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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?

Thanks

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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]
}
mean(betahat)

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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