Difference-in-difference (DD) estimator for non-discrete, longitudinal data?

Question

There are many examples of how to set up a DD model for two points in time and with dummy variables (0s and 1s) separating the treated from control units and before and after intervention periods. However, I wonder if there is a way to set up a DD model where instead of using dummies we use continuous variables?

For example, I would like to evaluate the impact of a price policy. I do have treated and control units observed over time (from 2008 to 2016). Would it be possible to replace the standard dummy variable for treated/control by a continuous (price) variable with values = zero before intervention and values = actual price after intervention? Similarly, would be possible to somehow use actual year information (i.e., 1,2,3, etc.) instead of a time-dummy for before/after intervention?

Does it make sense to think of a DD model for this problem or should I address this issue with a standard time-series model with a dummy variable for the treated/control units?

Answer 1

Would it be possible to replace the standard dummy variable for treated/control by a continuous (price) variable with values = zero before intervention and values = actual price after intervention?

Yes, you can use a continuous variable indicating treatment intensity instead of a dummy variable for treatment. I have seen people call this by various names. Hudson, Hull, and Liebersohn call it "instrumented difference-in-differences."

Similarly, would be possible to somehow use actual year information (i.e., 1,2,3, etc.) instead of a time-dummy for before/after intervention?

You need a time-dummy for before/after intervention whether or not you are using year-level data. The years after the treatment are all affected by the treatment.

You can use year-level data and possibly use year fixed-effects as controls, but you need to be careful about how you handle your standard errors if you are not collapsing the pre and post-treatment periods. Most outcome variables will be autocorrelated and the usual standard errors or even cluster-robust standard errors will not necessarily alleviate it. See Bertrand, Duflo, and Mullanaithan (2003) for details.