Sascha O. Becker and Marco Caliendo 3
2 Sensitivity Analysis with Rosenbaum Bounds
Checking the sensitivity of estimated treatment effects has become an increasingly im-
portant topic in the applied evaluation literature (see Caliendo and Kopeinig (2006) for
a recent survey of different methods to do so). Here, we are interested what happens
when there are deviations from the underlying identifying conditional independence
assumption.
2.1 The Model
Let us assume that the participation probability is given by Pi = P (xi, ui) = P (Di =
1 | xi, ui) = F (βxi + γui), where xi are the observed characteristics for individual i, ui
is the unobserved variable and γ is the effect of ui on the participation decision. Clearly,
if the study is free of hidden bias, γ will be zero and the participation probability will
solely be determined by xi . However, if there is hidden bias, two individuals with the
same observed covariates x have differing chances of receiving treatment. Let us assume
we have a matched pair of individuals i and j and further assume that F is the logistic
distribution. The odds that individuals receive treatment are then given by (1PP. ) and
(1PP ), and the odds ratio is given by:
TPPi _ Pi(1 — Pj) _ exp(βxi + γui)
1-Pp- Pj(1 — Pi) exp(βxj + γuj).
(1)
If both units have identical observed covariates - as implied by the matching procedure
- the x-vector cancels out implying that:
exp(βxi + γui)
exp(βxj + γuj )
= exp[γ(ui — uj )].
(2)
But still, both individuals differ in their odds of receiving treatment by a factor
that involves the parameter γ and the difference in their unobserved covariates u. So,
if there are either no differences in unobserved variables (ui = uj ) or if unobserved
variables have no influence on the probability of participating (γ = 0), the odds ratio
is one, implying the absence of hidden or unobserved selection bias. It is now the task
of sensitivity analysis to evaluate how inference about the programme effect is altered
by changing the values of γ and (ui — uj). We follow Aakvik (2001) and assume for the
sake of simplicity that the unobserved covariate is a dummy variable with ui ∈ {0, 1}.
Rosenbaum (2002) shows that (1) implies the following bounds on the odds-ratio that
either of the two matched individuals will receive treatment:
ɪ ≤ P(1
eγ ~ Pj (1
Pj ) ≤ eγ
Pi ) ≤ e .
(3)
Both matched individuals have the same probability of participating only if eγ = 1.
Otherwise, if for example eγ = 2, individuals who appear to be similar (in terms of x)