Set identification

In

Partial identification continues to be a major theme in research in econometrics. Powell (2017) named partial identification as an example of theoretical progress in the econometrics literature, and Bonhomme & Shaikh (2017) list partial identification as “one of the most prominent recent themes in econometrics.”

Definition

Let ${\displaystyle U\in {\mathcal {U}}\subseteq \mathbb {R} ^{d_{u}}}$ denote a vector of latent variables, let ${\displaystyle Z\in {\mathcal {Z}}\subseteq \mathbb {R} ^{d_{z}}}$ denote a vector of observed (possibly endogenous) explanatory variables, and let ${\textstyle Y\in {\mathcal {Y}}\subseteq \mathbb {R} ^{d_{y}}}$ denote a vector of observed endogenous outcome variables. A structure is a pair ${\displaystyle s=(h,{\mathcal {P}}_{U\mid Z})}$, where ${\displaystyle {\mathcal {P}}_{U\mid Z}}$ represents a collection of conditional distributions, and ${\displaystyle h}$ is a structural function such that ${\displaystyle h(y,z,u)=0}$ for all realizations ${\displaystyle (y,z,u)}$ of the random vectors ${\displaystyle (Y,Z,U)}$. A model is a collection of admissible (i.e. possible) structures ${\displaystyle s}$.^[2][3]

Let ${\displaystyle {\mathcal {P}}_{Y\mid Z}(s)}$ denote the collection of conditional distributions of ${\displaystyle Y\mid Z}$ consistent with the structure ${\displaystyle s}$. The admissible structures ${\displaystyle s}$ and ${\displaystyle s'}$ are said to be observationally equivalent if ${\displaystyle {\mathcal {P}}_{Y\mid Z}(s)={\mathcal {P}}_{Y\mid Z}(s')}$.^[2][3] Let ${\displaystyle s^{\star }}$ denotes the true (i.e. data-generating) structure. The model is said to be point-identified if for every ${\displaystyle s\neq s'}$ we have ${\displaystyle {\mathcal {P}}_{Y\mid Z}(s)\neq {\mathcal {P}}_{Y\mid Z}(s^{\star })}$. More generally, the model is said to be set (or partially) identified if there exists at least one admissible ${\displaystyle s\neq s^{\star }}$ such that ${\displaystyle {\mathcal {P}}_{Y\mid Z}(s)\neq {\mathcal {P}}_{Y\mid Z}(s^{\star })}$. The identified set of structures is the collection of admissible structures that are observationally equivalent to ${\displaystyle s^{\star }}$.^[4]

In most cases the definition can be substantially simplified. In particular, when ${\displaystyle U}$ is independent of ${\displaystyle Z}$ and has a known (up to some finite-dimensional parameter) distribution, and when ${\displaystyle h}$ is known up to some finite-dimensional vector of parameters, each structure ${\displaystyle s}$ can be characterized by a finite-dimensional parameter vector ${\displaystyle \theta \in \Theta \subset \mathbb {R} ^{d_{\theta }}}$. If ${\displaystyle \theta _{0}}$ denotes the true (i.e. data-generating) vector of parameters, then the identified set, often denoted as ${\displaystyle \Theta _{I}\subset \Theta }$, is the set of parameter values that are observationally equivalent to ${\displaystyle \theta _{0}}$.^[4]

Example: missing data

This example is due to

${\displaystyle \mathrm {P} (Y=1)}$

${\displaystyle Z=1}$

By the law of total probability,

${\displaystyle \mathrm {P} (Y=1)=\mathrm {P} (Y=1\mid Z=1)\mathrm {P} (Z=1)+\mathrm {P} (Y=1\mid Z=0)\mathrm {P} (Z=0).}$

The only unknown object is ${\displaystyle \mathrm {P} (Y=1\mid Z=0)}$, which is constrained to lie between 0 and 1. Therefore, the identified set is

${\displaystyle \Theta _{I}=\{p\in [0,1]:p=\mathrm {P} (Y=1\mid Z=1)\mathrm {P} (Z=1)+q\mathrm {P} (Z=0),{\text{ for some }}q\in [0,1]\}.}$

Given the missing data constraint, the econometrician can only say that ${\displaystyle \mathrm {P} (Y=1)\in \Theta _{I}}$. This makes use of all available information.

Statistical inference

Set estimation cannot rely on the usual tools for statistical inference developed for point estimation. A literature in statistics and econometrics studies methods for statistical inference in the context of set-identified models, focusing on constructing confidence intervals or confidence regions with appropriate properties. For example, a method developed by Chernozhukov, Hong & Tamer (2007) constructs confidence regions that cover the identified set with a given probability.

Notes

References

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• Tamer, Elie (2010). "Partial Identification in Econometrics".
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