Can I use an instrumental variable to control for treatment endogeneity in differences-in-differences model?

I have panel data with two time periods (pre and post) and two groups (treatment and control). I have a differences-in-differences model with the standard estimating equation:

$$$$y_{it} = \beta_0 + X_{it} \beta_1 + \gamma_1 T + \gamma_2 P + \theta T \cdot P + \epsilon_{it}$$$$ where $$T$$ is an indicator for the treatment group, $$P$$ is an indicator for the post period, $$T \cdot P$$ is the interaction, and $$X_{it}$$ are exogenous controls. Unfortunately, I have reason to be concerned about endogeneity between the assignment into treatment/control groups and the outcome.

I have an instrument $$z_i$$ (it does not vary over time) that satisfies the exclusion restriction in that it only affects $$y_{it}$$ through the assignment into the treatment or control group. Is there any way to combine the instrumental variables framework with a differences-in-differences framework to handle this endogeneity?

Is this even something I should be concerned about? Maybe it means the DID model won't be estimating the average treatment effect, but if it isn't possible to use IV here, should I be worried?

I searched through Woolridge's econometric text Econometric Analysis of Cross Section and Panel Data, specifically Chapter 18, which discusses average treatment effects and endogeneity, but there isn't any mention of DID models in the book. Other related questions (this one and this one) have been asked, but there are no answers thus far.

• If your endogeneity is time-invariant, the differencing would wipe it out. If it is not time-invariant, I am not sure you could handle it with a time-invariant IV. – Dimitriy V. Masterov Nov 20 '18 at 2:43
• The above comment about differencing assumes that you are fitting the DID using panel fixed effects approach. – Dimitriy V. Masterov Nov 22 '18 at 18:54