Treatment Effects

Treatment Effects The treatment effects literature is about how some outcome of interest, such as earnings, is affected by some treatment, such as a job training program. Evidently such treatment effects must be related to structural models, where the outcome of interest is the left hand side variable and the treatment is a right-hand side variable. Indeed, as we will see, treatment effects can be thought of as coming from a linear structural model with random coefficients. Treatment effect models do have a terminology and set up all their own though, so to help understand the literature it is important to set them up the way others do. To do so, let i index individuals and Di denote a treatment indicator, equal to 1 if a person is treated, and equal to 0 otherwise. For example, Di = 1 might correspond to enrollment in some training program or to some medical treatment. To describe the treatment effect, we need to define two other variables. Let Yi0 denote the potential outcome that would occur when person i is not treated (Di = 0) and Yi1 the potential outcome when they are treated (Di = 1). Clearly these are not both observed. One of them will be “counterfactual”, an outcome that would have occurred if a different treatment had been given. The observed outcome will be Yi = Di Yi1 + (1 − Di )Yi0 . The treatment effect for individual i is given by βi = Yi1 − Yi0 . This object is clearly not identified, because only one of the potential outcomes is observed. There are several objects that may be interesting that are identified under certain conditions. One of these is the average treatment effect, given by def AT E = E[βi ]. This describes the average over the entire population of the individual treatment effects. Another interesting object is the average effect of treatment on the treated, given by def AT T = E[βi |Di = 1]. This gives the average over the subpopulation of treated people of the treatment effect. A third important object that is also of interest in the literature is called the local average treatment effect. It will be described below. 1 To help understand the treatment framework and the various effects, it helps to relate this to a regression model with random coefficients. By the equation for Yi given above, Yi = Yi0 + (Yi1 − Yi0 )Di = αi + βi Di , αi = Yi0 , βi = Yi1 − Yi0 . Thus we see that Yi follows a linear model where the treatment effect βi is the coefficient of Di and the constant αi and slope βi may vary over individuals. The ATE is then the average of the slope over the entire population and the ATT is the average of the slope over the subset of the population where Di = 1. This random coefficient set up also helps place the treatment effects environ ment in a proper historical context. The coefficient βi = Yi1 − Yi0 is sometimes called a “counterfactual” because it describes how Yi would have been different if Di had been different. In the context of demand and supply systems we are familiar with such objects as “movements along a curve.” This kind of object was considered in economics as early as Wright (1928), who gives a nice explanation of “movements along a curve” in a supply and demand setting. Similarly, the average treatment effect is just the expected value of the random coefficient in a linear model, i.e., the average slope of the curve. The ATE and ATT will be identified and can be estimated under various as sumption. Here we will discuss various cases in which these objects are identified. The proofs of identification will consist of showing how the objects can be written in terms of expectations of the data. We begin with the simplest case. Constant Treatment Effects A simple special case of this model is constant treatment effects where βi = β̄, i.e. where the treatment effect is constant across individuals. Here the ATE and ATT is simply β̄. In this case, for ᾱ = E[αi ] and εi = αi − ᾱ, ¯ i + εi . Yi = α ¯ + βD Here the model reduces to a simple linear model with an additive disturbance and constant coefficients. In contrast, the general model is also a linear model with additive disturbance but random slope coefficient. Note here the equivalence between having a random αi and having a constant plus disturbance αi = ᾱ + εi . We can identify and estimate β̄ and ᾱ in the usual way if we have an instrument Zi that is uncorrelated with εi and correlated with Di , that is 0 = Cov(Zi , εi ) = Cov(Zi , αi ) = Cov(Zi , Yi0 ), Cov(Zi , Di ) = � 0. 