I am learning about the concept of cointegration and I found in various places the following claim about estimating the cointegration vector using OLS which is: Despite the fact that the OLS estimator of cointegrated variables is super-consistent it is biased in finite samples.

My question is why is the OLS estimator biased in finite samples?



It is not like it is necessarily biased. Consider this little simulation:

reps <- 10000
n <- 30

betahat <- rep(NA,reps)
for (i in 1:reps){
  u <- rnorm(n)
  x <- cumsum(rnorm(n))
  y <- 2*x + u
  betahat[i] <- summary(lm(y~x))$coefficients[2]

for which the bias is nonexistent. The distribution is also nicely normal:

enter image description here

Bias comes in when considering more general (one should/could also say: more realistic) cases: when the error terms driving x and the cointegration equation (here, u) are correlated and themselves exhibit short-run correlation. See e.g. the paper by Gonzalo, JoE 1994 for a more thorough simulation, or Chapter 19 in Hamilton, "Time Series Analysis", for an asymptotic analysis.


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