How did Dickey and Fuller know something was wrong?

Question

I am interested in testing if there is size distortion through simulations. I have recently been interested in replicating Dickey and Fuller (1979) and this source from another post helped a lot, here

However, whilst they are generating the correct critical values, how did Dickey and Fuller know that something was wrong in the first place.

From my understanding, the premise of the argument is the the t distribution was not effective when computing hypotheses tests when the AR(1) coefficient was 1, i.e.,

$$Y_t=\delta+Y_{t-1}+\varepsilon_t$$

So my question is, how would I go about simulating some data and testing the level of size distortion?

Whilst this may seem trivial for the DF research I would like to be able to understand this for a more complicated framework so any advice would be appreciated?

Cross post 2

Answer 1

They didn't know. The original work was performed by Mann and Wald in 1943 for the stationary case. John White solved the explosive root case for the method of maximum likelihood and Frequentist solutions, though not the Bayesian case. The unit root case is the intermediate case between the two. White had almost solved the unit root case and left it for a next paper, but never wrote it. I do not know if he died or what happened, but the follow on paper was never done.

It was a small leap from White to completion and so they did it.

Also, there is nothing wrong. They were not fixing anything, they were just completing the set.

The mistake that usually gets made is that White's paper effectively shows there is no non-Bayesian solution in the explosive root case. However, White's work implies a Bayesian solution. The estimator is the OLS estimator in the explosive root case, but the sampling distribution of the statistic is the Cauchy distribution.

Since the Cauchy distribution has no mean, the slope estimate is meaningless. Nonetheless, you can reverse engineer a Bayesian solution because White derived the proof by multiplying the unknown likelihood by the square root of Fisher information, which is the same as multiplying a Jeffreys prior by an unknown likelihood function. With a bit of extra work, you can show that the likelihood cannot be worse than the Cauchy distribution, which nicely leads to convergent solutions.