Prices (CRSP) of the University of Chicago. The files has four columns. They are dates, IBM, SP, and KO.
A 2-d list containing 612x4 observations. The files has four columns. They are dates, IBM, SP, and KO.
World Almanac and Book of Facts, 1975, page 406.
Perform estimation of a VARMA model specified by the Kronecker indices
Kronfit(da, kidx, include.mean = T, fixed = NULL, Kpar=NULL, seKpar=NULL, prelim = F, details = F, thres = 1)Kronfit(da, kidx, include.mean = T, fixed = NULL, Kpar=NULL, seKpar=NULL, prelim = F, details = F, thres = 1)
da |
Data matrix (T-by-k) of a k-dimensional time series |
kidx |
The vector consisting of Kronecker indices |
include.mean |
A logical switch for including the mean vector in estimation. Default is to include the mean vector. |
fixed |
A logical matrix used to set zero parameter constraints. This is used mainly in the command refKronfit. |
Kpar |
Parameter vectors for use in model simplification |
seKpar |
Standard errors of the parameter estimates for use in model simplification |
prelim |
A logical switch for a preliminary estimation. |
details |
A logical switch to control output. |
thres |
A threshold for t-ratios in setting parameter to zero. Default is 1. |
data |
The observed time series data |
Kindex |
Kronecker indices |
ARid |
Specification of AR parameters: 0 denotes fixing to zero, 1 denotes fixing to 1, and 2 denoting estimation |
MAid |
Specification of MA parameters |
cnst |
A logical variable: include.mean |
coef |
Parameter estimates |
se.coef |
Standard errors of the estimates |
residuals |
Residual series |
Sigma |
Residual covariance matrix |
aic, bic
|
Information criteria of the fitted model |
Ph0 |
Constant vector |
Phi |
AR coefficient matrices |
Theta |
MA coefficient matrices |
Ruey S. Tsay
Tsay (2014, Chapter 4). Multivariate Time Series Analysis with R and Financial Applications. John Wiley. Hoboken, NJ.
refKronfit, Kronspec
Find the Kronecker indices of a k-dimensional time series
Kronid(x, plag = 5, crit = 0.05)Kronid(x, plag = 5, crit = 0.05)
x |
Data matrix (T-by-k) of a k-dimensional time series |
plag |
The number of lags used to represent the past vector. Default is 5. |
crit |
Type-I error used in testing for zero canonical correlations. Deafult is 0.05. |
index |
Kronecker indices |
tests |
Chi-square test statistics |
Ruey S. Tsay
Tsay (2014, Chapter 4). Multivariate Time Series Analysis with R and Financial Applications. John Wiley. Hoboken, NJ.
phi=matrix(c(0.2,-0.6,.3,1.1),2,2); sigma=diag(2); theta=-0.5*sigma m1=VARMAsim(300,arlags=c(1),malags=c(1),phi=phi,theta=theta,sigma=sigma) zt=m1$series Kronid(zt)phi=matrix(c(0.2,-0.6,.3,1.1),2,2); sigma=diag(2); theta=-0.5*sigma m1=VARMAsim(300,arlags=c(1),malags=c(1),phi=phi,theta=theta,sigma=sigma) zt=m1$series Kronid(zt)
Compute forecasts of a fitted VARMA model via the command Kronfit
Kronpred(model,orig=0,h=1)Kronpred(model,orig=0,h=1)
model |
A model fitted by the Kronfit command |
orig |
Forecast origin. The default is 0, implying that the origin is the last observation |
h |
Forecast horizon. Default is h=1, 1-step ahead forecast |
For a model, which is the output of the command Kronfit, the command computes forecasts of the model starting at the forecast origin. !-step to h-step ahead forecasts are computed.
pred |
Forecasts |
se.err |
Standard errors of the forecasts |
orig |
Return the forecast origin |
Ruey S. Tsay
Tsay (2014). Multivariate Time Series Analysis with R and Financial Applications, John Wiley, Hoboken, New Jersey
For a given set of Kronecker indices, the program specifies a VARMA model. It gives details of parameter specification.
Kronspec(kdx, output = TRUE)Kronspec(kdx, output = TRUE)
kdx |
A vector of Kronecker indices |
output |
A logical switch to control output. Default is with output. |
PhiID |
Specification of the AR matrix polynomial. 0 denotes zero parameter, 1 denotes fixing parameter to 1, and 2 denotes the parameter requires estimation |
ThetaID |
Specification of the MA matrix polynomial |
Ruey S. Tsay
Tsay (2014, Chapter 4)
kdx=c(2,1,1) m1=Kronspec(kdx) names(m1)kdx=c(2,1,1) m1=Kronspec(kdx) names(m1)
Perform tests to check the conditional heteroscedasticity in a vector time series
MarchTest(zt, lag = 10)MarchTest(zt, lag = 10)
zt |
a nT-by-k data matrix of a k-dimensional financial time series, each column contains a series. |
lag |
The number of lags of cross-correlation matrices used in the tests |
Several tests are used. First, the vector series zt is transformed into rt = [t(zt) perform the test. The second test is based on the ranks of the transformed rt series. The third test is the multivariate Ljung-Box statistics for the squared vector series zt^2. The fourth test is the multivariate Ljung-Box statistics applied to the 5-percent trimmed series of the transformed series rt.
Various test statistics and their p-values
Ruey S. Tsay
Tsay (2014, Chapter 7). Multivariate Time Series Analysis with R and Financial Applications. John Wiley. Hoboken, NJ.
zt=matrix(rnorm(600),200,3) MarchTest(zt) function (zt, lag = 10) { if (!is.matrix(zt)) zt = as.matrix(zt) nT = dim(zt)[1] k = dim(zt)[2] C0 = cov(zt) zt1 = scale(zt, center = TRUE, scale = FALSE) C0iv = solve(C0) wk = zt1 %*% C0iv wk = wk * zt1 rt2 = apply(wk, 1, sum) - k m1 = acf(rt2, lag.max = lag, plot = F) acf = m1$acf[2:(lag + 1)] c1 = c(1:lag) deno = rep(nT, lag) - c1 Q = sum(acf^2/deno) * nT * (nT + 2) pv1 = 1 - pchisq(Q, lag) cat("Q(m) of squared series(LM test): ", "\n") cat("Test statistic: ", Q, " p-value: ", pv1, "\n") rk = rank(rt2) m2 = acf(rk, lag.max = lag, plot = F) acf = m2$acf[2:(lag + 1)] mu = -(rep(nT, lag) - c(1:lag))/(nT * (nT - 1)) v1 = rep(5 * nT^4, lag) - (5 * c(1:lag) + 9) * nT^3 + 9 * (c(1:lag) - 2) * nT^2 + 2 * c(1:lag) * (5 * c(1:lag) + 8) * nT + 16 * c(1:lag)^2 v1 = v1/(5 * (nT - 1)^2 * nT^2 * (nT + 1)) QR = sum((acf - mu)^2/v1) pv2 = 1 - pchisq(QR, lag) cat("Rank-based Test: ", "\n") cat("Test statistic: ", QR, " p-value: ", pv2, "\n") cat("Q_k(m) of squared series: ", "\n") x = zt^2 g0 = var(x) ginv = solve(g0) qm = 0 df = 0 for (i in 1:lag) { x1 = x[(i + 1):nT, ] x2 = x[1:(nT - i), ] g = cov(x1, x2) g = g * (nT - i - 1)/(nT - 1) h = t(g) %*% ginv %*% g %*% ginv qm = qm + nT * nT * sum(diag(h))/(nT - i) df = df + k * k } pv3 = 1 - pchisq(qm, df) cat("Test statistic: ", qm, " p-value: ", pv3, "\n") cut1 = quantile(rt2, 0.95) idx = c(1:nT)[rt2 <= cut1] x = zt[idx, ]^2 eT = length(idx) g0 = var(x) ginv = solve(g0) qm = 0 df = 0 for (i in 1:lag) { x1 = x[(i + 1):eT, ] x2 = x[1:(eT - i), ] g = cov(x1, x2) g = g * (eT - i - 1)/(eT - 1) h = t(g) %*% ginv %*% g %*% ginv qm = qm + eT * eT * sum(diag(h))/(eT - i) df = df + k * k } pv4 = 1 - pchisq(qm, df) cat("Robust Test(5%) : ", qm, " p-value: ", pv4, "\n") }zt=matrix(rnorm(600),200,3) MarchTest(zt) function (zt, lag = 10) { if (!is.matrix(zt)) zt = as.matrix(zt) nT = dim(zt)[1] k = dim(zt)[2] C0 = cov(zt) zt1 = scale(zt, center = TRUE, scale = FALSE) C0iv = solve(C0) wk = zt1 %*% C0iv wk = wk * zt1 rt2 = apply(wk, 1, sum) - k m1 = acf(rt2, lag.max = lag, plot = F) acf = m1$acf[2:(lag + 1)] c1 = c(1:lag) deno = rep(nT, lag) - c1 Q = sum(acf^2/deno) * nT * (nT + 2) pv1 = 1 - pchisq(Q, lag) cat("Q(m) of squared series(LM test): ", "\n") cat("Test statistic: ", Q, " p-value: ", pv1, "\n") rk = rank(rt2) m2 = acf(rk, lag.max = lag, plot = F) acf = m2$acf[2:(lag + 1)] mu = -(rep(nT, lag) - c(1:lag))/(nT * (nT - 1)) v1 = rep(5 * nT^4, lag) - (5 * c(1:lag) + 9) * nT^3 + 9 * (c(1:lag) - 2) * nT^2 + 2 * c(1:lag) * (5 * c(1:lag) + 8) * nT + 16 * c(1:lag)^2 v1 = v1/(5 * (nT - 1)^2 * nT^2 * (nT + 1)) QR = sum((acf - mu)^2/v1) pv2 = 1 - pchisq(QR, lag) cat("Rank-based Test: ", "\n") cat("Test statistic: ", QR, " p-value: ", pv2, "\n") cat("Q_k(m) of squared series: ", "\n") x = zt^2 g0 = var(x) ginv = solve(g0) qm = 0 df = 0 for (i in 1:lag) { x1 = x[(i + 1):nT, ] x2 = x[1:(nT - i), ] g = cov(x1, x2) g = g * (nT - i - 1)/(nT - 1) h = t(g) %*% ginv %*% g %*% ginv qm = qm + nT * nT * sum(diag(h))/(nT - i) df = df + k * k } pv3 = 1 - pchisq(qm, df) cat("Test statistic: ", qm, " p-value: ", pv3, "\n") cut1 = quantile(rt2, 0.95) idx = c(1:nT)[rt2 <= cut1] x = zt[idx, ]^2 eT = length(idx) g0 = var(x) ginv = solve(g0) qm = 0 df = 0 for (i in 1:lag) { x1 = x[(i + 1):eT, ] x2 = x[1:(eT - i), ] g = cov(x1, x2) g = g * (eT - i - 1)/(eT - 1) h = t(g) %*% ginv %*% g %*% ginv qm = qm + eT * eT * sum(diag(h))/(eT - i) df = df + k * k } pv4 = 1 - pchisq(qm, df) cat("Robust Test(5%) : ", qm, " p-value: ", pv4, "\n") }
Apply four portmanteau test statistics to check the validity of a fitted multivariate volatility model
MCHdiag(at, Sigma.t, m = 10)MCHdiag(at, Sigma.t, m = 10)
at |
A T-by-k matrix of residuals for a k-dimensional asset return series |
Sigma.t |
The fitted volatility matrices. The dimension is T-by-k^2 matrix |
m |
The number of lags used in the tests. Default is 10. |
The four test statistics are given in Tsay (2014, Chapter 7)
Four test statistics and their p-values
Ruey S. Tsay
Tsay (2014, Chapter 7). Multivariate Time Series Analysis with R and Financial Applications. John Wiley. Hoboken, NJ.
