I was wondering if the lagged dependent variable can be a valid instrument or not. In my data, policy intervention (x) is made depending on the outcome (y) in the past. (not depending on the effect (b) in the past, because they do not measure the effect itself). It sounds vague, but in other words, y in the past does not affect current y, but only affect through x like this:
Y_t = a + bX_t + u_t X_t = c + dY_t-1 + v_t-1
Does this occur any problem (biased, inconsistent or anything) to estimate? I encountered many literature using lagged explanatory variable as an instrument, but have not had a chance to see this type of estimation.
Could someone advise me or give me some example of this type of model? Thank you in advance.