A Stage-Based Identification of Policy Effects

Abstract
We develop a method that identifies the effects of policy implemented nationwide—i.e. across all regions at the same time. Starting point is the insight that outcome paths can be tracked over stages using a reference path. The stage of a regional outcome path is defined as its location on the support of a reference path. It is formally the result of a normalization that maps the time-path of regional outcomes onto a reference path using pre-policy data only. Intuitively, our normalization seeks to reshape the structural parameters that determine the outcome path of non-reference regions into those of a reference region—a phenomenon that we show with an example for which we can derive exact identification. Since regions can differ by stage at any point in time, our normalization uncovers heterogeneity in the stage at the time of policy implementation—even in instances where the implementation occurs at the same time across regions. We use this stage variation at the time of policy implementation to identify the policy effects: a stage leading region delivers the counterfactual path inside an identification window in which nonleading regions are subject to policy whereas the leading region is not. Our identification assumption is that the normalization conducted using pre-policy data holds post policy, i.e. the normalization coefficients reshaping the regional pre-policy outcome paths into those of a reference region are unaffected by policy. We validate our method with Monte-Carlo experiments on model-generated data that detect bounds for a successful identification. We us our method to evaluate the effectiveness of public health stay-home policies (i.e. the national lockdown against Covid-19 in Spain), the effects of oral contraceptives (i.e. the 1960 FDA nationwide approval of oral contraceptives in the U.S.) on women’s fertility and college education and the effects of growth policy (e.g. German Reunification). We further show how our method can be applied to non-nationwide policy—i.e. untreated regions and staggered rollouts—and discuss the implications of spillovers across regions.

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