In this section, we illustrate, through concrete examples, how the order relations developed above determine the toxicity of competitive selection’s effect on qualitative inference. First we consider the posterior problem of inferring the properties of the unconditional distributions from selection-conditioned observations. We show that in a classical linear regression model of treatment effects, the assumption of standard error law distributions, implies that the advantage of one treatment over another is fairly uniform over the range of possible responses. Thus, selection is innocuous in that it never reverses the stochastic ordering of the treatments, i.e., the sign of the treatment effect can be identified by simple, naive, statistical approaches that ignore the problem of selection bias. Next we illustrate the use of the ordering relations in prior problems. Here we derive selection conditioned predictions from unconditional predictions of two auction models.
9.1 Identifying the sign of treatment effects in standard regression mod-
els in the presence of self-selection.
In this section, we apply the results developed above to the problem of identifying the sign of treatment effects in the presence of self-selection bias. The development follows Manski (1990).13Manski analyzes the following problem: consider a population of subjects. Assume
that a random sample of the population is drawn and for each subject in the sample, the econo- metrician observes x and z, where x represents a vector of individual covariates of the subjects and z is a indicator function which equals 1 if the subject received drug treatment A and equals 0 if the subject received drug treatment B. Let yAbe a scalar measure of the outcome of treatment
A and let yBrepresent the outcome of treatment B. The econometrician’s difficulty is selectivity
bias: the econometrician observes ˜yA only when the subject receives treatment A and observes
˜yB only when the subject receives treatment B. The econometrician’s aim is to infer the dif-
ference in unconditional treatment effects, the difference in the average effects of the drugs on subjects with covariates x under random assignment, i.e., to infer E[ ˜yA| ˜x] − E[ ˜yB| ˜x]. One
version of the problem considered by Manski assumes that selection is based on self selection, i.e., treatment A is selected by a subject if and only if ˜yA> ˜yB.
Under self-selection, the problem of determining the sign of the treatment effect from the observed data becomes a problem of inferring E[ ˜yA| ˜x] − E[ ˜yB| ˜x] > 0 from E[ ˜yA| ˜x, ˜yA >
˜yB]− E[ ˜yB| ˜x, ˜yB > ˜yA]. Manski considers the question of how non-parametric restrictions on
the supports of the random variates can be used to identify the sign of the treatment effect. We consider a different question, in a standard parametric regression, what restrictions on the
error-term distribution permit identification of the sign of the treatment effect? In a standard parametric regression (linear or non-linear), the distribution of outcomes under treatment A and B can be expressed as
˜y = tB(˜x) + (tA(˜x) −tB(˜x))IA+Φ( ˜x) + ˜ε. (29)
whereΦ is a real-valued function of the covariates, x, and ˜ε, is an independent and identically distributed error term.
For any fixed vector of characteristics, x, the distributions of ˜yA and ˜yB are location trans-
lations of the distribution of the error term ˜ε, the translation being given by the average dif- ference in treatment effects given x. If the distribution of the error term comes from one of the translation-shift families of distributions identified in Table 8, e.g., ˜ε’s distribution is Normal, Logistic, or Laplace, then stochastic dominance is preserved by conditioning, i.e., E[ ˜yA| ˜x, ˜yA > ˜yB]− E[ ˜yB| ˜x, ˜yB> ˜yA] implies that E[ ˜yA| ˜x] − E[ ˜yB| ˜x] > 0. Thus the sign (but not
the magnitude) of the treatment effect can be identified by the standard regression model even in the presence of self-selection. Moreover, if the error term is Normally or Logistically dis- tributed, translations of the error term are ordered by geometric dominance, and thus, even, if in addition to self-selection bias, the self-selected sample is truncated based on observed values of ˜y, a simple regression would still correctly identify the sign of the treatment effect. Essentially, the same argument shows that the sign of the treatment effect in a standard Logit regression can be identified even in the presence of self selection bias.
Note also that identification of the sign of the treatment effect does not depend on informa- tion about the propensity of subjects to choose a given treatment. Thus, the average treatment effect could be determined from a study that sampled a predetermined number of patients re- ceiving each treatment, provided that the sample for each treatment was random. This contrasts from both standard parametric and nonparametric identification approaches where the propen- sity to choose the treatment is a key element in the identification strategy.
Of course, these results rest on the assumption that, save for self-selection and perhaps trun- cation, the regression model is correctly specified. As we demonstrated in Section 6, dispersion alone can induce geometric dominance. Thus, if the error term is not homoscedastic and varies with the treatment selected, the misspecified model might not identify the sign of the treatment effect. Even absent dispersion effects, our previous results point to a very simple case where selection preservation would fail. Suppose that the two drugs produce identically distributed effects for most subjects but for a fraction,ρ, of patients, drug B produces an allergic reaction with extremely negative side effects. In this case, allergic patients would choose drug A even if its effects were mediocre, while drug B would only be selected if it produced better effects than drug A. Thus, becasue of the “admission effects” discussed in Section 7.2, selection rever- sal would occur and the inferior drug B, would produce better results on average when it was selected.