Why Oster’s identified set can contain three coefficients
A small example with an exactly factorable cubic
The cubic in Masten and Poirier (2026), Section I.C, equation (4), can have three admissible real roots. Here is an example where, at the same sensitivity parameter \delta=1/2 and the same R_{\mathrm{long}}^2=1, the identified set is
\mathcal I_\beta(1/2,1)=\{-1,3,4\}.
We use population moments throughout, so there is no sampling uncertainty to account for.
1. A simple observed population
Let U,V,Z,E be independent, mean-zero normal random variables with
\operatorname{Var}(U)=\operatorname{Var}(Z)=1, \qquad \operatorname{Var}(V)=3, \qquad \operatorname{Var}(E)=12.
Define the observed treatment and outcome by
X=U+Z, \qquad Y=X+U+V+E=2U+V+Z+E.
The observed controls are W_1=(U,V)'. We observe (Y,X,U,V); the decomposition into Z and E is a device for specifying their joint distribution. Normality makes the population concrete but is unnecessary for the regression calculations.
The medium regression, which includes both observed controls, is already written for us:
Y=\underbrace{1}_{\beta_{\mathrm{med}}}X +\underbrace{1}_{\gamma_{U,\mathrm{med}}}U +\underbrace{1}_{\gamma_{V,\mathrm{med}}}V+E.
Also,
\operatorname{Var}(X)=2, \qquad \operatorname{Var}(Y)=4+3+1+12=20, \qquad \operatorname{Cov}(X,Y)=3.
Consequently, the short-regression coefficient and the two observed R-squared values are
\beta_{\mathrm{short}}=\frac32, \qquad R_{\mathrm{short}}^2=\frac{3^2}{2\cdot20}=\frac9{40}, \qquad R_{\mathrm{med}}^2=1-\frac{12}{20}=\frac25.
Fix R_{\mathrm{long}}^2=1. This endpoint is allowed by the paper’s assumptions and keeps the example especially simple.
2. Write the paper’s cubic using these numbers
Let b denote a candidate long-regression coefficient and define the omitted-variable bias
B=\beta_{\mathrm{med}}-b=1-b.
To make equation (4) readable, abbreviate its observed moments as
\begin{aligned} v&=\operatorname{Var}(X^{\perp W_1}), &s&=\operatorname{Var}(\gamma_{1,\mathrm{med}}'W_1),\\ c&=\operatorname{Cov}(X,\gamma_{1,\mathrm{med}}'W_1), &q&=\operatorname{Var}(\pi_1'W_1),\\ D&=(R_{\mathrm{long}}^2-R_{\mathrm{med}}^2)\operatorname{Var}(Y). \end{aligned}
Here \pi_1 is the coefficient vector from regressing X on W_1. In our example,
\pi_1=(1,0)',\qquad X^{\perp W_1}=Z, \qquad \gamma_{1,\mathrm{med}}'W_1=U+V,
so all five inputs are simple numbers:
v=1,\qquad s=4,\qquad c=1,\qquad q=1, \qquad D=(1-2/5)20=12.
The paper’s equation is f_0(B)+\delta f_1(B)=0, where
\begin{aligned} f_0(B)&=-Bv(s+2Bc+B^2q),\\ f_1(B)&=Dc+BDq+B^2vc+B^3vq. \end{aligned}
Substitution gives
-B(4+2B+B^2) +\delta(12+12B+B^2+B^3)=0.
The highest power is B^3. Its coefficient is \delta-1, so this is a cubic whenever \delta\ne1 in this example. A cubic need not have three real roots; our chosen numbers will make it do so.
3. Fix one delta and solve
Set \delta=1/2. Then
\begin{aligned} 0 &=-B(4+2B+B^2)+\frac12(12+12B+B^2+B^3)\\ &=-\frac12B^3-\frac32B^2+2B+6. \end{aligned}
Multiply by -2 and factor:
B^3+3B^2-4B-12 =(B+3)(B+2)(B-2)=0.
