Identification with Conditioning Instruments in Causal Systems

Transcription

1 Ientification wit onitioning Instruments in ausal ystems Karim alak Department of Economics Boston ollege Halbert Wite Department of Economics University of alifornia an Diego May bstract: We stuy te structural ientification of causal effects wit conitioning instruments witin te settable system framework. In particular we provie causal an preictive conitions sufficient for conitional exogeneity to ol. We provie two proceures base on exclusive of (~)-causality matrices an te irect causality matrix for inferring conitional causal isolation among vectors of settable variables an consequently conitional inepenence among corresponing vectors of ranom variables. imilarly we provie sufficient conitions for conitional stocastic isolation in terms of te σ-algebras generate by te conitioning variables. We buil on tese results to stuy te structural ientification of average effects an average marginal effects wit conitioning instruments. We istinguis between structural proxies an preictive proxies. cknowlegment: Te autors tank Julian Betts Graam Elliott live Granger Mark Macina Dimitris Politis an Rut Williams for teir elpful comments an suggestions. ll errors an omissions are te autors responsibility. Karim alak Department of Economics Boston ollege 40 ommonwealt venue estnut Hill M 0467 ( calak@bc.eu). Halbert Wite Department of Economics 0508 University of alifornia an Diego 9500 Gilman Drive La Jolla ( wite@ucs.eu).

2 . Introuction Researcers are often intereste in measuring causal relationsips an evaluating te effects of policies an interventions. In observational stuies tis is a particularly callenging task for several reasons incluing among oters te inability of te researcer to control relevant variables te fact tat some relevant causes may be unobserve an te unknown forms of response functions. Wite an alak (006) (W) iscuss te role tat conitional exogeneity can play in permitting informative causal inference in suc situations. We employ W s settable system framework to stuy te specification of proxies tat can function as conitioning instruments as iscusse in W an alak an Wite (007a) to support te structural ientification of causal effects of interest. In particular we buil on results of alak an Wite (007b) (W) to provie causal an preictive conitions sufficient for conitional exogeneity to ol. In te nomenclature of te treatment effect literature conitional exogeneity implies te property of ignorability or unconfouneness (see e.g. Rubin 974; Rosenbaum an Rubin 983). Neverteless te treatment effect literature is typically silent as to ow to construct vali covariates tat ensure conitional exogeneity in observational stuies. In stanar experimental settings te researcer may be able to control te assignment mecanism of te treatment collect pre-treatment measurements an perform iagnostic tests to ensure tat conitional exogeneity ols (see e.g. Rosenbaum 00). For example ranomization or conitional ranomization may be sufficient for useful causal inference (see W). In suc circumstances te etermination of suitable covariates is typically straigtforwar. On te oter an in observational stuies an certain experimental settings suc privileges are typically not available an te researcer as to resort to alternative means for ientification of effects of interest. In particular economic teory can play a central role in proviing guiance for constructing conitioning instruments tat ensure conitional exogeneity an tus support te structural ientification of causal effects of interest. main goal of te present analysis is to examine in etail ow information provie by unerlying economic (or oter) teory can be use to generate vali conitioning instruments tereby ientifying causal effects of interest in observational settings. We procee wit our analysis of structural ientification wit conitioning instruments uner minimal assumptions. In econometrics assumptions tat may permit structural inference are typically concerne wit te istribution of te unobserve variables or te functional form of te response functions or bot (see e.g. Heckman an Robb 985; Heckman an Honore 990; Blunell an Powell 003; Matzkin 003; Heckman an Vytlacil 005). Here we impose only te weakest possible assumptions on te istribution of te unobserve variables an te functional form of te response functions. Tis extens te analysis of alak an Wite (007a) wo stuy in etail te ientification of causal effects witin te structural equations framework uner te classic assumption of linearity. Teir analysis reveals several extension of te concept of instrumental variables.