Use Cholesky decomposition to obtain multivariate volatility models
MCholV(rtn, size = 36, lambda = 0.96, p = 0)MCholV(rtn, size = 36, lambda = 0.96, p = 0)
rtn |
A T-by-k data matrix of a k-dimensional asset return series. |
size |
The initial sample size used to start recursive least squares estimation |
lambda |
The exponential smoothing parameter. Default is 0.96. |
p |
VAR order for the mean equation. Default is 0. |
Use recursive least squares to perform the time-varying Cholesky decomposition. The least squares estimates are then smoothed via the exponentially weighted moving-average method with decaying rate 0.96. University GARCH(1,1) model is used for the innovations of each linear regression.
betat |
Recursive least squares estimates of the linear transformations in Cholesky decomposition |
bt |
The transformation residual series |
Vol |
The volatility series of individual innovations |
Sigma.t |
Volatility matrices |
Ruey S. Tsay
Tsay (2014, Chapter 7)
fGarch
Fit a multivariate multiple linear regression model via the least squares method
Mlm(y, z, constant=TRUE, output=TRUE)Mlm(y, z, constant=TRUE, output=TRUE)
y |
data matrix of dependent variable. Each column contains one variable. |
z |
data matrix of the explanatory variables. Each column contains one variable. |
constant |
A logical switch for including the constant term |
output |
A logical switch to print the output |
beta |
coefficient matrix |
se.beta |
standard errors of the coefficient matrix |
residuals |
The residual series |
sigma |
Residual covariance matrix |
Ruey S. Tsay
Computes the multivariate Ljung-Box statistics for cross-correlation matrices
mq(x, lag = 24, adj = 0)mq(x, lag = 24, adj = 0)
x |
The data matrix of a vector time series or residual series of a fitted multivariate model. |
lag |
The number of cross-correlation matrices used. Default is 24. |
adj |
Adjustment for the degrees of freedom for the Ljung-Box statistics. This is used for residual series. Default is zero. |
Computes the multivariate Ljung-Box statistics and their p-values. For model checking, the subcommand adj can be used to adjust the degrees of freedom of the Chi-square statistics.
The multivariate Q-statistics and their p-values. Also, it provides a plot of the p-values.
Ruey S. Tsay
Tsay (2014). Multivariate Time Series Analysis with R and Financial Applications. John Wiley. Hoboken, NJ.
x=matrix(rnorm(1500),500,3) mq(x)x=matrix(rnorm(1500),500,3) mq(x)
Compute the symmetric square root of a positive definite matrix
msqrt(M)msqrt(M)
M |
A positive definite matrix |
Use spectral decomposition to compute the square root of a positive definite matrix
mtxsqrt |
The square root matrix |
invsqrt |
The inverse of the square root matrix |
This command is used in some of the MTS functions.
Ruey S. Tsay
m=matrix(c(1,0.2,0.2,1),2,2) m1=msqrt(m) names(m1)m=matrix(c(1,0.2,0.2,1),2,2) m1=msqrt(m) names(m1)
Fits a t-copula to a k-dimensional standardized return series. The correlation matrices are parameterized by angles and the angles evolve over time via a DCC-type equation.
mtCopula(rt, g1, g2, grp = NULL, th0 = NULL, m = 0, include.th0 = TRUE, ub=c(0.95,0.049999))mtCopula(rt, g1, g2, grp = NULL, th0 = NULL, m = 0, include.th0 = TRUE, ub=c(0.95,0.049999))
rt |
A T-by-k data matrix of k standardized time series (after univariate volatility modeling) |
g1 |
lamda1 parameter, nonnegative and less than 1 |
g2 |
lambda2 parameter, nonnegative and satisfying lambda1+lambda2 < 1. |
grp |
a vector to indicate the number of assets divided into groups. Default means each individual asset forms a group. |
th0 |
initial estimate of theta0 |
m |
number of lags used to estimate the local theta-angles |
include.th0 |
A logical switch to include theta0 in estimation. Default is to include. |
ub |
Upper bound of parameters |
estimates |
Parameter estimates |
Hessian |
Hessian matrix |
rho.t |
Cross-correlation matrices |
theta.t |
Time-varying angel matrices |
Ruey S. Tsay
Tsay (2014, Chapter 7). Multivariate Time Series Analysis with R and Financial Applications. John Wiley. Hoboken, NJ.
MTS Internal Functions
These are not to be called by the user.
Performs model checking for a fitted multivariate time series model, including residual cross-correlation matrices, multivariate Ljung-Box tests for residuals, and residual plots
MTSdiag(model, gof = 24, adj = 0, level = F)MTSdiag(model, gof = 24, adj = 0, level = F)
model |
A fitted multivariate time series model |
gof |
The number of lags of residual cross-correlation matrices used in the tests |
adj |
The adjustment for degrees of freedom of Ljung-Box statistics. Typically, the number of fitted coefficients of the model. Default is zero. |
level |
Logical switch for printing residual cross-correlation matrices |
Various test statistics, their p-values, and residual plots.
Ruey S Tsay
phi=matrix(c(0.2,-0.6,0.3,1.1),2,2); sigma=diag(2) m1=VARMAsim(200,arlags=c(1),phi=phi,sigma=sigma) zt=m1$series m2=VAR(zt,1,include.mean=FALSE) MTSdiag(m2)phi=matrix(c(0.2,-0.6,0.3,1.1),2,2); sigma=diag(2) m1=VARMAsim(200,arlags=c(1),phi=phi,sigma=sigma) zt=m1$series m2=VAR(zt,1,include.mean=FALSE) MTSdiag(m2)
Provides time plots of a vector time series
MTSplot(data, caltime = NULL)MTSplot(data, caltime = NULL)
data |
data matrix of a vector time series |
caltime |
Calendar time. Default is NULL, that is, using time index |
Provides time plots of a vector time series. The output frame depends on the dimension of the time series
Time plots of vector time series
Ruey S. Tsay
xt=matrix(rnorm(1500),500,3) MTSplot(xt)xt=matrix(rnorm(1500),500,3) MTSplot(xt)
Compute the product of two polynomial matrices
Mtxprod(Mtx, sMtx, p, P)Mtxprod(Mtx, sMtx, p, P)
Mtx |
The coefficient matrix of a regular polynomial matrix |
sMtx |
The coefficient matrix of a seasonal polynomial matrix |
p |
Degree of the regular polynomial matrix |
P |
Degree of the seasonal polynomial matrix |
Coefficient matrix of the product. The product is in the form reg-AR * sAR, etc.
Ruey S. Tsay
Compute the product of two polynomial matrices
Mtxprod1(Mtx, sMtx, p, P)Mtxprod1(Mtx, sMtx, p, P)
Mtx |
The coefficient matrix of a regular polynomial matrix |
sMtx |
The coefficient matrix of a seasonal polynomial matrix |
p |
Degree of the regular polynomial matrix. p is less than P. |
P |
Degree of the seasonal polynomial matrix |
This polynomial product is used in seasonal VARMA modeling to check the multiplicative nature between the regular and seasonal polynomial matrices
Coefficient matrix of the product. The product matrix is in the form sAR * reg-AR, etc.
Ruey S. Tsay
Compute the Pi-weight matrices of a VARMA model
PIwgt(Phi = NULL, Theta = NULL, lag = 12, plot = TRUE)PIwgt(Phi = NULL, Theta = NULL, lag = 12, plot = TRUE)
Phi |
A k-by-kp matrix of VAR coefficients in the form [Phi1, Phi2, Phi3, ..., Phip] |
Theta |
A k-by-kq matrix of VMA coefficients in the form [Theta1, Theta2, ..., Thetaq] |
lag |
The number of Pi-weight matrices to be computed. |
plot |
A logical switch to plot the Pi-weight matrices |
The Pi-weight matrices for a VARMA model is Pi(B) = inverse(Theta(B)) times Phi(B).
pi.weight |
The matrix of Pi-weight coefficient |
Ruey S. Tsay
Tsay (2014, Chapters 2 and 3). Multivariate Time Series Analysis with R and Financial Applications. John Wiley. Hoboken, NJ.