Thus B\in\{-3,-2,2\}. Since b=1-B, the corresponding coefficients are
\boxed{b\in\{4,3,-1\}.}
Equivalently, the cubic written directly in the coefficient b is
(b-4)(b-3)(b+1)=0.
4. Check that these are possible regressions
The polynomial calculation alone does not establish that every root satisfies all the assumptions. Here we can explicitly construct the omitted variable for each root.
For a candidate bias B, define one scalar omitted variable
W_2^{(B)}=BZ+E.
Then the same observed outcome satisfies the identity
Y=(1-B)X+(1+B)U+V+W_2^{(B)}.
Because the regressors have a positive definite covariance matrix, these are the unique OLS coefficients in this long regression. Its residual is zero, giving R_{\mathrm{long}}^2=1 as required.
The controls are exogenous in the paper’s sense:
\operatorname{Cov}(U,W_2^{(B)}) =\operatorname{Cov}(V,W_2^{(B)})=0.
The observed-control coefficient vector is (1+B,1)', which never equals the zero vector. The omitted-variable coefficient is 1, also nonzero. Thus none of the three roots belongs to the exclusion set in equation (5).
We can check \delta directly. The two outcome-weighted control indices are
\gamma_{1,\mathrm{long}}'W_1=(1+B)U+V, \qquad \gamma_{2,\mathrm{long}}W_2^{(B)}=BZ+E.
Their selection coefficients, as defined in the paper’s equation (2), are
S_{\mathrm{obs}}(B) =\frac{1+B}{(1+B)^2+3}, \qquad S_{\mathrm{unobs}}(B) =\frac{B}{B^2+12}.
Their ratio is
\delta(B) =\frac{S_{\mathrm{unobs}}(B)}{S_{\mathrm{obs}}(B)} =\frac{B[(1+B)^2+3]}{(B^2+12)(1+B)}.
| Bias B | Long coefficient b=1-B | S_{\mathrm{obs}} | S_{\mathrm{unobs}} | \delta |
|---|---|---|---|---|
| -3 | 4 | -2/7 | -1/7 | 1/2 |
| -2 | 3 | -1/4 | -1/8 | 1/2 |
| 2 | -1 | 1/4 | 1/8 | 1/2 |
These are three different hypothetical completions of the same observed population, each with a different omitted variable. Each completion satisfies the same restrictions on \delta and R_{\mathrm{long}}^2. A single fully specified population still has a unique long-regression coefficient; multiplicity arises because the omitted variable is not observed.
5. See the three intersections
The left panel uses the orientation of the paper’s Figure 1: \delta on the horizontal axis and b on the vertical axis. The vertical line at \delta=1/2 intersects the curve at b=-1,3,4. The right panel shows the corresponding three zeros of the cubic.
Why use two observed controls?
This is the smallest number of observed controls that can give three admissible roots under these assumptions. With just one observed control W, write \gamma=\gamma_{1,\mathrm{med}}, \pi=\pi_1, and t=\operatorname{Var}(W)>0. Then
s=\gamma^2t,\quad c=\pi\gamma t,\quad q=\pi^2t,
and the paper’s polynomial factors as
f(B,\delta) =t(\gamma+B\pi) \left[-Bv(\gamma+B\pi)+\delta\pi(D+B^2v)\right].
The first factor gives \gamma+B\pi=0, which makes the long-regression coefficient on the observed control zero and violates Assumption 3. After excluding that root, the remaining equation is at most quadratic. In our two-control example, the coefficient on V remains 1, so no such root is excluded.
Source: Masten and Poirier (2026), “The Effect of Omitted Variables on the Sign of Regression Coefficients,” Sections I.B–I.C, equations (2), (4), and (5). The numerical example here is constructed for this explanation, rather than taken from the paper’s empirical application.
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