3 In brief te main contribution of tis paper is to provie formal an general causal an preictive conitions ensuring tat conitional exogeneity ols tus permitting informative causal inference. Our ope is tat once tese conitions ave been properly justifie by economic teory an expert knowlege tey can serve as elpful templates to guie empirical researcers in forming causal inferences. Te paper is organize as follows. In ection we buil on results of W to stuy conitioning instruments tat ensure conitional exogeneity in a canonical recursive settable system. In ection. we efine te irect causality matrix an for a given set of causal variables wose inexes belong to a set te exclusive of (~)- causality matrix. We ten provie two proceures for verifying te presence of ~ structural proxies using ~ an. Tese proceures rely on te fact tat conitional inepenence ols by conitioning on te common causes of te variables of interest or on variables tat fully meiate te effects of tese common causes. Tis inclues as a special case te notion of -separation introuce in te artificial intelligence literature (see e.g. Pearl 000 p.6-7). ection. stuies anoter special case of conitioning instruments tat we refer to as preictive proxies following W an alak an Wite (007a). In tis case conitional inepenence can ol ue to a particular preictive relationsip tat ols among te variables of interest. ection 3 buils on results of W to provie fully structural conitions ensuring tat conitioning instruments permit te structural ientification of covariate-conitione average effects an covariate-conitione average marginal ceteris paribus effect. Te structural ientification results in ection 3 tat employ structural proxies inclue as a special case metos base on te back oor criterion introuce in te artificial intelligence literature by Pearl ( ). We are not aware of results tat stuy structural ientification wit preictive proxies oter tan te results of W an alak an Wite (007a). ection 4 conclues an iscusses irections for future researc. Formal matematical proofs are collecte into te Matematical ppenix.. onitioning Instruments in ettable ystems We work wit te version of settable systems presente in efinition. of W. pecifically we consier canonical recursive settable systems {(Ω F) (Z Π r Π )} (W efinition.6). settable system as a stocastic component (Ω F) an a structural component (Z Π r Π ). W provie a etaile iscussion. settable system represents agents inexe by = eac governing settable variables inexe by j = represente as { j } were j : {0} Ω R. special settable variable te funamental settable variable is enote 0. ettable variables { } generate settings Z = {Z j = j (. )} an for te given partition Π {Π b } responses Y = {Y j = j (0. )} given by Y j = r Π j(z (b) ) for ( j) in Π b 0 were r Π jis te response function an Z (b) enotes settings of variables not belonging to partition block 3

4 Π b. By convention Π 0 = {0}. In recursive settable systems te blocks are orere suc tat responses in iger-level blocks epen only on settings in lower-level blocks. In canonical recursive settable systems we furter ave Z j = Y j for 0. By convention we also ave Z 0 = Y 0. For simplicity we sometimes consier finite settable systems tat is systems wit a finite number of agents an responses so tat j = J wit J < = H <.. ausally Isolating onitioning Instruments Next we provie efinitions of certain matrices tat summarize aspects of interest associate wit settable systems. We make use of tese efinitions sortly for inspecting conitional inepenence relationsips. Following W we write j ik wen j irectly causes ik in an we write j ik oterwise (see W efinition.3). For a settable system we efine te irect causality matrix associate wit. Definition.: Direct ausality Matrix Let {(Ω F) (Z Π r Π )} be a finite canonical recursive settable system. Te irect causality matrix associate wit enote by as elements given by: c = if j ( j)( i k) ik c = 0 if j ik. ( j)( i k) Tus as te form: = 0 (0 ) (0 ) H J (0 ) H 0 ( ) 0 ( ) 0 0 ( ) 0 0 H J H We note some properties of. Te iagonal entries are c = 0 for j = J ( j)( j) = H since j j by efinition. lso te entries of te first column = 0 for j = J = H since j 0 by efinition. Blank entries of c ( j)0 4

5 can take te values 0 or. Let G = (V E) be te irecte grap associate wit were V = { j : j = J ; = 0 H} is a non-empty finite set of vertices an E V V is a set of arcs suc tat an arc ( j ik ) belongs to E if an only if j ik. ince is recursive G amits an acyclic orering of its vertices. Bang-Jensen an Gutin s (00 p. 75) DF algoritm outputs an acyclic orering of te vertices of G. It follows tat tere exists a H J = strictly upper triangular. H J permutation matrix = M suc tat Te recursiveness of furter imposes te following constraints on M : is Proposition.: cyclicality Let {(Ω F) (Z Π r Π )} be a finite canonical recursive settable system an let be te irect causality matrix associate wit. Ten for all sets of n istinct elements say {( j ) ( n j n )} of {0 () (H H J H )} wit n J we ave: = c c... c ( ) ( ) ( ) ( j )( j ) ( j )( j ) ( j )( j ) 3 3 n n = 0. imilarly for a set of inex pairs we efine te causality matrix exclusive of (or ~causality matrix) ~ associate wit a recursive settable system. Following W we let I ( ):( ) enote te set of ( j ik ) ()-intercessors an we let in( I ( ):( )) enote j i k te set of inexes of te elements of I. For in( I ) we write ~ j ik wen j causes ik exclusive of (or j ~-causes ik ) wit respect to. We write Π 0 {0}. Definition.: ausality Matrix Exclusive of (~-ausality Matrix) Let {(Ω F) (Z Π r Π )} be a finite canonical recursive settable system. For given non-negative integers b an b let ( j) Π b an let (i k) Π b. Let Π Π 0 an let = in( I ). Te ~-causality matrix associate wit enote by ~ as elements given by: j i k c = if b < b ( j) (i k) an j ~ ( j)( i k) ~ ():() j ik ik c = 0 oterwise. ~ ( j)( i k) 5

