In
statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
and
econometrics
Econometrics is the application of Statistics, statistical methods to economic data in order to give Empirical evidence, empirical content to economic relationships.M. Hashem Pesaran (1987). "Econometrics," ''The New Palgrave: A Dictionary of ...
, set identification (or partial identification) extends the concept of
identifiability
In statistics, identifiability is a property which a model must satisfy for precise inference to be possible. A model is identifiable if it is theoretically possible to learn the true values of this model's underlying parameters after obtaining an ...
(or "point identification") in
statistical model
A statistical model is a mathematical model that embodies a set of statistical assumptions concerning the generation of Sample (statistics), sample data (and similar data from a larger Statistical population, population). A statistical model repres ...
s to situations where the distribution of observable variables is not informative of the exact value of a
parameter
A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
, but instead constrains the parameter to lie in a
strict subset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
of the parameter space. Statistical models that are set identified arise in a variety of settings in
economics
Economics () is the social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services.
Economics focuses on the behaviour and intera ...
, including
game theory
Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has appli ...
and the
Rubin causal model
The Rubin causal model (RCM), also known as the Neyman–Rubin causal model, is an approach to the statistical analysis of cause and effect based on the framework of potential outcomes, named after Donald Rubin. The name "Rubin causal model" was ...
.
Though the use of set identification dates to a 1934 article by
Ragnar Frisch
Ragnar Anton Kittil Frisch (3 March 1895 – 31 January 1973) was an influential Norwegian economist known for being one of the major contributors to establishing economics as a quantitative and statistically informed science in the early 20th ce ...
, the methods were significantly developed and promoted by
Charles Manski
Charles Frederick Manski (born November 27, 1948 in Boston), is Professor of Economics at Northwestern University, an econometrician in the realm of rational choice theory, and an innovator in the arena of parameter identification.Charles Mansk ...
starting in the 1990s. Manski developed a method of worst-case bounds for accounting for
selection bias
Selection bias is the bias introduced by the selection of individuals, groups, or data for analysis in such a way that proper randomization is not achieved, thereby failing to ensure that the sample obtained is representative of the population int ...
. Unlike methods that make additional statistical assumptions, such as
Heckman correction
The Heckman correction is a statistical technique to correct bias from non-randomly selected samples or otherwise incidentally truncated dependent variables, a pervasive issue in quantitative social sciences when using observational data. Conceptu ...
, the worst-case bounds rely only on the data to generate a range of supported parameter values.
Definition
Let
be a
statistical model
A statistical model is a mathematical model that embodies a set of statistical assumptions concerning the generation of Sample (statistics), sample data (and similar data from a larger Statistical population, population). A statistical model repres ...
where the parameter space
is either finite- or infinite-dimensional. Suppose
is the true parameter value. We say that
is set identified if there exists
such that
; that is, that some parameter values in
are not
observationally equivalent to
. In that case, the identified set is the set of parameter values that are observationally equivalent to
.
Example: missing data
This example is due to . Suppose there are two
binary random variable
Binary data is data whose unit can take on only two possible states. These are often labelled as 0 and 1 in accordance with the binary numeral system and Boolean algebra.
Binary data occurs in many different technical and scientific fields, wher ...
s, and . The econometrician is interested in
. There is a
missing data
In statistics, missing data, or missing values, occur when no data value is stored for the variable in an observation. Missing data are a common occurrence and can have a significant effect on the conclusions that can be drawn from the data.
Miss ...
problem, however: can only be observed if
.
By the
law of total probability
In probability theory, the law (or formula) of total probability is a fundamental rule relating marginal probabilities to conditional probabilities. It expresses the total probability of an outcome which can be realized via several distinct even ...
,
:
The only unknown object is
, which is constrained to lie between 0 and 1. Therefore, the identified set is
:
Given the missing data constraint, the econometrician can only say that
. This makes use of all available information.
Statistical inference
Set estimation
In statistics, a random vector ''x'' is classically represented by a probability density function.
In a set-membership approach or set estimation, ''x'' is represented by a set ''X'' to which ''x'' is assumed to belong. This means that the suppor ...
cannot rely on the usual tools for statistical inference developed for
point estimation
In statistics, point estimation involves the use of sample data to calculate a single value (known as a point estimate since it identifies a point in some parameter space) which is to serve as a "best guess" or "best estimate" of an unknown populat ...
. A literature in statistics and econometrics studies methods for
statistical inference
Statistical inference is the process of using data analysis to infer properties of an underlying probability distribution, distribution of probability.Upton, G., Cook, I. (2008) ''Oxford Dictionary of Statistics'', OUP. . Inferential statistical ...
in the context of set-identified models, focusing on constructing
confidence interval
In frequentist statistics, a confidence interval (CI) is a range of estimates for an unknown parameter. A confidence interval is computed at a designated ''confidence level''; the 95% confidence level is most common, but other levels, such as 9 ...
s or
confidence region In statistics, a confidence region is a multi-dimensional generalization of a confidence interval. It is a set of points in an ''n''-dimensional space, often represented as an ellipsoid around a point which is an estimated solution to a problem, al ...
s with appropriate properties. For example, a method developed by (and which describes as complicated) constructs confidence regions that cover the identified set with a given probability.
Notes
References
*
*
*
Further reading
*
*
*
*{{Cite book, publisher = Springer-Verlag, isbn = 978-0-387-00454-9, last = Manski, first = Charles F., author-link = Charles Manski , title = Partial Identification of Probability Distributions, location = New York, date = 2003
Econometric modeling
Estimation theory