Identified Sets, Sharpness, and Bound Interpretation

Why a Parameter May Not Have One Identified Value

In many econometric models, different descriptions of the unobserved part of the data-generating process can produce exactly the same distribution of what is observed. If the parameter of interest takes different values across those descriptions, the observed-data law alone does not determine one value. The appropriate conclusion is not that the parameter is unknowable under every possible assumption; it is that, under the model and restrictions currently imposed, the data identify a set of possible values.

This distinction separates identification from estimation. Identification asks what the observed-data law and model imply about a population parameter. Estimation asks how to use a finite sample to learn about that law or its implications. A precisely estimated observed-data distribution can still be compatible with multiple parameter values.

Definition: Let \(\mathcal{P}\) be a model, a set of admissible probability distributions for the full data. Let \(Q\) be the observed-data law, and let \(P_{\mathrm{obs}}\) denote the distribution of the observed variables implied by \(P\in\mathcal{P}\). The observationally equivalent distributions are \[ \mathcal{P}(Q)=\{P\in\mathcal{P}:P_{\mathrm{obs}}=Q\}. \] For a parameter \(\theta(P)\), its identified set given \(Q\) is \[ \Theta_I(Q)=\{\theta(P):P\in\mathcal{P}(Q)\}. \] The parameter is point identified if this set contains exactly one value. It is partially identified if the set contains more than one value.

The identified set is conditional on both the observed-data law and the chosen model. Adding credible restrictions can remove some observationally equivalent distributions and shrink the set; changing the restrictions can change it. A claim that a parameter is identified is therefore incomplete unless the relevant model and restrictions are understood.

Sharp Bounds and the Set They Describe

For a scalar parameter, it is often useful to summarize an identified set by its lower and upper endpoints. Define the lower endpoint as the infimum of all compatible parameter values and the upper endpoint as their supremum. These endpoints describe the narrowest interval containing the identified set. The infimum or supremum need not itself be attained by an admissible distribution.

Definition: Bounds \([L,U]\) are valid for \(\theta\) if every value in the identified set lies between them: \(L\leq\theta(P)\leq U\) for every \(P\in\mathcal{P}(Q)\). The sharp lower and upper bounds are \[ L^*=\inf_{P\in\mathcal{P}(Q)}\theta(P), \qquad U^*=\sup_{P\in\mathcal{P}(Q)}\theta(P). \] No valid lower bound can exceed \(L^*\), and no valid upper bound can be less than \(U^*\). A bound that can be tightened while remaining valid is non-sharp.

Sharp endpoints do not always mean the identified set is the entire interval between them. If the set is disconnected, its lower and upper endpoints still give the narrowest containing interval, but values in the gaps are not compatible with the model. Thus it is important to distinguish the exact identified set from an interval summary of that set. When the exact set is an interval, sharp bounds specify its endpoints; whether those endpoints are included depends on attainability.

Worked Example: A Joint Probability From Marginals Alone

Suppose the observed data provide the marginal probabilities \(P(X=1)=0.4\) and \(P(Y=1)=0.5\), but do not reveal the joint distribution of the binary variables \(X\) and \(Y\). Consider the target \(\theta=P(X=1,Y=1)\). Treat these marginals as known exactly for this illustration, and impose no restrictions on the joint distribution beyond being a valid probability distribution with those marginals.

Let \(t=P(X=1,Y=1)\). The other three cell probabilities are forced by the two marginals and the requirement that probabilities sum to one:

\begin{aligned} P(X=1,Y=1)&=t,\\ P(X=1,Y=0)&=0.4-t,\\ P(X=0,Y=1)&=0.5-t,\\ P(X=0,Y=0)&=1-t-(0.4-t)-(0.5-t)=0.1+t. \end{aligned}

Every cell probability must be nonnegative. In particular, \(t\geq 0\), \(0.4-t\geq 0\), and \(0.5-t\geq 0\). These imply \(t\geq0\) and \(t\leq0.4\); the remaining cell \(0.1+t\) is nonnegative throughout that range. Therefore any compatible joint distribution must have \(0\leq t\leq0.4\).