6 ~ ~ an sare similar properties. In particular c ~ ( j)( j) = c ( j)0= 0 for j = J = H by efinition. imilarly since tere exists a non-recursive orering of te H H sequence {Π b } it follows tat tere exists a J J permutation matrix M suc tat M ~ is strictly upper triangular. Wen = we refer to te ~ - causality matrix simply as te causality matrix enote by. = = Let an B be nonempty isjoint subsets of Π Π 0 an enote by an B te corresponing vectors of settable variables. Following W we enote by I : B I \ ( B ) te set of ( B ) ()-intercessors an we let ( j) ( ik) B in( I : B ) enote te set of inexes of te elements of I : B. lso we aopt W s efinitions of irect inirect an -causality for vectors of settable variables. In particular for in( I : B ) we write ~ B wen causes B exclusive of (i.e. ~-causes B ) wit respect to. Teorem 4.6 of W provies necessary an sufficient conitions for stocastic conitional epenence among certain ranom vectors in settable systems. In particular it follows from tis result tat if an B are conitionally causally isolate given ten teir corresponing ranom vectors Y an Y B are conitionally inepenent given Y. Our next proposition emonstrates tat -conitional causal isolation an tus certain conitional inepenence relationsips can be immeiately verifie by inspecting te ~-causality matrix ~. For tis we let a = max{b: tere exists ( j) Π b } an b = max{b: tere exists (i k) Π b B}. s in W for any blocks a b 0 a b we let Π [ ab : ] Πa... Πb Πb. Proposition.3 Let {(Ω F P) (Z Π r Π )} be a finite canonical settable system. Let an B be nonempty isjoint subsets of Π Π 0 an let an B be corresponing vectors of settable variables. Let \ ( B) let :[max( ab ) ] be te corresponing vector of settable variables an let ~ be te ~-causality matrix associate wit. Let Y = (0. ) Y B = B (0. ) an Y = (0. ). Ten (a) an B are conitionally causally isolate given if an only if ~ (i) 0 an c 0( ) = 0 for all (i k) B; or ~ (ii) 0 B an c 0( j) = 0 for all ( j) ; or ~ (iii) 0 B an c ~ = 0 for all ( j) or c = 0 for all (i k) B. 0( j) 0( ) 6

7 (b) uppose tat eiter (a.i) (a.ii) or (a.iii) ol; ten for every probability measure P on (Ω F) Y Y B Y. practical benefit of tis result is tat it can be straigtforwarly converte into a computer algoritm tat can be use to verify conitional causal isolation an tus conitional inepenence for any triple Y Y B Y. Note owever tat te failure to verify conitional causal isolation oes not imply conitional epenence. We iscuss tis furter below. s iscusse in W etermining ~-causality relationsips among settable variables generally requires knowlege of te functional form of te response functions associate wit te settable variables uner stuy. In economics an oter fiels were observational stuies preominate it is often true tat te researcer oes not ave etaile information about te functional form of te response functions. Frequently te researcer may know only weter or not a given variable is a irect cause of anoter. Is it possible to euce conitional inepenence from knowlege only of a irect causality matrix? Our next results emonstrate tat te answer to tis question is inee positive. For brevity we may rop explicit reference to () in wat follows wen referring to c ( j)( i k) as well as te matrices tat obtain from an teir entries. For example we write c (j)(ik) to enote an element of. Definition.4: Pat onitional Matrix given Let {(Ω F) (Z Π r Π )} be a finite canonical settable system. Let Π Π 0 an let be te corresponing vector of settable variables. Te pat conitional matrix given enote P is given by P = p ( ) were te elements of p ( ) are efine as: p ( j)( i k) = if ( j) (i k) an ( j)( i k) c = or tere exists a subset {( j ) ( n j n )} of Π \ wit n c c... c = ( j)( j ) ( j )( j ) ( j )( i k) n n H J suc tat = p = 0 oterwise. ( j)( i k) Definition.4 can be conveniently visualize using te grap G associate wit. In fact ( j)( i k) p = if an only if tere exists in G an ( j ik )-pat of positive lengt tat oes not contain elements of. 7

