published in Statistics and Probability Letters, 78, , 2008 Michael Lechner * SIAW

Transcription

1 publihed in Statitic and Probability Letter, 78, 9-95, 28 A NOTE ON ENDOGENOUS CONTROL VARIABLES IN CAUSAL STUDIES Michael Lechner * SIAW Thi verion: March, 27 Date thi verion ha been printed: 8 May 27 Abtract The iue of potentially endogenou control variable in caual tudie baed on the aumption of no election bia conditional on obervable (CIA) i dicued. The paper how that the tandard formulation of the CIA obcure the endogeneity problem. It ugget a CIA baed on potential variable together with explicit exogeneity condition which allow a eparate aement of the endogeneity bia and the plauibility of the CIA. Keyword: Evaluation tudie, matching method, conditional independence aumption, endogeneity JEL claification: C2, C3 * Addre for correpondence: Michael Lechner, Profeor of Econometric, Swi Intitute for International Economic and Applied Economic Reearch (SIAW), Univerity of St. Gallen, Bodantr. 8, CH-9 St. Gallen, Switzerland, Michael.Lechner@unig.ch, The author i alo affiliated with CEPR, London, ZEW, Mannheim, IZA, Bonn, and PSI, London.

2 Introduction An example of caual tudie i the rapidly growing literature on the evaluation of labour market programme (ee for example the urvey by Heckman, LaLonde, and Smith, 999). Many evaluation tudie of labour market programme take account of election effect into the programme by collecting large and informative data with many (control) variable that help to explain participation in the programme a well a outcome variable of interet. Then, they proceed with matching method (ee a example Sianei, 24, and Gerfin and Lechner, 22, among many other) to etimate the effect of the programme on the pecific ubgroup of participant and nonparticipant. If conditional on the rich control variable, the participation and the outcome variable are independent, the o-called election on obervable, or conditional independence aumption (CIA) hold. Then, a major condition for the conitency of matching etimator i fulfilled (ee the urvey by Imben, 24, for etimation method). However, ometime, the reearcher may not be ure whether the control variable are influenced by the treatment. An example i an unclear timing of the control variable, o that ome of them may be actually meaured after the treatment. Or they are meaured prior to the treatment, but the future treatment tart i already known and behaviour change becaue of thi knowledge. Although there i ome notion in the literature that conditioning variable hould not be influenced by programme participation (not to be confued with the fact that they may be correlated with programme participation), which i pelled out for example in Frangaki and Rubin (22), Heckman, LaLonde, and Smith (999), Roenbaum (984) and Rubin (24), a clear definition applicable to thi cae i miing. Strangely, an anwer to a

3 imple quetion like Doe endogeneity of the control variable matter if the CIA hold? appear to be miing a well. Thi note provide an anwer to that quetion ('it doe not matter') and define the exogeneity condition required for identification. The main thrut of the paper i that the conditional independence aumption baed on obervable control variable i ometime a mileading vehicle for dicuing identification of caual effect in a framework baed on potential outcome. Therefore, I ugget a reformulation of the conditional independence aumption together with explicit exogeneity condition that formally eparate the dicuion of election bia coming from a et of control variable that i not rich enough, from the dicuion about their potential endogeneity. The paper i organied a follow: The next ection introduce the notation of the caual model baed on potential outcome and define the parameter of interet. Section 3 conider the endogeneity problem in the tandard formulation of the conditional independence aumption (CIA). Section 4 and 5 reformulate the tandard CIA in term of potential control variable and characterie the bia a well a the exogeneity aumption required to avoid it. 2 The prototypical binary treatment model The binary caual model of potential outcome i currently the workhore in applied caual analyi. It ha it root in the literature about experimental evaluation in agriculture tarting Since in econometric endogeneity i uually a key iue, it i omewhat urpriing that it ha not received much general attention in the field of evaluation tudie. The paper in thi field concentrate fully on the endogeneity problem coming from miing variable that determine the election proce. Note that we ue the term endogeneity (and exogeneity) omewhat caually to mean that the variable i (not) influenced by the treatment, which i not exactly in line with the common ue of thi language in econometric (e.g. Engle, Hendry, and Richard, 983). Alternatively, the exogeneity condition below may alo be termed 'non-cauality condition', although there are different ue for thi word a well. 2