PSIwgt
Phi1=matrix(0,2,2); Phi2=matrix(c(0.2,-0.6,0.3,1.1),2,2) Theta1=diag(c(-0.5,-0.4)) Phi=cbind(Phi1,Phi2) m1=PIwgt(Phi=Phi,Theta=Theta1) names(m1)Phi1=matrix(0,2,2); Phi2=matrix(c(0.2,-0.6,0.3,1.1),2,2) Theta1=diag(c(-0.5,-0.4)) Phi=cbind(Phi1,Phi2) m1=PIwgt(Phi=Phi,Theta=Theta1) names(m1)
Computes the psi-weight matrices of a VARMA model
PSIwgt(Phi = NULL, Theta = NULL, lag = 12, plot = TRUE, output = FALSE)PSIwgt(Phi = NULL, Theta = NULL, lag = 12, plot = TRUE, output = FALSE)
Phi |
A k-by-kp matrix of VAR coefficient matrix. Phi=[Phi1, Phi1, ..., Phip] |
Theta |
A k-by-kq matrix of VMA coefficient matrix. Theta=[Theta1, Theta2, ..., Thetaq] |
lag |
The number of psi-weight matrices to be computed. Deafult is 12. |
plot |
A logical switch to control plotting of the psi-weights. |
output |
A logical switch to control the output. |
psi.weight |
Psi-weight matrices |
irf |
Impulse response cofficient matrices |
Ruey S. Tsay
Tsay (2014, Chapter 3). Multivariate Time Series Analysis with R and Financial Applications. John Wiley. Hoboken, NJ.
phi=matrix(c(0.2,-0.6,0.3,1.1),2,2) theta=matrix(c(-0.5,0.2,0.0,-0.6),2,2) m1=PSIwgt(Phi=phi,Theta=theta)phi=matrix(c(0.2,-0.6,0.3,1.1),2,2) theta=matrix(c(-0.5,0.2,0.0,-0.6),2,2) m1=PSIwgt(Phi=phi,Theta=theta)
Quarterly real gross domestic products of United Kingdom, Canada, and the United States from the first quarter of 1980 to the second quarter of 2011. The UK and CA data were originally from OECD and the US data from the Federal Reserve Bank of St Louis.
A 2-d list containing 126x5 observations. The data set consists of 5 columns: name, year, month, UK, CA, and US.
The data were downloaded from the FRED of the Federal Reserve Bank of St Louis. The UK data were in millions of chained 2006 Pounds, the CA data were in millions of chained 2002 Canadian dollars, and the US data were in millions of chained 2005 dollars.
Refining an estimated ECM VAR model by setting insignificant estimates to zero
refECMvar(m1, thres = 1)refECMvar(m1, thres = 1)
m1 |
An object of the ECMvar command or the refECMvar command |
thres |
Threshold for individual t-ratio. The default is 1. |
Set simultaneously all estimates with t-ratio less than the threshold to zero (in modulus).
Constrained estimation results of a ECM VAR model
Ruey S. Tsay
Tsay (2014, Chapter 5). Multivariate Time Series Analysis with R and Financial Applications. John Wiley. Hoboken, NJ.
Performs constrained least squares estimation of a ECM VAR model with known co-integrated processes
refECMvar1(m1, thres = 1)refECMvar1(m1, thres = 1)
m1 |
An object of the ECMvar1 command or the refECMvar1 command |
thres |
Threshold for individual t-ratio. Default is 1. |
Setting all estimates with t-ration less than the threshold, in absoluate value, to zero simultaneously.
Constrained estimation results of an ECM VAR model
Ruey S. Tsay
Tsay (2014, Chapter 5). Multivariate Time Series Analysis with R and Financial Applications. John Wiley. Hoboken, NJ.
ECMvar1, refECMvar
This program performs model simplification of a fitted VARMA model via the Kronecker index approach
refKronfit(model, thres = 1)refKronfit(model, thres = 1)
model |
The name of a model from the command Kronfit or refKronfit |
thres |
A threshold for t-ratio of individual parameter estimate. The default is 1. |
For a given threshold, the program sets a parameter to zero if its t-ratio (in modulus) is less than the threshold.
Same as those of the command Kronfit.
Ruey S. Tsay
Tsay (2014, Chapter 4). Multivariate Time Series Analysis with R and Financial Applications. John Wiley. Hoboken, NJ.
Kronfit
Refines a fitted REGts by setting simultaneously parameters with t-ratios less than the threshold (in modulus) to zero
refREGts(m1, thres = 1)refREGts(m1, thres = 1)
m1 |
An output object from the REGts command or refREGts command |
thres |
Threshold value for individual t-ratio. Default is 1. |
The same as those of the command REGts.
Ruey S. Tsay
Tsay (2014, Chapter 6). Multivariate Time Series Analysis with R and Financial Applications. John Wiley. Hoboken, NJ.
refVAR, refVARMA
Refine estimation of a VARMA model specified via the SCM approach by removing insignificant parameters
refSCMfit(model, thres = 1)refSCMfit(model, thres = 1)
model |
Name of the model from the SCMfit command or the refSCMfit command |
thres |
Threshold for the t-ratio of individual coefficient. Default is 1. |
The same as those of the command SCMfit.
Ruey S. Tsay
Tsay (2014, Chapter 4)
SCMfit
Refines a fitted seasonal VARMA model by setting insignificant estimates to zero
refsVARMA(model, thres = 0.8)refsVARMA(model, thres = 0.8)
model |
An output object of the sVARMA command or the refsVARMA command |
thres |
Threshold for individual t-ratio. Default is 0.8. |
The command removes simultaneously all parameters with t-ratio less than the threshold in modulus.
The same as those of the command sVARMA
Ruey S. Tsay
Tsay (2014, Chapter 6)
sVARMA
Refine a fitted VAR model by removing simultaneously insignificant parameters
refVAR(model, fixed = NULL, thres = 1)refVAR(model, fixed = NULL, thres = 1)
model |
An output object of the command VAR or the refVAR command |
fixed |
A logical matrix for VAR polynomial structure |
thres |
Threshold used to set parameter to zero. Default is 1. |
Refine a VAR fitting by setting all estimates with t-ratio less than the threshold (in modulus) to zero.
The same as those of the command VAR
Ruey S. Tsay
Tsay (2014, Chapter 2)
VAR
data("mts-examples",package="MTS") gdp=log(qgdp[,3:5]) zt=diffM(gdp) m1=VAR(zt,3) m2=refVAR(m1,thres=1.0) names(m2)data("mts-examples",package="MTS") gdp=log(qgdp[,3:5]) zt=diffM(gdp) m1=VAR(zt,3) m2=refVAR(m1,thres=1.0) names(m2)
Refines a fitted VARMA model by setting insignificant estimates to zero
refVARMA(model, thres = 1.5)refVARMA(model, thres = 1.5)
model |
An output object from the command VARMA or the command refVARMA |
thres |
A threshold value for individual t-ratio of the estimates. |
The program simultaneously sets estimates with t-ratios less than the threshold (in modulus) to zero.
The same as those of the command VARMA.
Ruey S. Tsay
Tsay (2014, Chapter 3). Multivariate Time Series Analysis with R and Financial Applications. John Wiley. Hoboken, NJ.
VARMA
Refine a fitted VARX model by setting insignificant parameters to zero
refVARX(m1, thres = 1)refVARX(m1, thres = 1)
m1 |
An output object of the VARX command or the refVARX command |
thres |
A threshold for the individual t-ratio. Default is 1. |
The program sets simultaneously all estimates with t-ratio less than threshold (in modulus) to zero and re-estimate the VARX model.
The same as those of the command VARX.
Ruey S. Tsay
Tsay (2014, Chapter 6). Multivariate Time Series Analysis with R and Financial Applications. John Wiley. Hoboken, NJ.
VARX
Refines a fitted VMA model by setting insignificant parameters to zero
refVMA(model, thres = 1)refVMA(model, thres = 1)
model |
An output object from the command VMA or the refVMA command |
thres |
A threshold for individual t-ratio of parameter estimate. Default is 1. |
The program simultaneously sets all estimates with t-ratios less than the threshold (in modulus) to zero.
The same as those of the command VMA.
Ruey S. Tsay
Tsay (2014, Chapter 3). Multivariate Time Series Analysis with R and Financial Applications. John Wiley. Hoboken, NJ.
VMA
Refines a fitted VMA model via the VMAe command by setting insignificant parameters to zero
refVMAe(model, thres = 1)refVMAe(model, thres = 1)
model |
An output object of the command VMAe or the command refVMAe itself |
thres |
A threshold for individual t-ratio of parameter estimates. Default is 1. |
The program sets simultaneously all estimates with t-ratios less than the threshold (in modulus) to zero.
The same as those of the command VMAe.
Ruey S. Tsay
Tsay (2014, Chapter 3). Multivariate Time Series Analysis with R and Financial Applications. John Wiley. Hoboken, NJ.
VMAe, refVMA
Perform the maximum likelihood estimation of a multivariate linear regression model with time-series errors
REGts(zt, p, xt, include.mean = T, fixed = NULL, par = NULL, se.par = NULL, details = F)REGts(zt, p, xt, include.mean = T, fixed = NULL, par = NULL, se.par = NULL, details = F)
zt |
A T-by-k data matrix of a k-dimensional time series |
p |
The VAR order |
xt |
A T-by-v data matrix of independent variables, where v denotes the number of independent variables (excluding constant 1). |
include.mean |
A logical switch to include the constant term. Default is to include the constant term. |
fixed |
A logical matrix used to set parameters to zero |
par |
Initial parameter estimates of the beta coefficients, if any. |
se.par |
Standard errors of the parameters in par, if any. |
details |
A logical switch to control the output |
Perform the maximum likelihood estimation of a multivariate linear regression model with time series errors. Use multivariate linear regression to obtain initial estimates of regression coefficients if not provided
data |
The observed k-dimensional time series |
xt |
The data matrix of independent variables |
aror |
VAR order |
include.mean |
Logical switch for the constant vector |
Phi |
The VAR coefficients |
se.Phi |
The standard errors of Phi coefficients |
beta |
The regression coefficients |
se.beta |
The standard errors of beta |
residuals |
The residual series |
Sigma |
Residual covariance matrix |
coef |
Parameter estimates, to be used in model simplification. |
se.coef |
Standard errors of parameter estimates |
Ruey S. Tsay
Tsay (2014, Chapter 6). Multivariate Time Series Analysis with R and Financial Applications. John Wiley. Hoboken NJ.