8 Our next result emonstrates tat it is possible to verify certain ~-causality relationsips from knowlege of only an tus P witout furter information on te functional form of te response functions. Lemma.5 Let B an be as in Proposition.3. Let P be te irect pat conitional matrix given. Let :B = in( I : B ). uppose tat p ( j)( i k) = 0 for all ( j) an (i k) B. Ten ~ B : B. We now provie sufficient conitions for te conitional inepenence of certain ranom vectors in settable systems expresse in terms of. orollary.6 Let B be as Proposition.3 an let P be te pat conitional matrix given. Let Y = (0. ) Y B = B (0. ) an Y = (0. ). uppose tat eiter (i) 0 an p 0( ) = 0 for all (i k) B; or (ii) 0 B an p 0( j) = 0 for all ( j) ; or (iii) 0 B an p 0( j) = 0 for all ( j) or p 0( ) = 0 for all (i k) B. Ten an B are conitionally causally isolate given an for every probability measure P on (Ω F) Y Y B Y. Te grap G associate wit a system can play a particularly elpful role in verifying conitional inepenence via orollary.6. In fact wen 0 if for all ( j) tere oes not exist a ( 0 j )-pat tat oes not contain elements of ten Y Y B Y. parallel result ols wen 0 B. Wen furter assumptions are impose on te probability measure P orollary.6 inclue as a special case te notion of -separation iscusse in te artificial intelligence literature (see Pearl 000 p ; an W section 5). s is true for Proposition.3 orollary.6 provies te basis for a straigtforwar computer algoritm tat can be use to verify conitional causal isolation an tus conitional inepenence. orollary.6 provies sufficient but not necessary conitions for conitional inepenence. In fact it is possible to ave B even wen p ( j)( i k) = for all ( j) an (i k) B ue to a cancellation of irect an inirect effects. Neverteless suc inference requires etaile knowlege of te response functions. Tus it is possible for an B to be causally isolate given even wen conitions (i) (ii) an (iii) of ~ B : 8

9 orollary.6 fail. For example wen = {( j)} an B = {(i k)} we may ave p 0( j) p 0( ) = an Y j Y ik Y. Pearl (000 p ) an pirtes et al (993 p ) introuce te assumptions of stability or faitfulness on P to ensure tat te conitions in Proposition.3 an orollary.6 eliver te same conclusion. Here owever we see tat it is te beavior of te response functions an not te properties of te probability measure tat etermine weter or not te conclusions of Proposition.3 an orollary.6 coincie. Wen = we simply call P te pat matrix associate wit an orollary.6 ten provies conitions ensuring te stocastic inepenence of Y an Y B. Tis result is owever less interesting since te conitions of orollary.6 imply tat eiter Y or Y B (or bot) are constants (see W lemma 3.). For given nonempty isjoint subsets an B of Π Π 0 one can ask wat is te is te smallest set \ ( B) (if any) suc tat te realizations of Y :[max( ab ) ] = (0. ) are observe an conitionally causally isolates an B. In aition to supporting computer algoritms tat can verify conitional causal isolation Proposition.3 an orollary.6 provie te basis for constructing algoritms tat may output suc a set. We leave evelopment of tese algoritms to oter work. Tus Proposition.3 an orollary.6 provie causal conitions sufficient to ensure tat vectors of settable variables an B are causally isolate given an tus tat Y Y B Y. Tis conitional inepenence ols because te common causes of an B or variables tat fully meiate te effects of tese common causes on eiter (or bot) an B are elements of (see W section 5). In ection 3 we iscuss ow suc a causally isolating can play an instrumental role in ensuring conitional exogeneity. In tis case we refer to an its settings Z = Y as causally isolating conitioning instruments or structural proxies.. tocastically Isolating onitioning Instruments W emonstrate tat Y Y B Y can also ol wen an B are not conitionally causally isolate given provie tat P stocastically isolates an B given. We now stuy special cases in wic tis can appen. We buil on results of Van Putten an Van cuppen (985) wo stuy certain operations tat leave conitional inepenence relationsips invariant. In particular we focus on operations tat enlarge or reuce te conitioning σ-algebra an tat preserve te conitional inepenence relationsips of interest. First we aapt efinitions from Van Putten an Van cuppen (985) to our context. 9

