I am conducting a study to analyze the impact of a specific policy on the number of businesses in each region. The policy was implemented in a staggered manner across different regions, and I'm utilizing two different regression models, using flow or stock of businesses, to understand its implications.
Model I: flow variable is my outcome variable
In this model, I regress the net number of business incorporations in each region-time on a treatment variable, which acts as an indicator of policy existence. The result is a precisely estimated zero coefficient, indicating no significant effect. I use fixed effects for both region and time in this model.
$$ Y_{it}=\alpha+\beta⋅Treatment_{it}+\gamma_{i}+\delta_{t}+\epsilon_{it} $$
where:
$Y_{it}$: Flow of businesses (net creation of incorporations) in region i at time t
Model II: Stock variable as my outcome variable
Here, I regress the stock of businesses in each region at each time—constructed using the initial stock and the running sum of net incorporations—on the treatment variable. This model yields very large and statistically significant coefficients, indicating a substantial effect of the policy.
$$ S_{it}=\alpha+\beta⋅Treatment_{it}+\gamma_{i}+\delta_{t}+\epsilon_{it} $$
where:
$S_{it}$: Stock of businesses in region i at time t
I am puzzled by the contrasting results obtained from the two models. Why might the treatment variable indicate no effect on the the number of (net) new business incorporations but show significant effect on the stock of businesses? I don't think we have mechanical reason to believe that if there's no statistical relationship between the flow of a variable and the treatment, there shouldn't be a relationship between stock and the treatment. But, how can I conceptually wrap my head around it? What could be the potential reasons for these inconsistent outcomes, and how can they be reconciled? I have also run the event study design with leads and lags but have not found any evidence of policy having a lagged effect on business incorporations that is not captured by the flow model.