4 with Neyman (923). 2 The model i characteried by random variable decribing two potential tate of the world, alo called treatment and denoted by S {,} (e.g. participation or nonparticipation in the programme), the correponding outcome that would occur if one of the tate i realied ( Y, Y ) (e.g. employment tatu that would occur in cae of participation and nonparticipation), and the oberved outcome variable Y. 3 Oberved and potential outcome variable are related by the obervation rule, Y SY S Y = + ( ). Denote characteritic that influence treatment participation a well a potential outcome a X (called control variable in econometric, or confounder in the tatitic literature). To addre the iue of endogeneity we treat confounder and outcome in a ymmetric way and define random variable for their potential value ( X SX S X X, X ), a well a an obervation rule = + ( ). A uual, we are intereted in etimating the average treatment effect on the treated (ATET) θ () : = EY ( S= ) EY ( S= ), the average treatment effect on the nontreated (ATENT) θ () : = EY ( S= ) EY ( S= ) and the average treatment effect (ATE) θ (): = EY EY ( ) ( ). Note that θ ( ) = θ () P( S = ) + θ ()[ P( S = )]. PS= ( ) denote the probability of participation. To complete the model, we follow Rubin (98) and impoe the table unit treatment value aumption (SUTVA). SUTVA implie that treatment on the individual level are well defined and complete and that there i no interference between treated and control (e.g. no general equilibrium effect for a labour market training programme), and lead to the obervation rule. Furthermore, aume that for all value of the oberved confounding variable that are of interet ( x χ ), there i ome chance to end up in either of the two tate, i.e. < PS ( = X= x) < (common upport). 2 3 In the following we ue the term common in that literature. We ue the binary model for pedagogical reaon only. The extenion to a model with multiple treatment along the line of Imben (2) and Lechner (2, 22) i traightforward. 3

5 With repect to the data available, aume that there i a random ample of the obervable variable {,, } : y x =, that i large enough o that tudying identification in the population i i i i N intead of the ample i reaonable. Neverthele, EY ( S= ), EY ( S= ), EY ( ), and EY ( ), are not identified without further aumption. 3 I there an endogeneity problem when the conditional independence aumption hold? Aume that the data are informative enough, o that controlling for oberved variable i ufficient to remove any election bia (Rubin, 974, 977, 979). Thi i the o-called conditional independence (CIA) or election on obervable aumption. It i formalied a: Y, Y S X = x, x χ. (CIA-O) denote conditional independence a in Dawid (979). CIA-O implie FY ( S= X, = x) = FY ( S= X, = x), {,}. It uffice to identify the treatment effect becaue EY ( S= ) = E EY ( S= X, = x) = XS = E EY ( S= X, = x) = XS = E EY ( S= X, = x). XS = Other than ome tandard regularity aumption about the exitence of moment, no further aumption, in particular, no aumption about the exogeneity of the control variable are required. Therefore, the quetion arie what happen if ome of the control variable are influenced by the treatment (i.e. X X ). At thi level, the anwer i that it doe not matter at all, a long a (CIA-O) i atified. Thi eem to be odd at firt glance, becaue it i a well known fact that endogenou control variable may lead to tremendou biae of the reulting 4