10 Definition.7 Projection Operator for σ-lgebras Let {(Ω F P) (Z Π r Π )} be a canonical recursive settable system. (i) Let F {G F G is a σ-algebra tat contains all te P-null sets of F}. (ii) If H G F ten H G is te smallest σ-algebra in F tat contains H an G. (iii) For G F let L + (G) = {g : Ω + g is G-measurable}. (iv) For H G F let te projection of H on G be te σ-algebra σ(h G) σ({e[ G] for all L + (H)}) F wit te unerstaning tat te P-null sets of F are ajoine to it. Te next result elps to caracterize situations were P is stocastically isolating. Teorem.8 Let {(Ω F P) (Z Π r Π )} be a canonical settable system. Let an B be nonempty isjoint subsets of Π Π 0 an let an B be corresponing vectors of settable variables. Let an be subsets of \ ( B) an let :[max( ab ) ] an = B (0. ) enote te corresponing vectors of settable variables. Let Y = (0. ) Y B Y = (0. ) an F suc tat. Ten (a.i) isolates an B given Y = (0. ) generate σ-algebras B an causally isolates an B or (a.ii) P stocastically an (b) σ( ) if an only if (c.i) isolates an B or (c.ii) P stocastically isolates an B given B ) (B ). causally an () σ( Wen Y Y B an if an only if given Y are as in Teorem.8 teorem 4.6 in W gives tat Y Y B causally isolates an B or P stocastically isolates an B. Heuristically one can view a σ-algebra as representing information B as te smallest set containing te information in bot an B an σ( ) as te Y information about inferre from te information in. imilarly one can unerstan Y 0

11 Y B Y as te statement tat knowlege of information reners information in irrelevant for information B (an tus te information in B irrelevant for information ). Now suppose tat information is a subset of information. Teorem.8 states tat (a) knowlege of reners irrelevant for B an (b) te information about inferre from is a subset of if an only if (c) knowlege of reners irrelevant for B an () te information about inferre from B an is containe in information B an. Tus Teorem.8 provies necessary an sufficient conitions for preserving a conitional inepenence relationsip among vectors of ranom variables Y an Y B wen te information tat we conition on is eiter enlarge or reuce. Examples of tis enlargement or reuction relevant ere occur wen we know tat conitional inepenence ols wen conitioning on unobservables an we seek to fin observables tat we can conition on instea tat will preserve conitional inepenence. Next we state a elpful orollary of Teorem.8. orollary.9 Let B B (0. ) Y = uppose tat (a.i) an B given (0. ) an an Y = be as in Teorem.8. Let Y = (0. ) Y B = (0. ) generate σ-algebras B an F. causally isolates an B or (a.ii) P stocastically isolates an (b) σ( ) (B ). Ten (c.i) an B or (c.ii) P stocastically isolates an B given. causally isolates uppose tat Y Y B Y an Y are as in Teorem.8 an tat an B or tat P stocastically isolates an B given causally isolates so tat Y Y B Heuristically orollary.9 states tat if (a) knowlege of reners irrelevant for B (b.i) contains te information about inferre from an (b.ii) is containe in B an ten (c) knowlege of reners irrelevant for B. In oter wors (b.i) an (b.ii) state tat contains te information in relevant for but oes not contain Y. information not inclue in B an. Tus orollary.9 provies a boun on sufficient for Y Y B Y to ol. If te conitions of orollary.9 ol an if ten it must be tat P stocastically isolates an B given oes not causally isolate an B. In tis case Y Y B

