# Regression of Two Random Walks
# This illustrates how easy it is to get spurious regressions
# when you just pick up two time series which are both I(1) processes,
# as prices usually are.

module(finmetrics)

x=cumsum(rnorm(1000))
y=cumsum(rnorm(1000))

#Do the two plots --- they both do look like random walks (or price processes)

par(mfrow=c(2,1))  
plot(x)
plot(y)


#Now regress

xyfit=lm(y~x)

summary(xyfit)

#You'll get a significant alpha and significant beta damn near every time!

################

#Just think how weird this is --- x and y are independent of each other!
#This is an example of "spurious regression"
# It is easy to "smell a rat" in this simple simulation, but you would be
# amazed how many people do thinks like regress the price of SPY vs the price
# of some bond fund ---
# The problem here is PRICE. If one were to look at returns,
# this problem would not show up.

#What is the problem? The problem is that the residuals from the fit of
#two I(1) processes can be either an I(1) process or an I(0) process.
# If it is an I(0) process the regression is "legitimate" --- or at least no
# worse than usual.
# If the residual is an I(1) process --- or if you can't reject this possibilty
# then the regression is JUNK (aka SPURIOUS).

#Let's look at our RW regression residuals and see if the regression 
# was SPURIOUS. That is we 
# take z= y-beta*x and test it to see if we can reject that z is an I(1) process
# That is we test z for a "unit root" ---
# say using Augmented Dickey Fuller t-test

#First we extract the beta from xyfit

beta=coefficients(xyfit)[2]

#Check
beta

# Now we get our residuals --- we could just pulled them out
# from xyfit, but lets be explict:

 z= y-beta*x

#Now we use unitroot to do the augmented Dickey-Fuller t-tests
#We'll take 6 lags --- you can explore this choice if you like. 
#It seldom matters so long as you do have two or more lags.

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

#Most of the time (but not absolutely every time) you will fail to reject the hypothesis that
# the residual series has a unit root. That is, you will fair to reject the hypothesis that
# the z process is non-stationary, in fact I(1) --- or at least we can't say that it isn't I(1)