6 etimator. For example, uppoe the choen etimator i a pair matching etimator 4 and the oberved outcome variable i included among the control variable, then matched pair have the ame value of the outcome variable and the etimator will alway give zero a etimated treatment effect, whatever the true effect may be. Of coure, thi example i extreme ince nobody would actually ue outcome a control variable. However, there are many cae in practice in which there i at leat a reaonable upicion that ome of the control variable may actually be a function of the outcome variable. In thoe cae, qualitatively imilar biae may occur. To reolve thi puzzle, note that with an endogenou variable included among the control variable, the untetable aumption CIA may become hard to defend on a priori ground. Uing the obervation rule for the confounder, we ee the reaon: CIA implie FY ( S= X, = x) = ( FY S= X, = x). In thi cae the randomization interpretation of CIA ('if we oberve all variable jointly influencing the outcome and election, then, conditional on the ame value of thee variable, election i random') doe not hold any longer, becaue the comparion relate to the ame value, but for different control variable. Since we cannot oberve X for nonparticipant and vice vera, we mut condition on the oberved X. However, conditioning on the oberved X implie conditioning on X for the participant, and conditioning on X for the nonparticipant. Thi could become a very eriou problem when X and X differ. To conclude, endogeneity of the control variable doe not play any role once the CIA i aumed, but it make the CIA unlikely to hold. Thu, the endogeneity problem i obcured when tipulating the CIA in the uual way a in CIA-O See Imben (24) for an extenive urvey of relevant etimation method. Simonen and Skipper (24) are concerned with endogeneity problem in matching tudie a well. However, they aume that CIA i baed on exogenou variable only. The thrut of their 5

7 4 The explicit treatment of endogeneity Thi ection dicue potential biae due to endogeneity of the control variable (i.e. they are influenced by the treatment) and give condition under which no uch biae occur. Such variable may contain valuable information about the election proce a conidered in Roenbaum (984), they may be in the 'caual pathway' of the treatment, a in Rubin (25), or they may be intermediate outcome of a dynamic treatment not yet completed (Lechner and Miquel, 25). Firt, I retate the conditional independence aumption conditioning on variable that are by definition exogenou, namely the potential confounder: 6 Y, Y S X = x, {,}, x χ. (CIA-P) CIA-P implie FY ( S= X, = x) = ( FY S= X, = x). Since thi condition i a contrat for the ame conditioning variable, the requirement i that if all variable influencing outcome ( Y, Y ) and election (D) are included in X, CIA-P mut hold. Note that conditioning on X or X i ufficient, whichever appear to be more plauible in the intended application. CIA-P ha the virtue of uing conditioning variable that are by definition exogenou. However, it doe not allow identification of the effect, becaue there i only data on X for thoe obervation participating in tate S=. Thu, additional aumption are required linking the 6 potential X to the oberved X. For the tronget of uch aumption, i.e. that S ha no influpaper eem to be to uncover effect heterogeneity driven by endogenou variable a well a uncovering the effect of the endogenou variable together with S on the outcome, which i a different iue than the topic dicued here. Note that potential confounder are in principle the ame type of variable a potential outcome. However, the difference i that we are not intereted in the direct effect of the treatment on potential confounder. 6

8 ence on X, X = X = X, CIA-P and CIA-O are identical. 7 In fact, they amount to what Roenbaum and Rubin (983) call the aumption of trongly ignorable treatment aignment. 4. Bia Next, conider the potential aymptotic bia of a matching etimator due to the endogeneity of the control variable for the counterfactual EY ( S= ). Thi bia can be characteried by uing the obervation rule for outcome and control variable together with CIA-P. The characteriation depend on whether the potential outcome and potential confounder ued in CIA-P relate to the ame or a different potential tate. For the ame tate, we obtain: EY ( S= ) E EY ( S= X, = x) = E EY ( S= X, = x) E EY ( S= X, = x) XS = X S= X S= Bia = E E( Y S =, X = x) E E( Y S =, X = x) X S= X S= = E E( Y S =, X = x) E E( Y S =, X = x). X S= X S= The bia arie becaue X and X may have different ditribution in the population of interet (-). If F( X S = ) = F( X S = ) [ = F( X S = )], the bia i zero. Thi i an exogeneity condition, becaue it tipulate that the ditribution of the control i not affected by the treatment in a pecific ubpopulation. For example, if interet i in ATET ( θ () ) and conditioning i on X, then thi i all that i needed for identification. When the potential confounder related to the alternative tate, then we can rewrite the bia differently, uch that a econd interpretation become obviou: 7 See for example Aumption A6 in Heckman and Vytlacil (25). 7