Perform prediction of a REGts model
REGtspred(model,newxt,h=1,orig=0)REGtspred(model,newxt,h=1,orig=0)
model |
An output of the REGts command for a vector time series with exogenous variables |
newxt |
The new data matrix of the exogenous variables. It must be of the same dimension as the original exogenous variables and of length at least h (the forecast horizon). |
orig |
The forecast origin. The default is zero indicating that the origin is the last observation. |
h |
The forecast horizon. For a given h, it computes 1-step to h-step ahead forecasts. Default is 1. |
Perform prediction of a fitted REGts model
pred |
Forecasts |
se.err |
Standard errors of forecasts |
rmse |
Root mean squares of forecast errors |
rmse |
Root mean squared forecast errors |
orig |
Return the forecast origin |
Ruey S. Tsay
Tsay (2014). Multivariate Time Series Analysis with R and Financial Applications. John Wiley. Hoboken, NJ.
Compute recursive least squares estimation
RLS(y, x, ist = 30, xpxi = NULL, xpy0 = NULL)RLS(y, x, ist = 30, xpxi = NULL, xpy0 = NULL)
y |
data of dependent variable |
x |
data matrix of regressors |
ist |
initial number of data points used to start the estimation |
xpxi |
Inverse of the X'X matrix |
xpy0 |
Initial value of X'y. |
beta |
Time-varying regression coefficient estimates |
resi |
The residual series of recursive least squares estimation |
This function is used internally, but can also be used as a command.
Ruey S. Tsay
Compute the sample constrained correlation matrices
SCCor(rt,end,span,grp)SCCor(rt,end,span,grp)
rt |
A T-by-k data matrix of a k-dimensional time series |
end |
The time index of the last data point to be used in computing the sample correlations. |
span |
The size of the data span used to compute the correlations. |
grp |
A vector of group sizes. The time series in the same group are pooled to compute the correlation matrix. |
unconCor |
Un-constrained sample correlation matrix |
conCor |
Constrained sample correlation matrix |
This is an internal function, not intended to be a general command
Ruey S. Tsay
rt=matrix(rnorm(1000),200,5) grp=c(3,2) m1=SCCor(rt,200,200,grp) m1$unconCor m1$conCorrt=matrix(rnorm(1000),200,5) grp=c(3,2) m1=SCCor(rt,200,200,grp) m1$unconCor m1$conCor
Perform estimation of a VARMA model specified via the SCM approach
SCMfit(da, scms, Tdx, include.mean = T, fixed = NULL, prelim = F, details = F, thres = 1, ref = 0, SCMpar=NULL, seSCMpar=NULL)SCMfit(da, scms, Tdx, include.mean = T, fixed = NULL, prelim = F, details = F, thres = 1, ref = 0, SCMpar=NULL, seSCMpar=NULL)
da |
The T-by-k data matrix of a k-dimensional time series |
scms |
A k-by-2 matrix of the orders of SCMs |
Tdx |
A k-dimensional vector for locating "1" of each row in the transformation matrix. |
include.mean |
A logical switch to include the mean vector. Default is to include mean vector. |
fixed |
A logical matrix to set parameters to zero |
prelim |
A logical switch for preliminary estimation. Default is false. |
details |
A logical switch to control details of output |
thres |
Threshold for individual t-ratio when setting parameters to zero. Default is 1. |
ref |
A switch to use SCMmod in model specification. |
SCMpar |
Parameter estimates of the SCM model, to be used in model refinement |
seSCMpar |
Standard errors of the parameter estimates in SCMpar |
Perform conditional maximum likelihood estimation of a VARMA model specified by the scalar component model approach, including the transformation matrix.
data |
Observed time series |
SCMs |
The specified SCMs |
Tdx |
Indicator vector for the transformation matrix. The length of Tdx is k. |
locTmtx |
Specification of estimable parameters of the transformation matrix |
locAR |
Locators for the estimable parameters of the VAR coefficients |
locMA |
Locators for the estimable parameters of the VMA coefficients |
cnst |
A logical switch to include the constant vector in the model |
coef |
The parameter estimates |
secoef |
Standard errors of the parameter estimates |
residuals |
Residual series |
Sigma |
Residual covariance matrix |
aic, bic
|
Information criteria of the fitted model |
Ph0 |
Estimates of the constant vector, if any |
Phi |
Estimates of the VAR coefficients |
Theta |
Estimates of the VMA coefficients |
Ruey S. Tsay
Tsay (2014, Chapter 4). Multivariate Time Series Analysis with R and Financial Applications. John Wiley. Hoboken, NJ.
Find the overall order of a VARMA process via the scalar component model approach
SCMid(zt, maxp = 5, maxq = 5, h = 0, crit = 0.05, output = FALSE)SCMid(zt, maxp = 5, maxq = 5, h = 0, crit = 0.05, output = FALSE)
zt |
The T-by-k data matrix of a k-dimensional time series |
maxp |
Maximum AR order entertained. Default is 5. |
maxq |
Maximum MA order entertained. Default is 5. |
h |
The additional past lags used in canonical correlation analysis. Default is 0. |
crit |
Type-I error of the chi-square tests used. |
output |
A logical switch to control the output. |
Nmtx |
The table of the numbers of zero canonical correlations |
DDmtx |
The diagonal difference table of the number of zero canonical correlations |
Ruey S. Tsay
Tsay (2014, Chapter 4). Multivariate Time Series Analysis with R and Financial Applications. John Wiley. Hoboken, NJ.
phi=matrix(c(0.2,-0.6,0.3,1.1),2,2); sigma=diag(2) m1=VARMAsim(300,arlags=c(1),phi=phi,sigma=sigma) zt=m1$series m2=SCMid(zt)phi=matrix(c(0.2,-0.6,0.3,1.1),2,2); sigma=diag(2) m1=VARMAsim(300,arlags=c(1),phi=phi,sigma=sigma) zt=m1$series m2=SCMid(zt)
Provides detailed analysis of scalar component models for a specified VARMA model. The overall model is specified by SCMid.
SCMid2(zt, maxp = 2, maxq = 2, h = 0, crit = 0.05, sseq = NULL)SCMid2(zt, maxp = 2, maxq = 2, h = 0, crit = 0.05, sseq = NULL)
zt |
The T-by-k data matrix of a k-dimensional time series |
maxp |
Maximum AR order specified. Default is 2. |
maxq |
Maximum MA order specified. Default is 2. |
h |
The additional past lags used in canonical correlation analysis. Default is zero. |
crit |
Type-I error used in testing. Default is 0.05. |
sseq |
The search sequence for SCM components. Default sequence starts with AR order. |
Tmatrix |
The transformation matrix T |
SCMorder |
The orders of SCM components |
Ruey S. Tsay
Tsay (2014, Chapter 4). Multivariate Time Series Analysis with R and Financial Applications. John Wiley. Hoboken, NJ.
SCMid
phi=matrix(c(0.2,-0.6,0.3,1.1),2,2); sigma=diag(2) m1=VARMAsim(300,arlags=c(1),phi=phi,sigma=sigma) zt=m1$series m2=SCMid2(zt) names(m2)phi=matrix(c(0.2,-0.6,0.3,1.1),2,2); sigma=diag(2) m1=VARMAsim(300,arlags=c(1),phi=phi,sigma=sigma) zt=m1$series m2=SCMid2(zt) names(m2)
For a given set of SCMs and locator of transformation matrix, the program specifies a VARMA model via SCM approach for estimation
SCMmod(order, Ivor, output)SCMmod(order, Ivor, output)
order |
A k-by-2 matrix of the orders of SCM |
Ivor |
A k-dimensional vector indicating the location of "1" for each component in the transformation matrix. |
output |
A logical switch to control output. |
The command specified estimable parameters for a VARMA model via the SCM components. In the output, "2" denotes estimation, "1" denotes fixing the value to 1, and "0" means fixing the parameter to zero.
Tmtx |
Specification of the transformation matrix T |
ARpar |
Specification of the VAR parameters |
MApar |
Specification of the VMA parameters |
Ruey S. Tsay
Tsay (2014, Chapter 4). Multivariate Time Series Analysis with R and Financial Applications. John Wiley. Hoboken, NJ.
ord=matrix(c(0,1,1,0,0,1),3,2) Ivor=c(3,1,2) m1=SCMmod(ord,Ivor,TRUE)ord=matrix(c(0,1,1,0,0,1),3,2) Ivor=c(3,1,2) m1=SCMmod(ord,Ivor,TRUE)
Performs conditional maximum likelihood estimation of a seasonal VARMA model
sVARMA(da, order, sorder, s, include.mean = T, fixed = NULL, details = F, switch = F)sVARMA(da, order, sorder, s, include.mean = T, fixed = NULL, details = F, switch = F)
da |
A T-by-k data matrix of a k-dimensional seasonal time series |
order |
Regular order (p,d,q) of the model |
sorder |
Seasonal order (P,D,Q) of the model |
s |
Seasonality. s=4 for quarterly data and s=12 for monthly series |
include.mean |
A logical switch to include the mean vector. Default is to include the mean |
fixed |
A logical matrix to set zero parameter constraints |
details |
A logical switch for output |
switch |
A logical switch to exchange the ordering of the regular and seasonal VMA factors. Default is theta(B)*Theta(B). |
Estimation of a seasonal VARMA model
data |
The data matrix of the observed k-dimensional time series |
order |
The regular order (p,d,q) |
sorder |
The seasonal order (P,D,Q) |
period |
Seasonality |
cnst |
A logical switch for the constant term |
ceof |
Parameter estimates for use in model simplification |
secoef |
Standard errors of the parameter estimates |
residuals |
Residual series |
Sigma |
Residual covariance matrix |
aic, bic
|
Information criteria of the fitted model |
regPhi |
Regular AR coefficients, if any |
seaPhi |
Seasonal AR coefficients |
regTheta |
Regular MA coefficients |
seaTheta |
Seasonal MA coefficients |
Ph0 |
The constant vector, if any |
switch |
The logical switch to change the ordering of matrix product |
Ruey S. Tsay
Tsay (2014, Chapter 6). Multivariate Time Series Analysis with R and Financial Applications. John Wiley. Hoboken, NJ.
