How to estimate staggered difference in difference with continuous treatment?

[Chris Sean]

I want to estimate a staggered difference in difference with continuous treatment. The data looks something like this:

individual	year	CT
1	2000	0
1	2001	0
1	2002	0
1	2003	0
1	2004	0.3
1	2005	0.4
1	2006	0.42
1	2007	0.2
1	2008	0
1	2009	0
2	2000	0
2	2001	0
2	2002	0
2	2003	0
2	2004	0
2	2005	0
2	2006	0
2	2007	0
2	2008	0
2	2009	0
3	2000	0.1
3	2001	0.1
3	2002	0.1
3	2003	0.5
3	2004	0.6
3	2005	0.4
3	2006	0.2
3	2007	0.1
3	2008	0.3
3	2009	0.1
4	2000	0.3
4	2001	0.2
4	2002	0.4
4	2003	0.2
4	2004	0.3
4	2005	0.5
4	2006	0.1
4	2007	0.12
4	2008	0.13
4	2009	0.14

The generalized diff in diff equation can be specified as follows:

y_{i, t} = gamma_i + lambda_t + delta CT_{i, t} + epsilon_{i, t} ....(1) where i denotes some individual and t for year. gamma_i are individual fixed effects, and lambda_t are year fixed effects. CT_{i, t} is a continuous treatment variable that measures individual i's exposure to some "shock" in year t. Each of the individuals only experiences one treatment, i.e., individual 1 in 2004, individual 2 never, individual 3 in 2003, and individual 4 in 2006. For example:

Consider individual 1, his exposure to the shock is 0 until treatment occurs in year 2004, where his exposure to the shock has an intensity of 0.3. In the year after treatment, his exposure becomes 0.4, then 0.42, then 0.2 and dies out in 2008.

Individual 2 is "never treated".

Individual 3 has a constant exposure until he becomes treated in 2003, where his exposure jumps to 0.5. As a result of this treatment, his exposure then fluctuates around until the end of the sample period 2009.

Individual 4 has a fluctuating exposure until he becomes treated in 2006, where his exposure falls to 0.1, then fluctuates until the end of the sample period 2009.

My first question is whether Equation (1) is an appropriate generalized difference in differences equation that I can use to estimate the "treatment" effect?

My second question is how can I estimate a dynamic period by period coefficient version of Equation (1)? For example, in the usual case where treatment is staggered but binary (and where the treatment variable is 0 in the "pre treatment" period), one can easily estimate a dynamic version by using period by period dummy variables. However, how can I do that here? The continuous treatment variable (CT) is not always 0 in the pre-treatment period, nor does it take on a constant value post-treatment either.

[George Ford]

did_multiplegt allows for a continuous treatment, but it basically dichotomizes it by a threshold change from the prior period.

[Chris Sean]

did_multiplegt allows for a continuous treatment, but it basically dichotomizes it by a threshold change from the prior period.

Thank you, but as far as I know, did_multiplegt only allows for a continuous treatment that is "constant", and that prior to treatment the treatment variable always takes on a value of 0. But my example here violates both of those assumptions. Do you have any other suggestions?

[George Ford]

may have to go with the "event study" type. center everything on the treatment date. I think there are new approaches somewhat informed my the new staggered treatment design.

[George Ford]

look at Wooldridge's Mundlack paper. I think it does staggered treatments in a direct way.

[Chris Sean]

Thank you for the reference. I will take a look.