9 E Y S E E Y S X x E E Y S X x E E Y S X x ( = ) ( =, = ) = ( =, = ) ( =, = ) XS = X S= XS = = E E Y S = X = x E E Y S = X = x X S= (, ) (, ) XS = = = = = = E [ EY ( S X, x) EY ( S X, x)]. XS =. The bia arie becaue we hould evaluate the firt conditional expectation of Y a a function of the counterfactual conditioning variable ( X ), intead of the oberved one ( X ). Thu, if both expectation coincide, i.e. EY ( S= X, = x) = EY ( S= X, = x), then no bia arie. Thi i again an exogeneity condition in a pecific ubpopulation, although weaker than the previou one. It tipulate that the X may be influenced by the treatment, but thee influence hould not matter for the outcome. 4.2 Identification under explicit exogeneity condition The previou conideration ugget explicitly introducing two exogeneity condition for X: F( X S = ) = F( X S = ) ; (E.) E( Y S =, X = x) = E( Y S =, X = x). (E.2) If interet i in θ, ( ), then CIA-P (for with E., or CIA-P (for Y only) conditioning on If interet i in the average treatment effect, Y only) conditioning on, θ X in combination X in combination with E.2, i ufficient. 8 (), then CIA-P conditioning on X or X in combination with E. and E.2, or CIA-P conditioning on X and CIA-P conditioning on X in combination with either E. or E.2 are ufficient. The proof follow directly from the expreion of the biae given above. If the treatment doe not influence the control variable at all ( X = X ), E. and E.2 hold trivially anyway. Note that together with the two 8, By an obviou change of indice, a imilar reult i obtained for θ (). 8

10 exogeneity condition, CIA-P implie CIA-O, and vice vera. However, without thoe exogeneity condition, neither implie the other. It i intereting to compare the exogeneity condition uggeted above to condition already conidered by Heckman, LaLonde, and Smith (999). They require F( X Y, Y, S = ) = F X Y Y (, ), which implie F( X Y, Y, S = ) = F( X Y, Y, S = ). Although thi condition i imilar to E. and E.2, the two equation characteriing the bia preented in ection 4. how that thi doe not olve the problem. When ued together with CIA baed on obervable (which i of coure unneceary for identification), it ha however intuitive appeal, becaue it rule out ome dependence of the oberved X on S. The early paper by Roenbaum (984) conider a formal framework in which the endogeneity come from ome variable affected by the treatment that are, however, not neceary a control variable for the CIA. Thu, hi quetion i about the potential damage when uch variable are included in the conditioning et. Hi ufficient condition for no election bia impoe X = X. Thu, it eem to be tronger than neceary. 9 The idea to condition on potential confounder i imilar to an idea recently advanced in paper by Frangaki and Rubin (22) and Rubin (24). The aim of thee author i to arrive at valid reult for the effect baed on matching etimation depite the endogeneity problem of the control. They define what they call principle trata of the data. Thee trata are defined by the value of the potential confounding variable. They olve the problem that uch trata are inherently unobervable by, firtly, imputing the value for the potentially endogenou control variable. Then, they ue the etimated value of the potential control variable a control variable in the final etimation. Thi method that how ome imilarity to idea underlying the old parametric two tage leat quare etimator in econometric i 9

11 computationally and conceptionally attractive, but it hinge on additional trong aumption, which poe a new identification problem that may be a difficult to olve - convincingly - a the original one. 5 Concluion Thi paper howed that the fact that ome control variable may be influenced by the treatment doe not matter a long a the uual formulation of the conditional independence aumption hold. In other word, the uual formulation of the CIA obcure thi endogeneity problem. Therefore, an alternative formulation of the conditional independence aumption together with explicit exogeneity condition i propoed. The new condition allow eparating the dicuion of eliminating election bia by conditioning on the neceary control variable from the dicuion about their potential endogeneity with repect to outcome and election variable. Thu, thee condition hould provide ueful and in fact better guidance for applied work, in particular for the aement of the plauibility of the untetable aumption that are alway required for the identification of caual effect. Acknowledgment I thank Don Rubin, Jeff Smith, Marku Frölich, and an anonymou referee for helpful comment on previou verion of thi paper. The uual diclaimer applie. Reference Dawid, A. P. (979): "Conditional Independence in Statitical Theory", The Journal of the Royal Statitical Society, Serie B, 4, -3. Engle R., D. F. Hendry, and J.-F. Richard (983): "Exogeneity", Econometrica, 5, Due to the pecific framework he ue, he mut impoe a further condition that amount to a 'no election bia conditional on X aumption' for the affected conditioning variable a well.

