1. Chapter 2. Multivariate Analysis of
Stationary Time Series
Cheng-Jun Wang

2. Contents

3. w
No
2000s
C M
VE
1990s
History
e r &
ng
Gr a tion
gra tion
Inte tegra
in
Co
1987
Sim R
VA
1980

4. VAR

5. Matrix Multiplication
http://en.wikipedia.org/wiki/Matrix_multiplication

6. Matrix Addition

7. R Code 2.1
Simulation of Var (2)-process
## Simulate VAR(2)-data ## Obtaining the generated series
library(dse1) vardat <- matrix(varsim$output, nrow = 500, ncol = 2)
library(vars)
colnames(vardat) <- c("y1", "y2")
## Setting the lag-polynomial A(L)
Apoly <- array(c(1.0, -0.5, 0.3, 0, ## Plotting the series
0.2, 0.1, 0, -0.2, plot.ts(vardat, main = "", xlab = "")
0.7, 1, 0.5, -0.3) , ## Determining an appropriate lag-order
c(3, 2, 2)) infocrit <- VARselect(vardat, lag.max = 3,
## Setting Covariance to identity-matrix type = "const")
B <- diag(2)
## Setting constant term to 5 and 10
## Estimating the model
TRD <- c(5, 10) varsimest <- VAR(vardat, p = 2, type = "const",
## Generating the VAR(2) model season = NULL, exogen = NULL)
var2 <- ARMA(A = Apoly, B = B, TREND = TRD) ## Alternatively, selection according to AIC
## Simulating 500 observations varsimest <- VAR(vardat, type = "const",
varsim <- simulate(var2, sampleT = 500,
lag.max = 3, ic = "SC")
noise = list(w = matrix(rnorm(1000),
nrow = 500, ncol = 2)), rng = list(seed = c(123456))) ## Checking the roots
roots <- roots(varsimest)

8. R Code 2.2
Diagnostic tests of VAR(2)-process
∗ ## testing serial correlation ∗ ## class and methods for diganostic tests
∗ args(serial.test)
∗ ## Portmanteau-Test ∗ class(var2c.serial)
∗ var2c.serial <- serial.test(varsimest, lags.pt = 16, ∗ class(var2c.arch)
∗ type = "PT.asymptotic")
∗ var2c.serial ∗ class(var2c.norm)
∗ plot(var2c.serial, names = "y1") ∗ methods(class = "varcheck")
∗ plot(var2c.serial, names = "y2")
∗ ## testing heteroscedasticity
∗ ## Plot of objects "varcheck"
∗ args(arch.test) ∗ args(vars:::plot.varcheck)
∗
∗
var2c.arch <- arch.test(varsimest, lags.multi = 5, ∗ plot(var2c.serial, names = "y1")
multivariate.only = TRUE)
∗ var2c.arch ∗ ## Structural stability test
∗ ## testing for normality ∗ reccusum <- stability(varsimest,
∗ args(normality.test)
∗ var2c.norm <- normality.test(varsimest, ∗ type = "OLS-CUSUM")
∗ multivariate.only = TRUE) ∗ fluctuation <- stability(varsimest,
∗ var2c.norm
∗ type = "fluctuation")

9. Causality Analysis
∗ Granger causality
∗ Wald-type instantaneous causality

10. ∗ ## Causality tests
∗ ## Granger and instantaneous causality
∗ var.causal <- causality(varsimest, cause = "y2")

11. Forecasting
∗ ## Forecasting objects of class varest
∗ args(vars:::predict.varest)
∗ predictions <- predict(varsimest, n.ahead
= 25,
∗ ci = 0.95)
∗ class(predictions)
∗ args(vars:::plot.varprd)
∗ ## Plot of predictions for y1
∗ plot(predictions, names = "y1")
∗ ## Fanchart for y2
∗ args(fanchart)
∗ fanchart(predictions, names = "y2")

12. Impulse Response Function
∗ Causality test falls short of quantifying the impact of the
impulse variable on the response variable over time.
∗ The impulse response analysis is used to investigate these kinds
of dynamic interactions between the endogenous variables and
is based upon the Wold moving average representation of a
VAR(p)-process.