2 In this case the coefficient is identified from the usual IV equation β̄ = Cov(Zi , Yi )/Cov(Zi , Di ). This coefficient can be estimated in the usual way by replacing population covari ances by sample covariances. In summary, there is not much new here, except terminology of putting standard model with dummy endogenous variable in a treatment effects framework, as “constant treatment effect.” Constant treatment effects is too strong for many settings. It would say that effect of training on earnings or of smaller class size on education is the same for every individual. This seems unlikely to hold in practice. Instead we would like to allow βi to vary over individuals. Random Assigment Random assignment means that whether or not a person is treated does not depend on their outcomes. The specific statistical assumption that we make is that E[Yi0 |Di ] = E[Yi0 ], i.e. that the mean of the nontreated variable does not depend on treatment status. Equivalently we can say that E[αi |Di ] = 0. This is slightly more general than independence, because it allows the higher-order moments of Yi0 to depend on Di . However, it seems difficult to think of environments where the mean assumption would be true without full independence. To see what happens under this assumption note first that E[βi |Di ]Di � 0, Di = 0, = = E[βi |Di = 1]Di . E[βi |Di = 1], Di = 1 Then under the mean independence assumption, E[Yi |Di ] = E[αi + βi Di |Di ] = E[αi ] + E[βi |Di ]Di = E[αi ] + E[βi |Di = 1]Di . Here the dummy variable regression of Yi on a constant Di has its slope coefficient the ATT. If in addition we assume that the mean of Yi1 does not depend on Di , i.e. if we assume that E[Yi1 |Di ] = E[Yi1 ], then we find that AT E = AT T , since 3 E[βi |Di = 1] = E[Yi1 |Di = 1] − E[Yi0 |Di = 1] = E[Yi1 ] − E[Yi0 ] = E[βi ]. Summarizing , we find that when Yi0 is mean independent of Di that the AT T is identified as the dummy coefficient in a regression of the outcome variable Yi on a constant and the treatment dummy variable. We also find that if, in addition, Yi1 is mean independent of Di then the AT E is also this coefficient. Of course, this coefficient can be estimated by a linear regression of Yi on (1, Di ). Further, as always, that linear regression coefficient is just the difference of means of Yi for the treated and untreated observations. Discussion Random assignment is too strong for many applications. Often individuals can choose whether to accept the treatment or not, e.g. by dropping out of the sample if they don’t like the treatment conditions. They can opt out of training programs, or not take medical treatment. If these decisions are related to (αi , βi ) then we do not have independence of (αi , βi ) and Di . In terms of the linear model Yi = αi + βi Di we have possible endogeneity, where Di may be correlated with the random coefficients αi and βi . This is a more severe problem than the usual case because the slope βi also may be correlated with Di . There are two approaches to this problem. One (familiar) one is instrumental variables (IV). The second approach is called “selection on observables.” In that approach conditioning on some observable variables removes the correlation be tween Di and (αi , βi ). Because IV is a most familiar and common approach we will first consider IV. IV Identification of