Suppose I have a Poisson model which investigates the effect of a county level policy on robbery counts. Here is the basic specification:
$$ \text{log}(y_{it}) = \theta y_{i,t-1} + \sum_i\text{County}_i + \sum_t\text{Week}_t + \delta \text{Policy}_{it}, $$
where a lag of the outcome is included on the right-hand side of the equation. The model also includes county and week fixed effects, and a policy dummy equal to 1 if the county is subject to the policy and is in a post-treatment time period, 0 otherwise. Some counties were not considered for inclusion in the program.
(1) I understand that biased is introduced since this is technically a "within" model. Suppose $N = 60$ and $T = 520$. Is it safe to ignore the bias if I have very large $T$? This is tangentially related to this post.
(2) What potential issues must I address if I only include $y_{i,t-2}$ or $y_{i,t-3}$ on the right-hand side but completely drop the first-order term? I suspect selection of counties into the "treatment" is based upon past incidents of robbery in some of the counties, but I have evidence policy-makers used more distant time periods before the treatment was introduced.
