Diff-in-diff and time-varying covariates

[Lukas Lang]

I am aware there are issues when controlling for time-varying covariates in a diff-in-diff setting.
However, I am wondering whether it would make sense to include (modified) covariates which values vary over time only during the pre-treatment periods but then become fixed at the pre-treatment average in the post-treatment periods.
I am thinking of covariates that may drive the treatment assignment and at the same time may be affected by the treatment.
By replacing the post-treatment values with the pre-treatment average, shouldn't we avoid the bad control problem?
Tagging some diff-in-diff experts below
Clyde Schechter Jared Greathouse

[Lukas Lang]

To increase the chances of a reply I will make the following example:

*Load database
use "http://www.princeton.edu/~otorres/WDI.dta", clear

*Creating the before/after dummy variable: 0 = before, 1 =after
gen after = (year >= 2009) if !missing(year)

*For the example in this document, the treated countries were saved in a separate fake Stata dataset containing a variable "treated" = 1. Below we merge that file to have the treatment variable.
merge m:1 country using "http://www.princeton.edu/~otorres/Treated.dta", gen(merge1)

*The untreated units will have a missing value (".")
replace treated = 0 if treated == .

*Create the diff-in-diff indicator
gen did = after * treated

* Create a labeled numeric variable for the grouping or panel variable. This is needed for Stata commands to identify the panels in the data.
encode country, gen(country1)

*Set data as panel data (only for use with `xt' commands).
xtset country1 year

*did without covariate
xtdidregress (gdppc) (did), group(country1) time(year)

*did with covariate
xtdidregress (gdppc labor) (did), group(country1) time(year)

*pre-treatment time-varying covariate
bys country1: egen labor_mean=mean(labor) if after==0
gsort country -labor_mean
bys country1: replace labor_mean=labor_mean[_n-1] if labor_mean==.
sort country year
gen labor_pretreat=labor if after==0
replace labor_pretreat=labor_mean if after==1
drop labor_mean

*did with pre-treatment time-varying covariate
xtdidregress (gdppc labor_pretreat) (did), group(country1) time(year)

Does controlling for 'labor_pretreat' avoid the potential bad control problem because of the potential (post-treatment) effect of the treatment on labor?

Would you consider this an econometrically valid solution if you thought that labor pre-treatment trends could impact the chance of receiving the treatment?