Performs conditional maximum likelihood estimation of a seasonal VARMA model. This is the same function as sVARMA, with the likelihood function implemented in C++ for efficiency.
sVARMACpp(da, order, sorder, s, include.mean = T, fixed = NULL, details = F, switch = F)sVARMACpp(da, order, sorder, s, include.mean = T, fixed = NULL, details = F, switch = F)
da |
A T-by-k data matrix of a k-dimensional seasonal time series |
order |
Regular order (p,d,q) of the model |
sorder |
Seasonal order (P,D,Q) of the model |
s |
Seasonality. s=4 for quarterly data and s=12 for monthly series |
include.mean |
A logical switch to include the mean vector. Default is to include the mean |
fixed |
A logical matrix to set zero parameter constraints |
details |
A logical switch for output |
switch |
A logical switch to exchange the ordering of the regular and seasonal VMA factors. Default is theta(B)*Theta(B). |
Estimation of a seasonal VARMA model
data |
The data matrix of the observed k-dimensional time series |
order |
The regular order (p,d,q) |
sorder |
The seasonal order (P,D,Q) |
period |
Seasonality |
cnst |
A logical switch for the constant term |
ceof |
Parameter estimates for use in model simplification |
secoef |
Standard errors of the parameter estimates |
residuals |
Residual series |
Sigma |
Residual covariance matrix |
aic, bic
|
Information criteria of the fitted model |
regPhi |
Regular AR coefficients, if any |
seaPhi |
Seasonal AR coefficients |
regTheta |
Regular MA coefficients |
seaTheta |
Seasonal MA coefficients |
Ph0 |
The constant vector, if any |
switch |
The logical switch to change the ordering of matrix product |
Ruey S. Tsay
Tsay (2014, Chapter 6). Multivariate Time Series Analysis with R and Financial Applications. John Wiley. Hoboken, NJ.
sVARMA
Perform prediction of a seasonal VARMA model
sVARMApred(model,orig,h=1)sVARMApred(model,orig,h=1)
model |
An output of the sVARMA command |
orig |
The forecast origin. |
h |
The forecast horizon. For a given h, it computes 1-step to h-step ahead forecasts. Default is 1. |
Perform prediction of a fitted sVARMA model
data |
The original data matrix |
pred |
Forecasts |
se.err |
Standard errors of forecasts |
orig |
Return the forecast origin |
Ruey S. Tsay
Tsay (2014, chapter 6). Multivariate Time Series Analysis with R and Financial Applications. John Wiley. Hoboken, NJ.
Uses the diffusion index approach of Stock and Watson to compute out-of-sample forecasts
SWfore(y, x, orig, m)SWfore(y, x, orig, m)
y |
The scalar variable of interest |
x |
The data matrix (T-by-k) of the observed explanatory variables |
orig |
Forecast origin |
m |
The number of diffusion index used |
Performs PCA on X at the forecast origin. Then, fit a linear regression model to obtain the coefficients of prediction equation. Use the prediction equation to produce forecasts and compute forecast errors, if any. No recursive estimation is used.
coef |
Regression coefficients of the prediction equation |
yhat |
Predictions at the forecast origin |
MSE |
Mean squared errors, if available |
loadings |
Loading matrix |
DFindex |
Diffusion indices |
Ruey S. Tsay
Tsay (2014, Chapter 6). Multivariate Time Series Analysis with R and Financial Applications. John Wiley. Hoboken, NJ.
Monthly simple returns of ten U.S. stocks. The sample period is from January 2001 to December 2011. Tick symbols of the ten stocks are used as column names for the returns.
A 2-d list containing 132x11 observations.
The original data were from Center for Research in Security Prices (CRSP) of the University of Chicago. The first column denotes the dates.
Estimates a transform function model. This program does not allow rational transfer function model. It is a special case of tfm1 and tfm2.
tfm(y, x, b = 0, s = 1, p = 0, q = 0)tfm(y, x, b = 0, s = 1, p = 0, q = 0)
y |
Data vector of dependent (output) variable |
x |
Data vector of independent variable |
b |
deadtime or delay |
s |
The order of the transfer function polynomial |
p |
AR order of the disturbance |
q |
MA order of the disturbance |
The model entertained is y_t = c_0+ v(B)x_t + n_t. v(B) = 1- v1*B - ... - vs*B^s, and n_t is an ARMA(p,q) process.
coef |
Coefficient estimates of the transfer function |
se.coef |
Standard errors of the transfer function coefficients |
coef.arma |
Coefficient estimates of ARMA models |
se.arma |
Standard errors of ARMA coefficients |
nt |
The disturbance series |
residuals |
The residual series |
Ruey S. Tsay
Box, G. E. P., Jenkins, G. M., and Reinsel, G. C. (1994). Time Series Analysis: Forecasting and Control, 3rd edition, Prentice Hall, Englewood Cliffs, NJ.
Estimation of a general transfer function model. The model can only handle one input and one output.
tfm1(y, x, orderN, orderX)tfm1(y, x, orderN, orderX)
y |
Data vector of dependent variable |
x |
Data vector of input (or independent) variable |
orderN |
Order (p,d,q) of the disturbance component |
orderX |
Order (r,s,b) of the transfer function model, where r and s are the degrees of denominator and numerator polynomials and b is the delay |
Perform estimation of a general transfer function model
estimate |
Coefficient estimates |
sigma2 |
Residual variance sigma-square |
residuals |
Residual series |
varcoef |
Variance of the estimates |
Nt |
The disturbance series |
Ruey S. Tsay
Box, G. E. P., Jenkins, G. M., and Reinsel, G. C. (1994). Time Series Analysis: Forecasting and Control, 3rd edition, Prentice Hall, Englewood Cliffs, NJ.
tfm
##da=read.table("gasfur.txt") ##y=da[,2]; x=da[,1] ##m1=tfm1(y,x,orderX=c(1,2,3),orderN=c(2,0,0))##da=read.table("gasfur.txt") ##y=da[,2]; x=da[,1] ##m1=tfm1(y,x,orderX=c(1,2,3),orderN=c(2,0,0))
Estimation of a general transfer function model with two input variables. The model can handle one output and up-to 2 input variables. The time series noise can assume multiplicative seasonal ARMA models.
tfm2(y,x,x2=NULL,ct=NULL,wt=NULL,orderN=c(1,0,0),orderS=c(0,0,0), sea=12,order1=c(0,1,0),order2=c(0,-1,0))tfm2(y,x,x2=NULL,ct=NULL,wt=NULL,orderN=c(1,0,0),orderS=c(0,0,0), sea=12,order1=c(0,1,0),order2=c(0,-1,0))
y |
Data vector of dependent variable |
x |
Data vector of the first input (or independent) variable |
x2 |
Data vector of the second input variable if any |
ct |
Data vector of a given deterministic variable such as time trend, if any |
wt |
Data vector of co-integrated series between input and output variables if any |
orderN |
Order (p,d,q) of the regular ARMA part of the disturbance component |
orderS |
Order (P,D,Q) of the seasonal ARMA part of the disturbance component |
sea |
Seasonality, default is 12 for monthly data |
order1 |
Order (r,s,b) of the transfer function model of the first input variable, where r and s are the degrees of denominator and numerator polynomials and b is the delay |
order2 |
Order (r2,s2,b2) of the transfer function model of the second input variable, where 2r and s2 are the degrees of denominator and numerator polynomials and b2 is the delay |
Perform estimation of a general transfer function model with two input variables
estimate |
Coefficient estimates |
sigma2 |
Residual variance sigma-square |
residuals |
Residual series |
varcoef |
Variance of the estimates |
Nt |
The disturbance series |
rAR |
Regular AR coefficients |
rMA |
Regular MA coefficients |
sAR |
Seasonal AR coefficients |
sMA |
Seasonal MA coefficients |
omega |
Numerator coefficients of the first transfer function |
delta |
Denominator coefficients of the first transfer function |
omega2 |
Numerator coefficients of the 2nd transfer function |
delta2 |
Denominator coefficients of the 2nd transfer function |
Ruey S. Tsay
Box, G. E. P., Jenkins, G. M., and Reinsel, G. C. (1994). Time Series Analysis: Forecasting and Control, 3rd edition, Prentice Hall, Englewood Cliffs, NJ.
tfm, tfm1
Perform least squares estimation of a VAR model
VAR(x, p = 1, output = T, include.mean = T, fixed = NULL)VAR(x, p = 1, output = T, include.mean = T, fixed = NULL)
x |
A T-by-k matrix of k-dimensional time series |
p |
Order of VAR model. Default is 1. |
output |
A logical switch to control output. Default is with output. |
include.mean |
A logical switch. It is true if mean vector is estimated. |
fixed |
A logical matrix used in constrained estimation. It is used mainly in model simplification, e.g., removing insignificant estimates. |
To remove insignificant estimates, one specifies a threshold for individual t-ratio. The fixed matrix is then defined automatically to identify those parameters for removal.
data |
Observed data |
cnst |
A logical switch to include the mean constant vector |
order |
VAR order |
coef |
Coefficient matrix |
aic, bic, hq
|
Information criteria of the fitted model |
residuals |
Residuals |
secoef |
Standard errors of the coefficients to be used in model refinement |
Sigma |
Residual covariance matrix |
Phi |
AR coefficient polynomial |
Ph0 |
The constant vector |
Ruey S. Tsay
Tsay (2014, Chapter 3). Multivariate Time Series Analysis with R and Financial Applications. John Wiley. Hoboken, NJ.
refVAR command
data("mts-examples",package="MTS") gdp=log(qgdp[,3:5]) zt=diffM(gdp) m1=VAR(zt,p=2)data("mts-examples",package="MTS") gdp=log(qgdp[,3:5]) zt=diffM(gdp) m1=VAR(zt,p=2)
Performs conditional maximum likelihood estimation of a VARMA model. Multivariate Gaussian likelihood function is used.
