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We then created two subsequent series, which I will rename to p t and q t so that we do not
confuse the original names y t and x t with the conventional names for regression reponses and
predictors:
> p <- q <- rep(0, 1000)
> p <- 0.3*z + rnorm(1000)
> q <- 0.6*z + rnorm(1000)
At this stage we can make use of the lm function, which calculates a linear regression between
two vectors. In this instance we will set q t to be the independent variable and p t to be the
dependent variable:
> comb <- lm(p~q)
If we take a look at the comb linear regression model we can see that the estimate for the β
regression coefficient is approximately 0.5, which makes sense given that q t is twice dependent
on z t compared to p t (0.6 compared to 0.3):
> comb
Call:
lm(formula = p ~ q)
Coefficients:
(Intercept)
q
0.1749 0.4745
Finally, we apply the ADF test to the residuals of the linear model in order to test for
stationarity:
> adf.test(comb$residuals, k=1)
Augmented Dickey-Fuller Test
data: comb$residuals
Dickey-Fuller = -23.4463, Lag order = 1, p-value = 0.01
alternative hypothesis: stationary
Warning message:
In adf.test(comb$residuals, k = 1) : p-value smaller than printed p-value
The Dickey-Fuller test statistic is very low, providing us with a low p-value. We can likely
reject the null hypothesis of the presence of a unit root and conclude that we have a stationary
series and hence a cointegrated pair. This is clearly not surprising given that we simulated the
data to have these properties in the first place.
We are now going to apply the CADF procedure to multiple sets of historical financial data.
12.7 CADF on Financial Data
There are many ways of forming a cointegrating set of assets. A common source is to use ETFs
that track similar characteristics. A good example is an ETF representing a basket of gold mining