Treatment Effects In the usual linear model, of which the constant treatment effects is a special case, the assumptions that are needed for the identification of the slope is that the instrument is uncorrelated with the disturbance and correlated with Di . Similar conditions will be used for IV identification of treatment effects. Let Zi be an instrument. We will assume throughout that E[αi |Zi ] = E[Yi0 |Zi ] = E[Yi0 ] = E[αi ], i.e. that the outcome without treatment is mean independent of the instrument. We also will focus on the case where Zi is also a dummy variable, i.e. where Zi ∈ {0, 1}, with P = Pr(Zi = 1) and 0 < P < 1. (Question: Why do we assume 0 < P < 1?). For a dummy instrument there is a useful formula for the covariance between the instrument and any other random variable Wi . Specifically, we have 4 Cov(Wi , Zi ) = E[Wi Zi ] − E[Wi ]E[Zi ] = ( E[Wi Zi ] − E[Wi ])P P = (E[Wi |Zi = 1] − E[Wi ])P = {E[Wi |Zi = 1] − (P E[Wi |Zi = 1] + (1 − P )E[Wi |Zi = 0])} P = (E[Wi |Zi = 1] − E[Wi |Zi = 0])P (1 − P ). That is, the covariance between Wi and Zi is the difference of the conditional mean at the two values of Z times P (1 − P ). This formula has two useful implications. The first is that mean independence of Yi0 from Zi is equivalent to Yi0 being uncorrelated with Zi . This occurs since Cov(Zi , Yi0 ) = 0 if and only if E[Wi |Zi = 1] = E[Wi |Zi = 0]. A second useful implication is a formula for the limit of the IV estimator of the slope, given by Cov(Zi , Yi ) E[Yi |Zi = 1] − E[Yi |Zi = 0] = Cov(Zi , Di ) E[Di |Zi = 1] − E[Di |Zi = 0] This is often referred to the Wald IV formula, referring to work here Wald sug gested using dummy variables as an IV solution to the measurement error problem. In general, under mean independence of αi from Zi , it does not seem like the IV formula identifies the ATT, the ATE, or anything useful. Plugging in Yi = αi + βi Di , and using mean independence of αi we find Cov(Zi , Yi ) E[αi |Zi = 1] − E[αi |Zi = 0] + E[βi Di |Zi = 1] − E[βi Di |Zi = 0] = Cov(Zi , Di ) E[Di |Zi = 1] − E[Di |Zi = 0] E[βi Di |Zi = 1] − E[βi Di |Zi = 0] = . E[Di |Zi = 1] − E[Di |Zi = 0] In general, the problem is that βi and Di are correlated, so that (apparently) it is not possible separate them out in general. There are two interesting, specific cases though where something important is identified. They are random intention to treat and local average treatment effects. Random Intention to Treat A common occurrence in medical trials is that people are randomly assigned to treatment but that not all take the treatment. Here Zi represents the assignment to treatment, with Zi = 1 if individual i is assigned to be treated and Zi = 0 if they are not. In this setting, the only ones who are treated (i.e. for which Di = 1) will be those who were randomly assigned to treated. It turns out that in this case IV gives the ATT. This finding, due to Imbens and Rubin, has led to the widespread use of IV in biostatistics. 5 To show that IV gives the ATT, note that Zi = 0 will not be treated, i.e. Di = 0 when Zi = 0. Then Cov(Zi , Yi ) E[βi Di |Zi = 1] − 0 E[βi Di |Zi = 1] = = . Cov(Zi , Di ) E[Di |Zi = 1] − 0 E[Di |Zi = 1] Also, note that Di = 1 implies Zi = 1, so that {Di = 1} ⊂ {Zi = 1}. Therefore, E[βi |Di = 1, Zi = 1] = E[βi |Di = 1] = AT T . Also, it follows similarly to the reasoning above that Di E[βi |Di , Zi = 1] = Di E[βi |Di = 1, Zi = 1] = Di · AT T. By iterated expectations it follows that E[βi Di |Zi = 1] = E[Di E[βi |Di , Zi = 1]|Zi = 1] = AT T · E[Di |Zi = 1]. Then dividing gives Cov(Zi , Yi ) E[βi Di |Zi = 1] AT T · E[Di |Zi = 1] = = = AT T. Cov(Zi , Di ) E[Di |Zi = 1] E[Di |Zi = 1] The Local Average Treatment Effect A second case where an interesting treatment effect is identified by IV involves independence and monotonicity conditions. Consider the following conditions: Independence: Di = Π(Zi , Vi ) and (βi , Vi ) is independent