# 17 Cointegration Tests and Modelling Cointegrated Systems

library(fUnitRoots)
library (readxl)
library(vars)
library(tseries)
library(urca)

SandPhedge <- read_excel("D:/Programming Guide/data/SandPhedge.xls",
                          col_types = c("date","numeric","numeric"))
View(SandPhedge)
#head(SandPhedge)

# Log prices
SandPhedge$lspot = log(SandPhedge$Spot)
SandPhedge$lfutures = log(SandPhedge$Futures)

# Apply Phillips-Ouliaris test for cointegration
dat <- cbind(SandPhedge$lspot,SandPhedge$lfutures)
po.test(dat) # H0: no cointegration is rejected

# Log returns
SandPhedge$rspot = c(NA,100*diff(log(SandPhedge$Spot)))
SandPhedge$rfutures = c(NA,100*diff(log(SandPhedge$Futures)))

log_lm <- lm(SandPhedge$lspot ~ SandPhedge$lfutures)
summary(log_lm)

par(lwd =2,cex.axis = 2)
#par(mar = c(1,1,1,1)) # Changing area margins

plot(SandPhedge$Date,SandPhedge$lspot,type = "l",xlab = "",ylab = "",col ="red")
lines(SandPhedge$Date,log_lm$fitted.values)

par(new=T)
plot(SandPhedge$Date,log_lm$residuals,col="blue",axes=F,type ="l")
axis(side=4, at = pretty(range(log_lm$residuals)))
legend ("bottomleft",legend=c("Actual","Fitted"),col =c("black","red"),lty=1)
legend ("bottomright",legend=c("Resid"),col=c("blue"),lty = 1)

# Unitroot test applied to residuals
urersTest(log_lm$residuals,doplot=F,type="DF-GLS",model="trend",lag.max=12)@test$test@teststat
urersTest(log_lm$residuals,doplot=F,type="DF-GLS",model="trend",lag.max=12)@test$test@cval
# Unit root in the residuals cannot be tejected -> no cointegration

summary(lm(SandPhedge$rspot[-1] ~ SandPhedge$rfutures[-1]+log_lm$residuals[-247]))
# ECM is statisrically significant -> cointegration

### Johansen's method

fred <- read_excel("D:/Programming Guide/data/fred.xls",
                         col_types = c("date",rep("numeric",6)))
View(fred)
head(fred)

summary(fred[c("GS3M","GS6M","GS1","GS3","GS5","GS10")])

plot (fred$Date,fred$GS3M, type ="l",xlab ="",ylab ="")
lines(fred$Date,fred$GS6M,col="red")
lines(fred$Date,fred$GS1,col="blue")
lines(fred$Date,fred$GS3,col="brown")
lines(fred$Date,fred$GS5,col="orange")
lines(fred$Date,fred$GS10,col="darkgreen")

# Select lags in the VAR model
VARselect(fred[c("GS3M","GS6M","GS1","GS3","GS5","GS10")],lag.max=12)

# Johansen's test
summary(ca.jo(fred[c("GS3M","GS6M","GS1","GS3","GS5","GS10")],K=2,ecdet="const",type="trace"))
# At most 3 cointegrating vectors at the 1% significance level

# Estimate VECM
vecm <- (ca.jo(fred[c("GS3M","GS6M","GS1","GS3","GS5","GS10")],K=2,ecdet="const",type="trace"))

cajorls(vecm,r=3)