To check that these are sharp bounds, we must show compatibility at every value in the range, not just derive necessary inequalities. For each \(t\in[0,0.4]\), the four displayed probabilities are nonnegative and sum to one, and their marginals are \(0.4\) and \(0.5\). Thus each such \(t\) corresponds to an admissible joint distribution, and the exact identified set is \([0,0.4]\). At \(t=0\), the variables never equal 1 together; at \(t=0.4\), every unit with \(X=1\) also has \(Y=1\). Both endpoint distributions preserve the observed marginals.

The observed marginal law alone therefore does not identify the joint probability. It identifies the sharp interval from 0 to 0.4. The endpoint statements are about possible population distributions compatible with the specified marginals, not about sampling uncertainty.

A Short Derivation: Why Compatibility Gives Sharp Bounds

The logic in the binary example has two distinct parts. First, nonnegativity and the model restrictions give necessary constraints: every observationally equivalent distribution must satisfy them. Second, to establish sharpness, one must show that the proposed endpoints—and, when claiming an interval identified set, all values between them—are attainable.

Translate the observed law into restrictions.
Write the unknown full-data probabilities in terms of the observed probabilities and the target. Marginalization and adding-up restrictions determine some cells.

Impose feasibility.
Require all probabilities to be nonnegative and all model restrictions to hold. This yields a set of necessary inequalities for the target.

Construct compatible distributions.
For each proposed target value, verify that a full-data distribution exists that satisfies the restrictions and reproduces the observed law. This sufficiency check establishes attainability.

For the binary example, feasibility gave \(0\leq t\leq0.4\). Conversely, for any \(t\) in that interval, the four cell probabilities in the worked example form a valid distribution with the required marginals. Necessity plus attainability proves that the identified set is exactly \([0,0.4]\). Merely deriving inequalities would establish an outer bound, not necessarily a sharp one.

Valid but Non-Sharp Bounds

A valid bound contains every value in the identified set, but it may include values that no admissible distribution can produce. Such a bound is informative only to the extent that it restricts the parameter; calling it “a bound” does not establish that it is sharp. The following example makes the distinction explicit using the compatibility result just derived.

Worked Example: Tightening a Loose Interval

Continue with the same observed marginals and target \(t=P(X=1,Y=1)\). Suppose an analysis reports \([-0.1,0.5]\) as a range for \(t\). This interval is valid: every compatible value \(t\in[0,0.4]\) lies inside it. But it is not sharp.

A probability cannot be negative, and the joint probability cannot exceed either marginal probability. Hence the observed law requires \(t\geq0\) and \(t\leq\min(0.4,0.5)=0.4\). These restrictions tighten the reported interval to \([0,0.4]\). The earlier construction shows that every value in the tightened interval is attainable. Thus \([0,0.4]\), not \([-0.1,0.5]\), gives the sharp bounds.

The non-sharp interval is not false; it is unnecessarily wide. Its endpoints do not have the strong extremal interpretation of sharp endpoints: the lower endpoint \(-0.1\) is not the smallest parameter value compatible with the observed law, and the upper endpoint \(0.5\) is not the largest. In applied work, a loose valid bound may be a useful intermediate result, but it should not be presented as if no further tightening were possible.

Endpoint Interpretation and Attainability

An endpoint answers an extremal question: how low or high can the parameter be across distributions consistent with the model and observed law? It does not, by itself, say that the parameter equals that endpoint. Nor does the phrase “sharp bound” guarantee that a distribution attains the endpoint: an infimum or supremum can be sharp even when it is not reached.

Worked Example: A Sharp Upper Bound That Is Not Attained

Consider a hypothetical population with a continuous quantity \(Y\in[0,2]\). The observation records only which of two categories contains \(Y\): category 1 means \(Y\in[0,1)\), and category 2 means \(Y\in[1,2]\). Suppose the observed category probabilities are each \(0.5\). The target is the population mean \(E[Y]\). Assume only the stated support and category restrictions; in particular, do not impose a distributional shape within either category.