of Zi ; Monotonicity: Π(1, Vi ) ≥ Π(0, Vi ) and Pr(Π(1, Vi ) > Π(0, Vi )) > 0. The independence condition says that there is a reduced form Π(z, v) with a disturbance Vi that may be a vector and enters nonlinearly. An example is a threshold crossing model where Di = 1(Zi + Vi > 0). The monotonicity condition changing the instrument only moves the treatment one direction. This condition is satisfied in a threshold crossing model. The reduced form is sometimes called the selection equation, with a person being selected into treatment when Π(z, v) = 1. Under these conditions it turns out that IV identifies an average of βi over a subpopulation that is referred to as the Local Average Treatment Effect (LATE). This effect is defined as LAT E = E[βi |Π(1, Vi ) > Π(0, Vi )]. This object is the average of the treatment effect over the individuals whose be havior would be different if the instrument were changed. This object may often 6 be a parameter of interest. For example, in a model where Yi is the log of earn ings, Di is completing high school, and Zi is a quarter of birth dummy, LATE is the average effect of a high school education over all those dropouts who would have remained in school had their quarter of birth been different and for those who remained in school but would have dropped out if their quarter of birth were different. Thus, IV estimates the average returns to completing high school for potential dropouts. This is an interesting parameter, although it is not the returns to schooling over the whole population. To show that IV give LATE under independence and monotonicity, let Ti = Π(1, Vi ) − Π(0, Vi ). Then we have E[βi Di |Zi = 1] − E[βi Di |Zi = 0] = E[βi Π(1, Vi )|Zi = 1] − E[βi Π(0, Vi )|Zi = 0] = E[βi Π(1, Vi )] − E[βi Π(0, Vi )] = E[βi Ti ]. It follows similarly that E[Di |Zi = 1] − E[Di |Zi = 0] = E[Ti ]. By monotonicity, Ti is a dummy variable, taking the value zero or one. Therefore we have Cov(Zi , Yi ) E[βi Ti ] = = E[βi |Ti = 1] = E[βi |Π(1, Vi ) > Π(0, Vi )]. Cov(Zi , Di ) E[Ti ] LATE Empirical Example An empirical example is provided by the Angrist and Krueger (1991) study of the returns to schooling using quarter of birth as an instrument. We consider data drawn from the 1980 U.S. Census for males born in 1930-1939, as in Donald and Newey (2001, “Choosing the Number of Instruments,” Econometrica). The 2SLS estimator with 3 instruments is .1077 with standard error .0195 and the FULL estimator with 180 instruments is .1063 with standard error .0143 (corrected for many instruments). Thus we find returns to schooling of “potential dropouts” is about 11 percent. Selection on Observables The other kind of model that has been used to identify treatment effects is one where conditioning on observable (or identifiable) variables Xi makes the treat ment behave as if it were randomly assigned. This is like removing endogeneity in a linear equation by adding regressors. The conditioning variables are like omitted 7 regressors, which remove the endogeneity when they are included. The specific assumption that is made is E[Yi0 |Xi , Di ] = E[Yi0 |Xi ]. In word, it is assumed that Yi0 is mean independent of Di conditional on Xi . This assumption is analogous to the previous one that E[Yi0 |Di ] = E[Yi0 ], being a conditional version of that hypothesis. One concern with this kind of assumption is the source for the variables Xi . There are some economic models where such variables are implied by the model. However in many cases in applications these variables Xi are chosen without reference to a model. In those cases identification is fragile, requiring specifying just the right Xi . Conditional mean independence that holds for X, need not hold for a subset of Xi nor when additional variables are added to Xi . This assumption allows identification of the ATT, with one additional condi tion. Let χ denote the support of Xi (the smallest