Set identification

Let ${\displaystyle {\mathcal {P}}=\{P_{\theta }:\theta \in \Theta \}}$ be a statistical model where the parameter space ${\displaystyle \Theta }$ is either finite- or infinite-dimensional. Suppose ${\displaystyle \theta _{0}}$ is the true parameter value. We say that ${\displaystyle \theta _{0}}$ is set identified if there exists ${\displaystyle \theta \in \Theta }$ such that ${\displaystyle P_{\theta }\neq P_{\theta _{0}}}$; that is, that some parameter values in ${\displaystyle \Theta }$ are not observationally equivalent to ${\displaystyle \theta _{0}}$. In that case, the identified set is the set of parameter values that are observationally equivalent to ${\displaystyle \theta _{0}}$.[1]

This example is due to Tamer (2010). Suppose there are two binary random variables, Y and Z. The econometrician is interested in ${\displaystyle \mathrm {P} (Y=1)}$. There is a missing data problem, however: Y can only be observed if ${\displaystyle Z=1}$.

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.

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) (and which Lewbel (2019) describes as complicated) constructs confidence regions that cover the identified set with a given probability.

• Chernozhukov, Victor; Hong, Han; Tamer, Elie (2007). "Estimation and Confidence Regions for Parameter Sets in Econometric Models". Econometrica. The Econometric Society. 75 (5): 1243–1284. doi:10.1111/j.1468-0262.2007.00794.x. hdl:1721.1/63545. ISSN 0012-9682.
• Lewbel, Arthur (2019-12-01). "The Identification Zoo: Meanings of Identification in Econometrics". Journal of Economic Literature. American Economic Association. 57 (4): 835–903. doi:10.1257/jel.20181361. ISSN 0022-0515.
• Tamer, Elie (2010). "Partial Identification in Econometrics". Annual Review of Economics. 2 (1): 167–195. doi:10.1146/annurev.economics.050708.143401.

• Ho, Kate; Rosen, Adam M. (2017). "Partial Identification in Applied Research: Benefits and Challenges" (PDF). In Honore, Bo; Pakes, Ariel; Piazzesi, Monika; Samuelson, Larry (eds.). Advances in Economics and Econometrics (PDF). Cambridge: Cambridge University Press. pp. 307–359. doi:10.1017/9781108227223.010. ISBN 978-1-108-22722-3.
• Manski, Charles F. (May 1990). "Nonparametric Bounds on Treatment Effects". The American Economic Review: Papers and Proceedings. 80 (2): 319–323. ISSN 0002-8282. JSTOR 2006592.
• Manski, Charles F.; Pepper, John V. (July 2000). "Monotone Instrumental Variables: With an Application to the Returns to Schooling" (PDF). Econometrica. 68 (4): 997–1010. doi:10.1111/1468-0262.00144. ISSN 0012-9682. JSTOR 2999533.
• Manski, Charles F. (2003). Partial Identification of Probability Distributions. New York: Springer-Verlag. ISBN 978-0-387-00454-9.