12 Y ols but not because te common causes of an B (or variables tat fully meiate te effects of tese common causes on eiter (or bot) an B ) are elements of. Instea conitional inepenence ols because of a preictive relationsip tat relates Y Y B Y an Y motivating our terminology esignating Z = Y as preictive proxies. Parallel to our nomenclature above we may also call Z = Y stocastically isolating conitioning instruments. Observe tat in suc situations Y Y B Y but tat an B are not -separate given in te corresponing grap G. orollary.9 plays a particularly elpful role wen iscussing ientification via preictive proxies in ection Ientification wit onitioning Instruments In tis work our focus is on te total causal effects of specific settings on a response of interest. We leave to oter work te stuy of irect an inirect effects as well as of more refine measures of effects specifically effects via an exclusive of subsets of causal variables. We buil on results of W to iscuss te structural ientification of effects of interest. In particular we employ te results of ection to provie causal an preictive conitions sufficient to ensure tat W s conitional exogeneity assumption (.) ols. In turn tese conitions provie significant insigt into te generation an construction of covariates. Te next lemma constructs te total response function associate wit certain settings an a response of interest in canonical systems. s in W we write values of settings corresponing to Π [ ab : ] as z [a:b]. In particular wen (i k) Π b we can express response values for ik as r Π ( z ). [0: b ] Lemma 3. Let {(Ω F P) (Z Π r Π )} be a canonical settable system. For given non-negative integers b an b wit b < b let Π b let (i k) ik enote te corresponing settable variables. Let Π = in( I :{( )} inices in ) ) an enote by [0: b ]( :( i k)) {(g l) Π [0: b ]( :( i k)) [0: b ]( :( i k)) Π b an let an Π \ ( [0: b ] : gl ik } te set of Π associate wit all irect causes of ik. Let enote te vector of settable variables corresponing to for settings of an let z enote a vector of values. Ten tere exists a measurable function r tat we call te total response function of ik wit respect to suc tat r ( z z ) = r Π ( z ). [0: b ]

13 Te total response function of ik wit respect to represents values of a response of interest Y ik as a function of te setting value z an values of settings corresponing to all irect causes of ik tat are not ( ik ) intercessors. Tus te ifference r ( z z ) r ( z ) serves as a measure of te total effect on ik of an intervention z z z to setting Z to z. In applications we are often intereste in comparable groups of agents inexe by =. In particular we may be intereste in measuring te effect of settings on corresponing responses associate wit tese agents. ssumption.(i) caracterizes settable systems of interest to us ere. ssumption.(i) Let {(Ω F P) (Z Π r Π )} be a canonical settable system. (a) Fix inices k an {j j } suc tat for all = an non-negative integers b an b wit b < b {( j ) ( j ) } Π b an ( k) Π b. Let D an Y.k enote te corresponing settable variables. (b) uppose tat for all = {( l) k Π b k [0: ]( :( )) : l k } = ( B ) = {( l ) ( l ) ( l ) ( l ) } an enote by (Z U ) ( B ) te corresponing vector k of settable variables. (c) Furter for all = let {( m ) ( m ) } Π [0: b ]( :( k)) \ k an let W enote te corresponing vector of settable variables. Put ( ) an ( ) = (Z W ). Tus j j inex causes of interest an k inexes a response of interest associate wit a comparable group of agents inexe by =. For given we enote by D agent s vector of settable variables generating causes of interest an by Y.k is settable variable generating te response of interest. Furter for all = we let {( l ) ( l ) ( l ) ( l ) } inex elements of Π [0: b ]( :( k)) corresponing to all irect causes of k oter tan tose generate by D. For given we let (Z U ) = enote te corresponing vector of settable variables. We k istinguis between Z an U sortly. From.(i) we ave tat Z Π an Z Y. It follows tat for all = [0: b ]( :( k)) D Z. For all = we treat inexes corresponing to variables tat 3