12 Frangaki, C. E., and D. B. Rubin (22): "Principal Stratification in Caual Inference," Biometric, 58, Gerfin, M., and M. Lechner (22): "Microeconometric Evaluation of the Active Labour Market Policy in Switzerland", The Economic Journal, 2, Heckman, J., R. LaLonde and J. Smith (999): "The Economic and Econometric of Active Labor Market Program", in: O. Ahenfelter and D. Card (ed.), Handbook of Labour Economic, Vol. 3, , Amterdam: North-Holland. Heckman, Jame J, and Edward Vytlacil (25): "Caual Parameter, Structural Equation, Treatment Effect and Randomized Evaluation of Social Program", Econometrica. Imben, G. W. (2): "The Role of the Propenity Score in Etimating Doe-Repone Function", Biometrika, 87, Imben, G. W. (24): "Nonparametric Etimation of Average Treatment Effect under Exogeneity: A Review", The Review of Economic and Statitic, 86, Lechner, M. (2): "Identification and Etimation of Caual Effect of Multiple Treatment under the Conditional Independence Aumption", in: M. Lechner and F. Pfeiffer (ed.), Econometric Evaluation of Active Labour Market Policie, 43-58, Heidelberg: Phyica. Lechner, M. (22): "Programme Heterogeneity and Propenity Score Matching: An Application to the Evaluation of Active Labour Market Policie", Review of Economic and Statitic, 84, Lechner, M., and R. Miquel (25): "Identification of Effect of Dynamic Treatment by Sequential Conditional Independence Aumption", Dicuion Paper, Department of Economic, Univerity of St. Gallen, 22, revied 25. Neyman, J. (923): "On the Application of Probability Theory to Agricultural Experiment. Eay on Principle. Section 9," tranlated in Social Science, 5, Roenbaum, P. R. (984): "The Conequence of Adjutment for a Concominat Variable That Ha Been Affected by the Treatment", The Journal of the Royal Statitical Society, Serie A, 47, Roenbaum, P.R., and D.B. Rubin (983): "The Central Role of the Propenity Score in Obervational Studie for Caual Effect", Biometrica, 7, 4-5. Robin, J. M. (986): A new approach to caual inference in mortality tudie with utained expoure period - Application to control of the healthy worker urvivor effect. Mathematical Modelling, 7, ,

13 Rubin, D. B. (974): "Etimating Caual Effect of Treatment in Randomized and Nonrandomized Studie", Journal of Educational Pychology, 66, Rubin, D. B. (977): "Aignment to Treatment Group on the Bai of a Covariate", Journal of Educational Statitic, 2 (), -26. Rubin, D. B. (979): "Uing Multivariate Matched Sampling and Regreion Adjutment to Control Bia in Obervational Studie", Journal of the American Statitical Aociation, 74, Rubin, D. B. (98): "Dicuion of 'Randomization analyi of experimental data in the Fiher randomiation tet by Bau", Journal of the American Statitical Aociation, 75, Rubin, D. B. (24): "Direct and Indirect Caual Effect via Potential Outcome", Scandinavian Journal of Statitic, 3, 6-7. Rubin, D. B. (25): "Caual Inference Uing Potential Outcome: Deign, Modeling, Deciion", Journal of the American Statitical Society,, Sianei, B. (24): "An Evaluation of the Swedih Sytem of Active Labor Market Program in the 99", The Review of Economic and Statitic, 86, Simonen, M., and L. Skipper (24): "The Cot of Motherhood: An Analyi Uing Matching Etimator", mimeo. 2