Why we need at least three pre-treatment time periods for a two-year difference-in-difference model? From reading a discussion in this forum, I am wondering what is "two-year difference-in-difference". From my understanding, the basic DID is we examine the change of treatment compared to the change of control samples after an event. Whether "two-year difference-in-difference" means there are two event years in this setting?
And from the answer of an expert here, I am wondering about one thing that why he said

You cannot possibly demonstrate this, visually or statistically, with
less than three pre-treatment time periods

I do not understand what does "three pre-treatment time periods" mean. From my basic DID knowledge, only one pre-treatment period, maybe this phrase link to the "two-year difference-in-difference".
 A: In the canonical Difference in Difference setup two groups are observed at two different times. One of the groups is treated is second period while the other it treated in none.
The treatment histories of the two groups are therefore
$$Treated: \ (0,1)$$
and
$$Untreated: \ (0,0).$$
To calculate the DD-estimator you only need  two periods of observations and this is what is meant by "two-year difference-in-difference". However, it is still assumed that had the treated not been treated they would have evolved just like the untreated. The untreated is a valid counterfactual for the treated.
Now imagine you were given several observations in time before treatment kicked in
$$Treated: \ (0,0,0,0,0,1)$$
and
$$Untreated: \ (0,0,0,0,0,0),$$
then you could graphically check whether the untreated and the treated developed similarly on the dependent variable for the first 5 years before treatment. Hopefully, it is intuitively clear that if pretreatment patterns of development on dependent variable are not similar then that indicates that the two groups are not really following the same trend before treatment. But if this is the case then why should the treatment group have evolved just like the untreated for the laters years had the treatment group not been treated?
Unlike pretreatment trends casts doubt about the assumption that the treatment group have evolved just like the untreated without treatment for the year after treatment.
