The logic of partial-risk aversion: Paradox lost

One rational individual may be willing to pay less than another to insure a risk % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf% gDOfdaryqr1ngBPrginfgDObYtUvgaiuaacuWF1oG8gaacaaaa!41B4!\[\tilde \varepsilon \] when another risk % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4Dayaaia% aaaa!36F7!\[\tilde w\] is present even though he would pay more to insure any isolated risk, and even though % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyraiGacI% cadaabcaqaamrr1ngBPrwtHrhAXaqeguuDJXwAKbstHrhAG8KBLbac% faGaf8xTdiVbaGaaaiaawIa7aiqadEhagaacaiGacMcaciGG9aGaai% imaaaa!47F3!\[E(\left. {\tilde \varepsilon } \right|\tilde w) = 0\] for all w. Noticing this, Ross (1981) proposed excluding such reversals and gave equivalent analytical conditions. Reconsidering, we explain why some reversals are natural and show that prohibiting them has peculiar and undesirable properties. Although we also simplify the conditions and prove them necessary for partial-risk portfolio results, we conclude that they represent revealing restrictions on comparative statics rather than natural implications of increased aversion to risk.     

Inclusion of binary proxy variables in logistic regression improves treatment effect estimation in observational studies in the presence of binary unmeasured confounding variables

We present a simulation study and application that shows inclusion of binary proxy variables related to binary unmeasured confounders improves the estimate of a related treatment effect in binary logistic regression. The simulation study included 60,000 randomly generated parameter scenarios of sample size 10,000 across six different simulation structures. We assessed bias by comparing the probability of finding the expected treatment effect relative to the modeled treatment effect with and without the proxy variable. Inclusion of a proxy variable in the logistic regression model significantly reduced the bias of the treatment or exposure effect when compared to logistic regression without the proxy variable. Including proxy variables in the logistic regression model improves the estimation of the treatment effect at weak, moderate, and strong association with unmeasured confounders and the outcome, treatment, or proxy variables. Comparative advantages held for weakly and strongly collapsible situations, as the number of unmeasured confounders increased, and as the number of proxy variables adjusted for increased.