closed set having probability one), and χ0 and χ1 the support of Xi conditional on Di = 0 and Di = 1 respec tively. The additional condition is the common support condition that χ = χ0 = χ1 . This assumption is necessary and sufficient for E[Yi |Xi , Di = 1] and E[Yi |Xi , Di = 0] to be well defined for all Xi . It is verifiable and may or may not be satisfied in practice. The common support condition and conditional mean independence give E[Yi |Xi , Di = 1] − E[Yi |Xi , Di = 0] = E[αi |Xi , Di = 1] − E[αi |Xi , Di = 0] + E[βi |Xi , Di = 1] = E[βi |Xi , Di = 1]. The object E[βi |Xi , Di = 1] is a conditional version of the ATT. By iterated expectations the ATT is then identified as the expectation over Xi of this difference given Di = 1, that is AT T = E[βi |Di = 1] = E [{E[Yi |Xi , Di = 1] − E[Yi |Xi , Di = 0]} |Di = 1] . The ATE can also be obtained if we assume that Yi1 is conditional mean indepen dent of Di conditional on Xi . In that case E[βi |Xi , Di = 1] = E[Yi1 |Xi , Di = 1] − E[Yi0 |Xi , Di = 1] = E[Yi1 |Xi ] − E[Yi0 |Xi ] = E[βi |Xi ]. Therefore 8 AT E = E[βi ] = E [{E[Yi |Xi , Di = 1] − E[Yi |Xi , Di = 0]}] Unlike the unconditional case the ATE is a different function of the data dis tribution than the ATT. The ATE is obtained by averaging E[Yi |Xi , Di = 1] − E[Yi |Xi , Di = 0] over all Xi while the ATT is obtained by averaging over just Di = 1. Estimating the ATT and ATE under these conditional restrictions is a chal lenge. Notice that they depend on conditional expectations. Usually we will not want to assume that these conditional expectations have any particular functional form. Consequently, we will want to use nonparametric regression estimators, which will be discussed later in the course. Nonparametric estimation is difficult when the dimension of Xi is large. This is often referred to as the “curse of dimensionality.” Some have tried to reduce the curse of dimensionality using the “propensity score” P (X), which is defined as the conditional probability of being treated (or “selected”) given X, i.e. P (Xi ) = Pr(Di = 1|Xi ) = E[Di |Xi ]. It turns out that the conditional mean independence of Yi0 given Xi implies con ditional mean independence given P (Xi ). Thus, if P (Xi ) were known, it would be possible to identify and estimate the ATE and ATT using a one dimensional conditioning variable rather than a multidimensional variable Xi . Specifically, if E[Yi0 |Xi , Di ] = E[Yi0 |Xi ] and 0 < P (Xi ) < 1 with probability one then E[Yi0 |P (Xi ), Di ] = E[Yi0 |P (Xi )], so that reasoning like that above gives AT T = E [{E[Yi |P (Xi ), Di = 1] − E[Yi |P (Xi ), Di = 0]} |Di = 1] . If, in addition, E[Yi0 |Xi , Di ] = E[Yi0 |Xi ] then AT E = E [{E[Yi |P (Xi ), Di = 1] − E[Yi |P (Xi ), Di = 0]}] . Thus, ATE and ATT are expectations of nonparametric functions of two variables, P (Xi ), and Di . If P (Xi ) is completely unknown and unrestricted there is no known advantage for conditioning on the propensity score, since P (Xi ) is also a function of a highdimensional argument. Thus, it appears that any advantage for using the propen sity score will depend on knowing more about P (X) than about E[Yi |Xi , Di ]. It remains to prove that independence conditional on X implies independence conditional on P (Xi ). For notational simplicity let Pi = P (Xi ). We will prove the result for a general variable Wi . The general result will then apply to both Yi0 and Yi1 . To prove that E[Wi |Xi , Di ] = E[Wi |Xi ] implies E[Wi |Pi , Di ] = E[Wi |Pi ], note that by iterated expectations, E[Di |Pi ] = E[E[Di |Xi ]|Pi ] = Pi . By iterated expectations again, 9 E[Wi |Pi , Di = 1] = E[E[Wi |Xi , Di = 1]|Pi , Di = 1] = E[E[Wi |Xi ]|Pi , Di = 1] E[Di E[Wi |Xi ]|Pi ] E[Pi E[Wi |Xi ]|Pi ] = = E[Di |Pi ] Pi = E[E[Wi |Xi ]|Pi ] = E[Wi |Pi ]. By similar reasoning we also have E[Wi |Pi , Di = 0] = E[Wi |Pi ], so the conclusion follows by the previous equation. 10