Identification of Nonparametric Simultaneous Equations Models With a Residual Index Structure

Abstract

Berry and Haile (2018) considered identification in a class of nonparametric simultaneous equations models, providing several combinations of sufficient conditions on the joint density of structural errors and the support of instruments. We show here that, even when the instruments vary only over a small open ball, the requirements on the joint density may be viewed as mild in at least one formal sense.

Unobserved Worker Quality and Inter‐Industry Wage Differentials*

“…The multiplicatively separable case where f (𝜉, 𝜖) = f 1 (𝜉) × f 2 (𝜖) and f 2 (𝜖) ≠ 0 for every 𝜖 would also work if f 1 (𝜉) is strictly monotone. Berry and Haile [2018] extend the identification results of Matzkin [2008] for general additive separable structures and document their applicability in many settings, including differentiated product markets. We discuss identification of unobserved worker quality below.…”

Identification With Additively Separable Heterogeneity

“… Alternative forms of separability have been studied in the recent literature. Examples include random coefficients models (Ichimura and Thompson (), Gautier and Kitamura ()), quantile models (Matzkin ()), simultaneous equations models (Matzkin (, ), Berry and Haile ()), and triangular models (Imbens and Newey (), Torgovitsky (), Hoderlein, Holzmann, Kasy, and Meister ()). …”

“…Identification may then be established by leveraging binary choice identification results.4 In discrete choice, the conditional distribution of Y | X = x and conditional means E[Y | X = x] contain the same information, and so these distinctions are not important. SeeBerry and Haile (2014) for identification of discrete choice models from market-level data. In contrast, in the bundles model ofGentzkow (2007), conditional means contain potentially less information and so the distinctions we mention are nontrivial.…”

Identification of a class of index models: A topological approach

“…The image being connected then ensures that local identi…cation extends to all of the image. Like us, Berry and Haile (2018) and Evdokimov (2010), among others, rely on connectness to achieve global identi…cation but in these papers the restriction is imposed directly on the support of the covariates thereby implicity restricting the covariates to be continuous. In contrast, we impose connectedness on the image of a (W; X) and so allow for both X and W to contain discrete components.…”

“…There is also a nascent literature on nonparametric identi…cation of so-called BLP models (Berry et al, 1995) as used in industrial organization; see, for example, Berry and Haile (2018) and Chiappori et al (2018). The setting of the BLP model is somewhat di¤erent, though, since there the observed choice probabilities contain unobserved product characteristics that have to be controlled for.…”