# R 3.3.1
# example for the multivariate Ljung Box test and 
# cross correlations

library("zoo")

## Required
source("mq_R.txt")    # multivariate Ljung-Box Test / Call: mq(xx,16) 
                      # xx <- as.matrix(data) #e.g.

## Generated data 
n <- 200; 
#AR(1)
x <- c(1:n)*0; 
aa <- 0.5; x[1]=0.1; for (i in (2:n)) { x[i]=aa*x[i-1] + rnorm(1) }; 

x2 <- as.ts(x);  
x1 <- 2*lag(x2,-3) + as.ts(rnorm(n),0,0.5);      # note lag(x,k=-1) ... one lag 
x3 <- as.ts(rnorm(n)); x4 <- as.ts(rnorm(n)); 

xx <- merge.zoo(x1,x2,x3,x4, all=FALSE)     # NAs excluded                  
                                         # data.class(xx) is "zooreg"
 
## Plots
plot(xx)                                                       #plot the single series
plot(xx[,1],xx[,2]); plot(xx[,1],xx[,3]); plot(xx[,1],xx[,4]); # plot 2-dim scatters
                     plot(xx[,2],xx[,3]); plot(xx[,2],xx[,4]);
                                          plot(xx[,3],xx[,4]);
plot( lag(xx[,2],-3), xx[,1] )            # this scatter makes the data structure visible

## Univariate tests of white noise                                    
acf(xx[,1]);     
acf(xx[,2]);  # compare with acf(xx[,1])
acf(xx[,3]); acf(xx[,4]);

Box.test(xx[,1], lag = 10, type = "Ljung-Box", fitdf = 0)
Box.test(xx[,2], lag = 10, type = "Ljung-Box", fitdf = 0)
Box.test(xx[,3], lag = 10, type = "Ljung-Box", fitdf = 0)
Box.test(xx[,4], lag = 10, type = "Ljung-Box", fitdf = 0)

## Cross-correlations: 
## ccf(x1,x2) := { corr(x_{1,t},x_{2,t-ll}), ll= ...,-1,0,1,... }
## if ll=5:  corr(x_{1,t},x_{2,t-5})
# try: ccf(xx[,1], xx[,2], lag.max = 10, type = "correlation", plot = FALSE)
ccf(xx[,1], xx[,2], lag.max = 10, type = "correlation", plot = TRUE)
ccf(xx[,1], xx[,3], lag.max = 10, type = "correlation", plot = TRUE)
ccf(xx[,1], xx[,4], lag.max = 10, type = "correlation", plot = TRUE)

ccf(xx[,2], xx[,3], lag.max = 10, type = "correlation", plot = TRUE)
ccf(xx[,2], xx[,4], lag.max = 10, type = "correlation", plot = TRUE)

ccf(xx[,3], xx[,4], lag.max = 10, type = "correlation", plot = TRUE)

## Multivariate Ljung-Box test (up to lag 15)
mq(xx,15) 


## EXERCISE:
# (1) Look at the data set. Verify that x1 is essentially x2 lagged by 3 periods. 
# (2) Plot the univariate data, test for univariate white noise.
# (3) Plot the data in bivariate scattergrams.
# (4) Calculate and interprete the ccf's. 
# (4.a) Which series leads the other?
# (4.b) What is rho_{1,2}(3), rho_{1,2}(-3)?
# (4.c) What is rho_{2,1}(3), rho_{2,1}(-3)? 
# (4.d) Build the CCM matrix Rho(3).
# (5) Perform the mv Ljung-Box test. What is H0, what is the alternative? 
# Do the analysis for 
# (A) x2 is an AR(1) as above
# (B) x2 is white noise.

# Check with
library(vars)
mod <- VAR(xx,p=3)
summary(mod)