#==========================================
#Example of Augmented DF Test and exchange rate data
#Based on an example from Z&W 
#=================================================

#===============================================
#First Get the series for Canadian/US log spot rates
#Then get a picture of the series and its ACF 
#Then do the same for the first difference
#=================================================

uscn.spot = lexrates.dat[,"USCNS"]
uscn.spot@title = "Log US/CN spot exchange rate"
par(mfrow=c(2,2))
plot.timeSeries(uscn.spot, reference.grid=F,main="Log of US/CN spot exchange rate")
xx = acf(uscn.spot)
plot.timeSeries(diff(uscn.spot), reference.grid=F,main="First difference of log US/CN spot exchange rate")
xx = acf(diff(uscn.spot))

#==============================================
#Now look at the Augmented DF t-test, allowing for a constant and 6 lags
#==============================================

adft.out = unitroot(uscn.spot, trend="c", statistic="t",method="adf", lags=6)

#======================================
#Check the class of the object we have created.
#======================================

class(adft.out)

#=======================================
#Use one of the "generic methods" to look at this object.
#Both print() and summary() are defined for objects of this class.
#We can automactically call print() just by typing the name of our object.
#==================================================

adft.out

#============================================
#We see that we "fail to reject" the hypothesis that our series has a unti root.
#This is NOT quite the same as having positive evidence in FAVOR of our hypothesis;
#Z&W are not quite precise in their language.
# 
#From summary() we get information on the significance of the lagged differences in the
#test model
#===========================================


summary(adft.out)

#===============================================
#The message here is that lags after the second look insignificant.
#Thus, we try out the ADF test with fewer lags.
#================================================

adft.out = unitroot(uscn.spot, trend="c", statistic="t",method="adf", lags=2)

#===============================================================
#This time we see even less significance of the test, less evidence against the
#null hypothesis. This is not the same as evidence for the hypothesis, but it 
#does point in that direction.
#==================================================================

#============================================================
#Let's do the second example from Z&W; 
#It is more generic; most stock price data has looks more like an I(1)
#process than an I(0) process. That's why we always study returns
#for stock prices.
#========================================================

lnp = log(singleIndex.dat[,1])
lnp@title = "Log of S&P 500 Index"
par(mfrow=c(2,2))
plot.timeSeries(lnp, reference.grid=F,main="Log of S&P 500 index")
acf.plot(acf(lnp,plot=F))
plot.timeSeries(diff(lnp), reference.grid=F,main="First difference of log S&P 500 Index")
acf.plot(acf(diff(lnp),plot=F))

#================================================
#The picture argues for "nonstationarity" though I would not be as confident of
#a positive trend as Z&W seem to be. If we could spot trends by looking at
#a plot of the prices, we would be rich beyond the dreams of avarice. Still,
#log stock price having a trend is reasonalbe --- one expects a return after all.
#Let's check out an ADF test with a constant and trend and 4 lagged differences
#===================================================

adft.out = unitroot(lnp, trend="ct", lags=4)
summary(adft.out)

#=====================================
#Here we also "fail to reject."
#You'll get a similar story for almost any price series
#====================================

#=======================================
#Final story about unit root tests?
#
#They are lame in many respects, but they do give us something.
#For example, in statistical arbitrage, one tries like hell to 
#construct a series that is mean reverting.
#For such a series, we definitly want to run unit root tests
#and keep our fingers crossed for a STRONG REJECTION.
#That would be good news indeed.