VARMA(da, p = 0, q = 0, include.mean = T, fixed = NULL, beta=NULL, sebeta=NULL, prelim = F, details = F, thres = 2)VARMA(da, p = 0, q = 0, include.mean = T, fixed = NULL, beta=NULL, sebeta=NULL, prelim = F, details = F, thres = 2)
da |
Data matrix (T-by-k) of a k-dimensional time series with sample size T. |
p |
AR order |
q |
MA order |
include.mean |
A logical switch to control estimation of the mean vector. Default is to include the mean in estimation. |
fixed |
A logical matrix to control zero coefficients in estimation. It is mainly used by the command refVARMA. |
beta |
Parameter estimates to be used in model simplification, if needed |
sebeta |
Standard errors of parameter estimates for use in model simplification |
prelim |
A logical switch to control preliminary estimation. Default is none. |
details |
A logical switch to control the amount of output. |
thres |
A threshold used to set zero parameter constraints based on individual t-ratio. Default is 2. |
The fixed command is used for model refinement
data |
Observed data matrix |
ARorder |
VAR order |
MAorder |
VMA order |
cnst |
A logical switch to include the mean vector |
coef |
Parameter estimates |
secoef |
Standard errors of the estimates |
residuals |
Residual matrix |
Sigma |
Residual covariance matrix |
aic, bic
|
Information criteria of the fitted model |
Phi |
VAR coefficients |
Theta |
VMA coefficients |
Ph0 |
The constant vector |
Ruey S. Tsay
Tsay (2014, Chapter 3). Multivariate Time Series Analysis with R and Financial Applications. John Wiley. Hoboken, NJ.
refVARMA
phi=matrix(c(0.2,-0.6,0.3,1.1),2,2); theta=matrix(c(-0.5,0,0,-0.5),2,2) sigma=diag(2) m1=VARMAsim(300,arlags=c(1),malags=c(1),phi=phi,theta=theta,sigma=sigma) zt=m1$series m2=VARMA(zt,p=1,q=1,include.mean=FALSE)phi=matrix(c(0.2,-0.6,0.3,1.1),2,2); theta=matrix(c(-0.5,0,0,-0.5),2,2) sigma=diag(2) m1=VARMAsim(300,arlags=c(1),malags=c(1),phi=phi,theta=theta,sigma=sigma) zt=m1$series m2=VARMA(zt,p=1,q=1,include.mean=FALSE)
Uses psi-weights to compute the autocovariance matrices of a VARMA model
VARMAcov(Phi = NULL, Theta = NULL, Sigma = NULL, lag = 12, trun = 120)VARMAcov(Phi = NULL, Theta = NULL, Sigma = NULL, lag = 12, trun = 120)
Phi |
A k-by-kp matrix consisting of VAR coefficient matrices, Phi = [Phi1, Phi2, ..., Phip]. |
Theta |
A k-by-kq matrix consisting of VMA coefficient matrices, Theta = [Theta1, Theta2, ..., Thetaq] |
Sigma |
Covariance matrix of the innovations (k-by-k). |
lag |
Number of cross-covariance matrices to be computed. Default is 12. |
trun |
The lags of pis-weights used in calculation. Default is 120. |
Use psi-weight matrices to compute approximate autocovariance matrices of a VARMA model.
autocov |
Autocovariance matrices |
ccm |
Auto correlation matrices |
Ruey S. Tsay
Tsay (2014, Chapter 3). Multivariate Time Series Analysis with R and Financial Applications. John Wiley. Hoboken, NJ.
Phi=matrix(c(0.2,-0.6,0.3,1.1),2,2) Sig=matrix(c(4,1,1,1),2,2) VARMAcov(Phi=Phi,Sigma=Sig)Phi=matrix(c(0.2,-0.6,0.3,1.1),2,2) Sig=matrix(c(4,1,1,1),2,2) VARMAcov(Phi=Phi,Sigma=Sig)
Performs conditional maximum likelihood estimation of a VARMA model. Multivariate Gaussian likelihood function is used. This is the same function as VARMA, with the likelihood function implemented in C++ for efficiency.
VARMACpp(da, p = 0, q = 0, include.mean = T, fixed = NULL, beta=NULL, sebeta=NULL, prelim = F, details = F, thres = 2)VARMACpp(da, p = 0, q = 0, include.mean = T, fixed = NULL, beta=NULL, sebeta=NULL, prelim = F, details = F, thres = 2)
da |
Data matrix (T-by-k) of a k-dimensional time series with sample size T. |
p |
AR order |
q |
MA order |
include.mean |
A logical switch to control estimation of the mean vector. Default is to include the mean in estimation. |
fixed |
A logical matrix to control zero coefficients in estimation. It is mainly used by the command refVARMA. |
beta |
Parameter estimates to be used in model simplification, if needed |
sebeta |
Standard errors of parameter estimates for use in model simplification |
prelim |
A logical switch to control preliminary estimation. Default is none. |
details |
A logical switch to control the amount of output. |
thres |
A threshold used to set zero parameter constraints based on individual t-ratio. Default is 2. |
The fixed command is used for model refinement
data |
Observed data matrix |
ARorder |
VAR order |
MAorder |
VMA order |
cnst |
A logical switch to include the mean vector |
coef |
Parameter estimates |
secoef |
Standard errors of the estimates |
residuals |
Residual matrix |
Sigma |
Residual covariance matrix |
aic, bic
|
Information criteria of the fitted model |
Phi |
VAR coefficients |
Theta |
VMA coefficients |
Ph0 |
The constant vector |
Ruey S. Tsay
Tsay (2014, Chapter 3). Multivariate Time Series Analysis with R and Financial Applications. John Wiley. Hoboken, NJ.
VARMA
phi=matrix(c(0.2,-0.6,0.3,1.1),2,2); theta=matrix(c(-0.5,0,0,-0.5),2,2) sigma=diag(2) m1=VARMAsim(300,arlags=c(1),malags=c(1),phi=phi,theta=theta,sigma=sigma) zt=m1$series m2=VARMA(zt,p=1,q=1,include.mean=FALSE)phi=matrix(c(0.2,-0.6,0.3,1.1),2,2); theta=matrix(c(-0.5,0,0,-0.5),2,2) sigma=diag(2) m1=VARMAsim(300,arlags=c(1),malags=c(1),phi=phi,theta=theta,sigma=sigma) zt=m1$series m2=VARMA(zt,p=1,q=1,include.mean=FALSE)
Compute and plot the impulse response function of a given VARMA model
VARMAirf(Phi = NULL, Theta = NULL, Sigma = NULL, lag = 12, orth = TRUE)VARMAirf(Phi = NULL, Theta = NULL, Sigma = NULL, lag = 12, orth = TRUE)
Phi |
A k-by-kp matrix of VAR coefficients in the form Phi=[Phi1, Phi2, ..., Phip]. |
Theta |
A k-by-kq matrix of VMA coefficients in the form Theta=[Theta1, Theta2, ..., Thetaq] |
Sigma |
Covariance matrix (k-by-k) of the innovations. |
lag |
Number of lags of impulse response functions to be computed |
orth |
A logical switch to use orthogonal innovations. Deafult is to perform orthogonalization of the innovations. |
psi |
The Psi-weight matrices |
irf |
Impulse response functions |
Ruey S. Tsay
Tsay (2014, Chapter 3). Multivariate Time Series Analysis with R and Financial Applications. John Wiley. Hoboken, NJ.
VARMApsi command
p1=matrix(c(0.2,-0.6,0.3,1.1),2,2) th1=matrix(c(-0.5,0.2,0.0,-0.6),2,2) Sig=matrix(c(4,1,1,1),2,2) m1=VARMAirf(Phi=p1,Theta=th1,Sigma=Sig)p1=matrix(c(0.2,-0.6,0.3,1.1),2,2) th1=matrix(c(-0.5,0.2,0.0,-0.6),2,2) Sig=matrix(c(4,1,1,1),2,2) m1=VARMAirf(Phi=p1,Theta=th1,Sigma=Sig)
Compute forecasts and their associate forecast error covariances of a VARMA model
VARMApred(model, h = 1, orig = 0)VARMApred(model, h = 1, orig = 0)
model |
A fitted VARMA model |
h |
Number of steps of forecasts, i.e., forecast horizon. |
orig |
Forecast origin. Default is the end of the sample. |
pred |
Predictions |
se.err |
Standard errors of forecasts |
orig |
Forecast origin |
Ruey S. Tsay
Tsay (2014, Chapter 3). Multivariate Time Series Analysis with R and Financial Applications. John Wiley. Hoboken, NJ.
Performs simulation of a given VARMA model
VARMAsim(nobs, arlags = NULL, malags = NULL, cnst = NULL, phi = NULL, theta = NULL, skip = 200, sigma)VARMAsim(nobs, arlags = NULL, malags = NULL, cnst = NULL, phi = NULL, theta = NULL, skip = 200, sigma)
nobs |
Sample size |
arlags |
The exact lags of the VAR matrix polynomial. |
malags |
The exact lags of the VMA matrix polynomial. |
cnst |
Constant vector, Phi0 |
phi |
Matrix of VAR coefficient matrices in the order of the given arlags. |
theta |
Matrix of VMA coefficient matrices in the order of the given malags. |
skip |
The number of initial data to be omitted. Default is 200. |
sigma |
Covariance matrix (k-by-k, positive definite) of the innovations |
Use multivariate Gaussian distribution to generate random shocks. Then, generate a given VARMA model. The first skip data points were discarded.
series |
Generated series |
noises |
The noise series |
Ruey S. Tsay
Tsay (2014, Chapter 3). Multivariate Time Series Analysis with R and Financial Applications. John Wiley. Hoboken, NJ.
p1=matrix(c(0.2,-0.6,0.3,1.1),2,2) sig=matrix(c(4,0.8,0.8,1),2,2) th1=matrix(c(-0.5,0,0,-0.6),2,2) m1=VARMAsim(300,arlags=c(1),malags=c(1),phi=p1,theta=th1,sigma=sig) zt=m1$seriesp1=matrix(c(0.2,-0.6,0.3,1.1),2,2) sig=matrix(c(4,0.8,0.8,1),2,2) th1=matrix(c(-0.5,0,0,-0.6),2,2) m1=VARMAsim(300,arlags=c(1),malags=c(1),phi=p1,theta=th1,sigma=sig) zt=m1$series
Computes information criteria and the sequential Chi-square statistics for a vector autoregressive process
VARorder(x, maxp = 13, output = T)VARorder(x, maxp = 13, output = T)
x |
Data matrix of dimension T-by-k with T being the sample size and k the number of time series |
maxp |
The maximum VAR order entertained. Default is 13. |
output |
A logical switch to control the output. Default is to provide output |
For a given maxp, the command computes Akaike, Bayesian and Hannan-Quinn information criteria for various VAR models using the data from t=maxp+1 to T. It also computes the Tiao-Box sequential Chi-square statistics and their p-values.
aic |
Akaike information criterion |
bic |
Bayesian information criterion |
hq |
Hannan and Quinn information criterion |
aicor, bicor, hqor
|
Orders selected by various criteria |
Mstat |
Chi-square test statistics |
Mpv |
p-values of the Mstat |
Ruey S. Tsay
Tsay (2014). Multivariate Time Series Analysis with R and Financial Applications. John Wiley. Hoboken, NJ.
