This question involves a questionable hypothetical scenario, but please bear with me.

Suppose I ran an experiment in a coffee stand where the treatment was playing country music instead of the usual soft jazz while people wait. I have panel data on expenditures from the loyalty program, and I am interested in whether playing country music has a negative effect on the subsequent transaction. Suppose there's no long-term effect or any fade-out of the effect to simplify things. The data for each person is at transaction level, so I see how much she spends the first time she came in, the second time, and so on.

Let $i$ index people, and $t$ index transactions: \begin{equation} \overbrace{\ln y_{it}}^\text{Logged \$} =\overbrace{\alpha_i}^\text{Customer Fixed Effect} + \underbrace{\gamma \cdot C_{it-1}}_\text{Country Music at Previous Trans.} +\overbrace{x_{it}^\prime \beta}^\text{Covariates} + \underbrace{\varepsilon_{it}}_\text{Idiosyncratic Error} \end{equation}

The null hypothesis is that $\gamma<0$. Consistent estimation of $\gamma$ requires eliminating $\alpha_i$ (otherwise $\hat \gamma >0$) because being a regular is correlated with treatment. I first-difference the transactions data to get rid of the fixed effect, and I am left with:

\begin{equation} \ln y_{it} - \ln y_{it-1} = \gamma \cdot \left( C_{it-1} - C_{it-2} \right) + \left(x_{it} - x_{it-1}\right)^{\prime}\beta + \left(\varepsilon_{it}-\varepsilon_{it-1}\right) \end{equation}

I run this regression and $\hat \gamma_{FD}<0$. I go to get an espresso. One the way to the coffee stand I realize that the people who don't return to the coffee shop are not in the data. There's no row because that transaction does not exist. If you spend less, you're in the data. If you miss a day because the memories of Garth Brooks haunt you (so the effect is -100%), you're not in the data. So the FD approach estimates an intensive margin effect, ignoring the extensive margin response. So $\hat \gamma_{FD}$ is an upper bound on the true effect.

I could aggregate the data to weekly level and get some zeros that way. I can also model time between transactions. But I want the bi-marginal effect on the next transaction only. Are there any solutions to this? Hacks are perfectly acceptable if they are well-motivated.


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