Identification and Estimation in Panel Data Models with Nonparametric Two-Way Fixed Effects

We study identification and estimation of panel data models that are characterised by a common, potentially, infinite-dimensional parameter, $\theta$ and an unknown transformation $g(\alpha _{i},\gamma _{t})$ of two-way fixed effects $(\alpha _{i},\gamma _{t})$ that are themselves unknown. We treat $g$ as a nonparametric object to be identified/estimated together with $\theta$ and $(\alpha _{i},\gamma _{t})$. We first provide high-level assumptions under which $\theta$ together with the image of $g$, $\{g(\alpha _{i},\gamma _{t})\}$, is identified. Next, we show that, under weak regularity conditons on $g$ and the underlying distribution of $(\alpha _{i},\gamma _{t})$, this image in turn allows us to separately identify normalised versions of g and the fixed effects. This is achieved by using a finite number of leading singular functions of $g$ as proxies (normalised versions) of the fixed effects. Our identification argument is constructive in the sense that it leads to natural estimators of the parameters of interest.