VARorderI
data("mts-examples",package="MTS") zt=diffM(log(qgdp[,3:5])) VARorder(zt,maxp=8)data("mts-examples",package="MTS") zt=diffM(log(qgdp[,3:5])) VARorder(zt,maxp=8)
This program is similar to VARorder, but it uses observations from t=p+1 to T to compute the information criteria for a given VAR(p) model.
VARorderI(x, maxp = 13, output = T)VARorderI(x, maxp = 13, output = T)
x |
A T-by-k data matrix of vector time series |
maxp |
The maximum VAR order entertained |
output |
A logical switch to control output |
For a given VAR(p) model, the program uses observations from t=p+1 to T to compute the information criteria. Therefore, different numbers of data points are used to estimate different VAR models.
aic |
Akaike information criterion |
aicor |
Order selected by AIC |
bic |
Bayesian information criterion |
bicor |
Order selected by BIC |
hq |
Hannan and Quinn information criterion |
hqor |
Order selected by hq |
Mstat |
Step-wise Chi-square statistics |
Mpv |
p-values of the M-statistics |
Ruey S Tsay
Tsay (2014)
VARorder
Computes the forecasts of a VAR model, the associated standard errors of forecasts and the mean squared errors of forecasts
VARpred(model, h = 1, orig = 0, Out.level = FALSE, output = TRUE)VARpred(model, h = 1, orig = 0, Out.level = FALSE, output = TRUE)
model |
An output object of a VAR or refVAR command |
h |
Forecast horizon, a positive integer |
orig |
Forecast origin. Default is zero meaning the forecast origin is the last data point |
Out.level |
Boolean control for details of output |
output |
Boolean control for printing forecast results |
Computes point forecasts and the associated variances of forecast errors
pred |
Point predictions |
se.err |
Standard errors of the predictions |
mse |
Mean-square errors of the predictions |
Ruey S. Tsay
Tsay (2014, Chapter 2). Multivariate Time Series Analysis with R and Financial Applications. John Wiley. Hoboken, NJ.
data("mts-examples",package="MTS") gdp=log(qgdp[,3:5]) zt=diffM(gdp) m1=VAR(zt,p=2) VARpred(m1,4)data("mts-examples",package="MTS") gdp=log(qgdp[,3:5]) zt=diffM(gdp) m1=VAR(zt,p=2) VARpred(m1,4)
Computes the psi-weight matrices of a VAR model
VARpsi(Phi, lag = 5)VARpsi(Phi, lag = 5)
Phi |
A k-by-kp matrix of VAR coefficients in the form Phi=[Phi1, Phi2, ..., Phip] |
lag |
Number of psi-weight lags |
Psi-weights of a VAR model
Ruey S. Tsay
Tsay (2014, Chapter 2). Multivariate Time Series Analysis with R and Financial Applications. John Wiley. Hoboken, NJ.
p1=matrix(c(0.2,-0.6,0.3,1.1),2,2) m1=VARpsi(p1,4) names(m1)p1=matrix(c(0.2,-0.6,0.3,1.1),2,2) m1=VARpsi(p1,4) names(m1)
This is a modified version of VAR command by allowing the users to specify which AR lags to be included in the model.
VARs(x, lags, include.mean = T, output = T, fixed = NULL)VARs(x, lags, include.mean = T, output = T, fixed = NULL)
x |
A T-by-k data matrix of k-dimensional time series with T observations |
lags |
A vector of non-zero AR lags. For instance, lags=c(1,3) denotes a VAR(3) model with Phi2 = 0. |
include.mean |
A logical switch to include the mean vector |
output |
A logical switch to control output |
fixed |
A logical matrix to fix parameters to zero. |
Performs VAR estimation by allowing certain lag coefficient matrices being zero.
data |
Observed time series data |
lags |
The selected VAR lags |
order |
The VAR order |
cnst |
A logical switch to include the mean vector |
coef |
Parameter estimates |
aic, bic
|
Information criteria of the fitted model |
residuals |
Residual series |
secoef |
Standard errors of the estimates |
Sigma |
Residual covariance matrix |
Phi |
VAR coefficient matrix |
Ph0 |
A constant vector |
Ruey S. Tsay
Tsay (2014, Chapter 2). Multivariate Time Series Analysis with R and Financial Applications. John Wiley. Hoboken, NJ.
VAR command
data("mts-examples",package="MTS") zt=log(qgdp[,3:5]) m1=VARs(zt,lags=c(1,2,4))data("mts-examples",package="MTS") zt=log(qgdp[,3:5]) m1=VARs(zt,lags=c(1,2,4))
Estimation of a VARX model
VARX(zt, p, xt = NULL, m = 0, include.mean = T, fixed = NULL, output = T)VARX(zt, p, xt = NULL, m = 0, include.mean = T, fixed = NULL, output = T)
zt |
A T-by-k data matrix of a k-dimensional time series |
p |
The VAR order |
xt |
A T-by-kx data matrix of kx exogenous variables |
m |
The number of lags of exogenous variables |
include.mean |
A logical switch to include the constant vector. Default is to include the constant. |
fixed |
A logical matrix for setting parameters to zero. |
output |
A logical switch to control output |
Performs least squares estimation of a VARX(p,s) model
data |
The observed time series |
xt |
The data matrix of explanatory variables |
aror |
VAR order |
m |
The number of lags of explanatory variables used |
Ph0 |
The constant vector |
Phi |
VAR coefficient matrix |
beta |
The regression coefficient matrix |
residuals |
Residual series |
coef |
The parameter estimates to be used in model simplification |
se.coef |
Standard errors of the parameter estimates |
include.mean |
A logical switch to include the mean vector |
Ruey S. Tsay
Tsay (2014, Chapter 6). Multivariate Time Series Analysis with R and Financial Applications. John Wiley. Hoboken, NJ.
Compute the impulse response functions and cumulative impulse response functions of a fitted VARX model
VARXirf(model,lag=12,orth=TRUE)VARXirf(model,lag=12,orth=TRUE)
model |
An output of the VARX (or refVARX) command for a vector time series with exogeneous variables |
lag |
The number of lags of the impulse response function to be computed. Default is 12. |
orth |
The control variable for using orthogonal innovations. This command applies to the impulse response functions of the VAR part only. |
Compute the impulse response functions and cumulative impulse response functions of a fitted VARX model. The impulse response function of the exogeneous variables are also given. The plots of impulse response functions are shown.
irf |
Impulse response functions of the VAR part, original innovations used |
orthirf |
Impulse response functions of the VAR part using orthogonal innovations |
irfX |
Impulse response function of the exogenous variables |
Ruey S. Tsay
Tsay (2014). Multivariate Time Series Analysis with R and Financial Applications. John Wiley. Hoboken, NJ.
Specifies the orders of a VARX model, including AR order and the number of lags of exogenous variables
VARXorder(x, exog, maxp = 13, maxm = 3, output = T)VARXorder(x, exog, maxp = 13, maxm = 3, output = T)
x |
A T-by-k data matrix of a k-dimensional time series |
exog |
A T-by-v data matrix of exogenous variables |
maxp |
The maximum VAR order entertained |
maxm |
The maximum lags of exogenous variables entertained |
output |
A logical switch to control output |
Computes the information criteria of a VARX process
aic |
Akaike information criterion |
aicor |
Order selected by AIC |
bic |
Bayesian information criterion |
bicor |
Order selected by BIC |
hq |
Hannan and Quinn information criterion |
hqor |
Order selected by hq |
Ruey S. Tsay
Tsay (2014, Chapter 6). Multivariate Time Series Analysis with R and Financial Applications. John Wiley. Hoboken, NJ.
Computes point forecasts of a VARX model. The values of exogenous variables must be given.
VARXpred(m1, newxt = NULL, hstep = 1, orig = 0)VARXpred(m1, newxt = NULL, hstep = 1, orig = 0)
m1 |
An output object of VARX or refVARX command |
newxt |
The data matrix of exogenous variables needed in forecasts. |
hstep |
Forecast horizon |
orig |
Forecast origin. Default is 0, meaning the last data point. |
Uses the provided exogenous variables and the model to compute forecasts
Point forecasts and their standard errors
Ruey S. Tsay
Tsay (2014, Chapter 6). Multivariate Time Series Analysis with R and Financial Applications. John Wiley. Hoboken, NJ.