Let \(m_1\) and \(m_2\) be the conditional means in categories 1 and 2. Then \(0\leq m_1<1\) and \(1\leq m_2\leq2\). The law of total expectation gives

$$ E[Y]=0.5m_1+0.5m_2. $$

It follows that \(E[Y]\geq0.5(0)+0.5(1)=0.5\). The value 0.5 is attainable by setting \(Y=0\) for every member of category 1 and \(Y=1\) for every member of category 2. Also, \(E[Y]<0.5(1)+0.5(2)=1.5\), because category 1 requires \(Y<1\). Therefore 1.5 is an upper bound but cannot be attained.

It is nevertheless the sharp upper bound: for any \(\varepsilon\) with \(0<\varepsilon\leq1\), set \(Y=1-\varepsilon\) in category 1 and \(Y=2\) in category 2. This is compatible with the observed category probabilities and gives \(E[Y]=0.5(1-\varepsilon)+0.5(2)=1.5-0.5\varepsilon\), which approaches 1.5 as \(\varepsilon\) approaches zero. In fact, all values from 0.5 up to, but not including, 1.5 are attainable by choosing category-specific values in the allowed ranges. The exact identified set is \([0.5,1.5)\), with sharp infimum 0.5 and sharp supremum 1.5.

Thus a statement that the mean is “bounded above by 1.5” is correct, but it would be incorrect to say that some admissible distribution has mean exactly 1.5. Endpoint inclusion depends on the support and restrictions, not on whether an endpoint is written next to a bracket in a summary.

Why the Distinctions Matter

Partial identification makes explicit how conclusions depend on information and assumptions. If observationally equivalent processes disagree about the parameter, selecting one value requires additional identifying information or restrictions. Reporting the set keeps the remaining ambiguity visible. Sharpness also matters: a loose bound may make a conclusion appear less informative than the model permits, while a bound mistakenly called sharp can conceal an incomplete feasibility analysis.

Keep three questions separate when interpreting a result:

What is the model? Which full-data distributions and restrictions are admissible?
What is the exact identified set? Which parameter values can those distributions produce while matching the observed law?
Are the reported endpoints attainable? Are they minima and maxima, or only infima and suprema?

Identification bounds are population implications of the model and observed-data law. A finite sample introduces a separate source of uncertainty because the observed-data law must be estimated. A confidence procedure for bounds addresses that sampling uncertainty; it does not turn an unidentified parameter into a point-identified one. Conversely, an interval describing the identified set is not automatically a confidence interval, and its endpoints are not automatically confidence limits.

Common Pitfalls

Calling every containing interval sharp. “It contains all compatible values” establishes validity only. To establish sharpness, show that no endpoint can be tightened and verify attainability, or approachability, as appropriate.
Assuming every value between sharp endpoints is feasible. The interval from the infimum to the supremum is the interval hull. The exact identified set can have gaps or exclude an endpoint.
Reading an endpoint as the parameter value. A sharp endpoint describes an extremum over compatible processes, not necessarily the process that generated the data.
Confusing identification with precision. More precise estimation of the observed law does not remove ambiguity that remains even when that law is known exactly.

Key takeaway: Define the admissible model and observed-data law first. The identified set contains exactly the parameter values generated by observationally equivalent admissible processes. Sharp bounds are the best possible lower and upper limits, but endpoint attainment and the feasibility of intermediate values must be checked separately.

Check Your Understanding

Answer these questions using the definitions and examples above.

What does it mean for two full-data distributions to be observationally equivalent?
Why does deriving necessary inequalities for a parameter not, by itself, prove that the resulting bounds are sharp?
In the binary-marginal example, which construction verifies that every value of \(t\) between 0 and 0.4 is attainable?
Can sharp lower and upper bounds enclose values that are not in the exact identified set? Explain.
In the coarsening example, why is 1.5 a sharp upper bound even though no admissible distribution attains it?
Why is an identified set not automatically a confidence interval?