13. ## Forecast error variance
## Impulse response analysis decomposition
∗ irf.y1 <- irf(varsimest, impulse = "y1", ∗ fevd.var2 <- fevd(varsimest,
∗ response = "y2", n.ahead = 10,
∗ ortho = FALSE, cumulative = n.ahead = 10)
FALSE, ∗ args(vars:::plot.varfevd)
∗ boot = FALSE, seed = 12345)
∗ args(vars:::plot.varirf) ∗ plot(fevd.var2, addbars = 2)
∗ plot(irf.y1)
∗ irf.y2 <- irf(varsimest, impulse = "y2",
∗ response = "y1", n.ahead = 10,
∗ ortho = TRUE, cumulative = TRUE,
∗ boot = FALSE, seed = 12345)
∗ plot(irf.y2)

14. SVAR
∗ An SVAR-model can be used to identify shocks and trace these
out by employing IRA and/or FEVD through imposing
restrictions on the matrices A and/or B.

15. SVAR: A-model
∗ library(dse1) ∗ ## Obtaining the generated series
∗ library(vars) ∗ svardat <- matrix(svarsim$output, nrow = 500, ncol =
∗ ## A-model 2)
∗ colnames(svardat) <- c("y1", "y2")
∗ Apoly <- array(c(1.0, -0.5, 0.3, 0.8,
∗ ## Estimating the VAR
∗ 0.2, 0.1, -0.7, -0.2, ∗ varest <- VAR(svardat, p = 2, type = "none")
∗ 0.7, 1, 0.5, -0.3) , ∗ ## Setting up matrices for A-model
∗ c(3, 2, 2)) ∗ Amat <- diag(2)
∗ ## Setting covariance to identity-matrix ∗ Amat[2, 1] <- NA
∗ B <- diag(2) ∗ Amat[1, 2] <- NA
∗ ## Generating the VAR(2) model ∗ ## Estimating the SVAR A-type by direct
∗ svarA <- ARMA(A = Apoly, B = B) maximisation
∗ ## Simulating 500 observations ∗ ## of the log-likelihood
∗ svarsim <- simulate(svarA, sampleT = 500, ∗ args(SVAR)
∗ svar.A <- SVAR(varest, estmethod = "direct",
∗ rng = list(seed = c(123456)))
∗ Amat = Amat, hessian = TRUE)

16. SVAR: B-model
∗ library(dse1) ∗ ## Simulating 500 observations
∗ library(vars) ∗ svarsim <- simulate(svarB, sampleT = 500,
∗ ## B-model ∗ rng = list(seed = c(123456)))
∗ svardat <- matrix(svarsim$output, nrow = 500,
∗ Apoly <- array(c(1.0, -0.5, 0.3, 0, ncol = 2)
∗ 0.2, 0.1, 0, -0.2, ∗ colnames(svardat) <- c("y1", "y2")
∗ 0.7, 1, 0.5, -0.3) , ∗ varest <- VAR(svardat, p = 2, type = "none")
∗ c(3, 2, 2)) ∗ ## Estimating the SVAR B-type by scoring
algorithm
∗ ## Setting covariance to identity-matrix ∗ ## Setting up the restriction matrix and vector
∗ B <- diag(2) ∗ ## for B-model
∗ B[2, 1] <- -0.8 ∗ Bmat <- diag(2)
∗ ## Generating the VAR(2) model ∗ Bmat[2, 1] <- NA
∗ svarB <- ARMA(A = Apoly, B = B) ∗ svar.B <- SVAR(varest, estmethod = "scoring",
∗ Bmat = Bmat, max.iter = 200)

17. ## Impulse response analysis of ## FEVD analysis of SVAR B-type
SVAR A-type model model
∗ args(vars:::irf.svarest) ∗ args(vars:::fevd.svarest)
∗ irf.svara <- irf(svar.A, impulse = ∗ fevd.svarb <- fevd(svar.B, n.ahead
"y1", response = "y2", boot = = 5)
FALSE) ∗ class(fevd.svarb)
∗ args(vars:::plot.varirf) ∗ methods(class = "varfevd")
∗ plot(irf.svara) ∗ plot(fevd.svarb)

18. 20121209