Obtain the half-stacking vector of a symmetric matrix
Vech(mtx)Vech(mtx)
mtx |
A symmetric matrix |
Stacking a matrix into a vector using data on and below the diagonal.
a vector consisting of stacked elements of a symmetric matrix
Ruey S. Tsay
m1=matrix(c(1:9),3,3) m2=(m1+t(m1))/2 v1=Vech(m2)m1=matrix(c(1:9),3,3) m2=(m1+t(m1))/2 v1=Vech(m2)
Restores the symmetric matrix from the Vech command
VechM(vec)VechM(vec)
vec |
A vector representing the half-stacking of a symmetric matrix |
This command re-construct a symmetric matrix from output of the Vech command
A symmetric matrix
Ruey S. Tsay
Tsay (2014, Appendix A)
Vech
v1=c(2,1,3) m1=VechM(v1) m1v1=c(2,1,3) m1=VechM(v1) m1
Performs VMA estimation using the conditional multivariate Gaussian likelihood function
VMA(da, q = 1, include.mean = T, fixed = NULL, beta=NULL, sebeta=NULL, prelim = F, details = F, thres = 2)VMA(da, q = 1, include.mean = T, fixed = NULL, beta=NULL, sebeta=NULL, prelim = F, details = F, thres = 2)
da |
Data matrix of a k-dimensional VMA process with each column containing one time series |
q |
The order of VMA model |
include.mean |
A logical switch to include the mean vector. The default is to include the mean vector in estimation. |
fixed |
A logical matrix used to fix parameter to zero |
beta |
Parameter estimates for use in model simplification |
sebeta |
Standard errors of parameter estimates for use in model simplification |
prelim |
A logical switch to select parameters to be included in estimation |
details |
A logical switch to control the amount of output |
thres |
Threshold for t-ratio used to fix parameter to zero. Default is 2. |
data |
The data of the observed time series |
MAorder |
The VMA order |
cnst |
A logical switch to include the mean vector |
coef |
Parameter estimates |
secoef |
Standard errors of the parameter estimates |
residuals |
Residual series |
Sigma |
Residual covariance matrix |
Theta |
The VAR coefficient matrix |
mu |
The constant vector |
aic, bic
|
The information criteria of the fitted model |
Ruey S. Tsay
Tsay (2014, Chapter 3).
theta=matrix(c(0.5,0.4,0,0.6),2,2); sigma=diag(2) m1=VARMAsim(200,malags=c(1),theta=theta,sigma=sigma) zt=m1$series m2=VMA(zt,q=1,include.mean=FALSE)theta=matrix(c(0.5,0.4,0,0.6),2,2); sigma=diag(2) m1=VARMAsim(200,malags=c(1),theta=theta,sigma=sigma) zt=m1$series m2=VMA(zt,q=1,include.mean=FALSE)
Performs VMA estimation using the conditional multivariate Gaussian likelihood function. This is the same function as VMA, with the likelihood function implemented in C++ for efficiency.
VMACpp(da, q = 1, include.mean = T, fixed = NULL, beta=NULL, sebeta=NULL, prelim = F, details = F, thres = 2)VMACpp(da, q = 1, include.mean = T, fixed = NULL, beta=NULL, sebeta=NULL, prelim = F, details = F, thres = 2)
da |
Data matrix of a k-dimensional VMA process with each column containing one time series |
q |
The order of VMA model |
include.mean |
A logical switch to include the mean vector. The default is to include the mean vector in estimation. |
fixed |
A logical matrix used to fix parameter to zero |
beta |
Parameter estimates for use in model simplification |
sebeta |
Standard errors of parameter estimates for use in model simplification |
prelim |
A logical switch to select parameters to be included in estimation |
details |
A logical switch to control the amount of output |
thres |
Threshold for t-ratio used to fix parameter to zero. Default is 2. |
data |
The data of the observed time series |
MAorder |
The VMA order |
cnst |
A logical switch to include the mean vector |
coef |
Parameter estimates |
secoef |
Standard errors of the parameter estimates |
residuals |
Residual series |
Sigma |
Residual covariance matrix |
Theta |
The VAR coefficient matrix |
mu |
The constant vector |
aic, bic
|
The information criteria of the fitted model |
Ruey S. Tsay
Tsay (2014, Chapter 3).
VMA
theta=matrix(c(0.5,0.4,0,0.6),2,2); sigma=diag(2) m1=VARMAsim(200,malags=c(1),theta=theta,sigma=sigma) zt=m1$series m2=VMACpp(zt,q=1,include.mean=FALSE)theta=matrix(c(0.5,0.4,0,0.6),2,2); sigma=diag(2) m1=VARMAsim(200,malags=c(1),theta=theta,sigma=sigma) zt=m1$series m2=VMACpp(zt,q=1,include.mean=FALSE)
Estimation of a VMA(q) model using the exact likelihood method. Multivariate Gaussian likelihood function is used.
VMAe(da, q = 1, include.mean = T, coef0 = NULL, secoef0 = NULL, fixed = NULL, prelim = F, details = F, thres = 2)VMAe(da, q = 1, include.mean = T, coef0 = NULL, secoef0 = NULL, fixed = NULL, prelim = F, details = F, thres = 2)
da |
Data matrix (T-by-k) for a k-dimensional VMA process |
q |
The order of a VMA model |
include.mean |
A logical switch to include the mean vector in estimation. Default is to include the mean vector. |
coef0 |
Initial estimates of the coefficients used mainly in model refinement |
secoef0 |
Standard errors of the initial estimates |
fixed |
A logical matrix to put zero parameter constraints |
prelim |
A logical switch for preliminary estimation |
details |
A logical switch to control output in estimation |
thres |
The threshold value for zero parameter constraints |
data |
The observed time series |
MAorder |
The VMA order |
cnst |
A logical switch to include the mean vector |
coef |
Parameter estimates |
secoef |
Standard errors of parameter estimates |
residuals |
Residual series |
Sigma |
Residual covariance matrix |
Theta |
VMA coefficient matrix |
mu |
The mean vector |
aic, bic
|
The information criteria of the fitted model |
Ruey S. Tsay
Tsay (2014). Multivariate Time Series Analysis with R and Financial Applications. John Wiley. Hoboken, NJ.
VMA
Performs multivariate Ljung-Box tests to specify the order of a VMA process
VMAorder(x, lag = 20)VMAorder(x, lag = 20)
x |
Data matrix of the observed k-dimensional time series. Each column represents a time series. |
lag |
The maximum VMA order entertained. Default is 20. |
For a given lag, the command computes the Ljung-Box statistic for testing rho_j = ... = rho_lag = 0, where j = 1, 2, ..., lag. For a VMA(q) process, the Ljung-Box statistics should be significant for the first q lags, and insignificant thereafter.
The Q-statistics and p-value plot
Ruey S. Tsay
Tsay (2014). Multivariate Time Series Analysis with R and Financial Applications. John Wiley. Hoboken, NJ.
zt=matrix(rnorm(600),200,3) VMAorder(zt)zt=matrix(rnorm(600),200,3) VMAorder(zt)
Performs the conditional maximum likelihood estimation of a VMA model with selected lags in the model
VMAs(da, malags, include.mean = T, fixed = NULL, prelim = F, details = F, thres = 2)VMAs(da, malags, include.mean = T, fixed = NULL, prelim = F, details = F, thres = 2)
da |
A T-by-k matrix of a k-dimensional time series with T observations |
malags |
A vector consisting of non-zero MA lags |
include.mean |
A logical switch to include the mean vector |
fixed |
A logical matrix to fix coefficients to zero |
prelim |
A logical switch concerning initial estimation |
details |
A logical switch to control output level |
thres |
A threshold value for setting coefficient estimates to zero |
A modified version of VMA model by allowing the user to select non-zero MA lags
data |
The observed time series |
MAlags |
The VMA lags |
cnst |
A logical switch to include the mean vector |
coef |
The parameter estimates |
secoef |
The standard errors of the estimates |
residuals |
Residual series |
aic, bic
|
The information criteria of the fitted model |
Sigma |
Residual covariance matrix |
Theta |
The VMA matrix polynomial |
mu |
The mean vector |
MAorder |
The VMA order |
Ruey S. Tsay
Tsay (2014, Chapter 3). Multivariate Time Series Analysis with R and Financial Applications. John Wiley. Hoboken, NJ.
VMA
Assuming that the model is known, this program estimates the value of a missing data point. The whole data point is missing.
Vmiss(zt, piwgt, sigma, tmiss, cnst = NULL, output = T)Vmiss(zt, piwgt, sigma, tmiss, cnst = NULL, output = T)
zt |
A T-by-k data matrix of a k-dimensional time series |
piwgt |
The pi-weights of a VARMA model defined as piwgt=[pi0, pi1, pi2, ....] |
sigma |
Positive definite covariance matrix of the innovations |
tmiss |
Time index of the missing data point |
cnst |
Constant term of the model |
output |
A logical switch to control output |
Use the least squares method to estimate a missing data point. The missing is random.
Estimates of the missing values
Ruey S. Tsay
Tsay (2014, Chapter 6). Multivariate Time Series Analysis with R and Financial Applications. John Wiley. Hoboken, NJ.
Vpmiss
data("mts-examples",package="MTS") gdp=log(qgdp[,3:5]) m1=VAR(gdp,3) piwgt=m1$Phi; Sig=m1$Sigma; cnst=m1$Ph0 m2=Vmiss(gdp,piwgt,Sig,50,cnst)data("mts-examples",package="MTS") gdp=log(qgdp[,3:5]) m1=VAR(gdp,3) piwgt=m1$Phi; Sig=m1$Sigma; cnst=m1$Ph0 m2=Vmiss(gdp,piwgt,Sig,50,cnst)
Assuming that the data is only partially missing, this program estimates those missing values. The model is assumed to be known.
Vpmiss(zt, piwgt, sigma, tmiss, mdx, cnst = NULL, output = T)Vpmiss(zt, piwgt, sigma, tmiss, mdx, cnst = NULL, output = T)
zt |
A T-by-k data matrix of a k-dimensional time series |
piwgt |
pi-weights of the model in the form piwgt[pi0, pi1, pi2, ....] |
sigma |
Residual covariance matrix |
tmiss |
Time index of the partially missing data point |
mdx |
A k-dimensional indicator with "0" denoting missing component and ""1" denoting observed value. |
cnst |
Constant term of the model |
output |
values of the partially missing data |
Estimates of the missing values
Ruey S. Tsay
Tsay (2014, Chapter 6). Multivariate Time Series Analysis with R and Financial Applications. John Wiley. Hoboken, NJ.
Vmiss
#data("mts-examples",package="MTS") #gdp=log(qgdp[,3:5]) #m1=VAR(gdp,1) #piwgt=m1$Phi; cnst=m1$Ph0; Sig=m1$Sigma #mdx=c(0,1,1) #m2=Vpmiss(gdp,piwgt,Sig,50,mdx,cnst)#data("mts-examples",package="MTS") #gdp=log(qgdp[,3:5]) #m1=VAR(gdp,1) #piwgt=m1$Phi; cnst=m1$Ph0; Sig=m1$Sigma #mdx=c(0,1,1) #m2=Vpmiss(gdp,piwgt,Sig,50,mdx,cnst)