14 succee but are not ( k ) intercessors as elements of Πb Π... b + witout loss of generality. Tus we also ave tat for all = D W. Let Y D Z U an W enote settings (equivalently responses for canonical systems) of Y D Z U an W respectively. By Lemma 3. we can write Y c = r (D Z U ) for all = were r enotes te unknown total response function of Y wit respect to D. (Following W we use te c = symbol instea of te usual equality sign to empasize tat tis is a causal relationsip.) ince we consier a comparable group of agents inexe by we assume tat r = r for all =. We continue to suppress explicit mention of attributes from our analysis keeping in min tat attributes can be incorporate into tis framework as iscusse in W (section.6) to permit eterogeneity of responses across agents among oter tings. We assume tat we only observe a sample from te population as is often te case in economics an oter fiels were observational stuies are of interest. Following W we treat sampling ere as generating ranom positive integers say {H i } governe by te probability measure P. Tus te sample responses are given by: Y = c r ( D H i H i Z H i U H i ) i =. Engaging in a mil abuse of notation we write Y i = an W i =. In particular we write: W H i Y H i D i = D H i Z i = Z H i U i = U H i Y i c = r (D i Z i U i ) i =. Because agents are comparable we rop explicit reference to te subscript i wen referring to te settable variables involve an we write (Y D Z U W) keeping in min tat tese pertain to a representative agent. Following W we say tat tat generates a sample {(Y i D i Z i U i W i )} involving settable variables (Y D Z U W). We now specify ow te ata are generate. We let N enote te natural numbers + incluing zero by convention N te positive integers an N = N { }. ssumption.(ii) Let a canonical settable system generate a sample {(Y i D i Z i U i W i )} involving settable variables (Y D Z U W). (a) uppose tat te joint istribution 4

15 of (D i i ) (D i Z i W i ) is H an te conitional istribution of U i given (D i i ) = ( x) k + k is G( x) for all i = were D i is is -value k N Z i is -value k k4 k3 N U i is -value k 4 N an W i is -value k 3 N. (b) Let te responses {Y i } of Y be given by : Y i c = r (D i Z i U i ) i = were r enotes te unknown scalar-value total response function of Y wit respect to D. (c) We assume tat te realizations of Y i D i Z i an W i are observe but tat tose of U i are not. Te ientical istribution assumption.(a b) is not necessary to our analysis but significantly simplifies our notation. We employ it in wat follows to rop reference to te subscript i from te sample variables. Furter we let x (z w) an u enote values of settings D (Z W) an U respectively. Wit r te total response function of Y wit respect to D te total effect on Y of te intervention to D given Z = z an U = u is r ( z u) r ( z u). Neverteless we cannot measure tis ifference as te function r is not known. Furter even if r were known te fact tat te realizations of U are not observe prevents us from measuring te total effect on Y of an intervention to D. One accessible measurement is te conitional average response over te istribution of te unobserve variables given te observe variables. Following W we refer to = (Z W) as covariates. In fact proposition 3. in W sows tat wen E(Y) < te average response given (D ) = ( x) exists is finite an for eac ( x) in supp(d ) (te support of (D )) it is given by µ ( x) = rzu ( ) Gu ( x ). Te average response function µ is informative about te expecte response given realizations ( x) of (D ). However witout furter assumptions it oes not permit measurement of te total effect on Y of te intervention to D. Wen E(r( Z U)) exists an is finite for eac in supp(d) W efine te average counterfactual response at given = x as ρ ( x) E (r( Z U) = x) = rzu ( ) Gu ( x) were G( x) is te conitional istribution of U given = x. W sow tat uner suitable assumptions tis conitional expectation as a clear counterfactual 5

16 interpretation. In particular uner assumption.(i.b i.c) we ave tat D Z an D W; tus ifferent values of o not necessitate ifferent realizations of = (W Z). Tis permits evaluating counterfactual quantities associate wit te intervention to D. In particular W efine te covariate-conitione average effect on Y of te intervention to D given = x enote by ρ ( x) ρ ( x) ρ ( x) an te covariate-conitione average marginal ceteris paribus effect on Y of D j given (D ) = ( x) as ξ j ( x) D rzugu j ( ) ( x ) were D j = ( / j ) enotes te partial erivative wit respect to j te jt element of. Wen conitional exogeneity ols tat is wen D U we ave G( u x ) = G( u x ) an tus ρ ( x) is structurally ientifie as ρ ( x) = µ ( x) µ ( x) µ ( x). imilarly wen in aition regularity conitions permitting intercange of erivative an integral ol ξ j ( x) is structurally ientifie as ξ j ( x) = D j ρ ( x) = D j µ ( x). We now examine conitions sufficient to ensure tat conitional exogeneity ols. In particular suppose tat (Z W) causally isolates D an U or tat P stocastically isolates D an U given (Z W). Ten conitional exogeneity ols an we can tus measure te covariate conitione average an marginal effects on Y of te intervention to D given = x. Teorem 3. buils on a result of W (teorem 3.3) to formalize tis. Teorem 3. tructural Ientification wit onitioning Instruments uppose tat assumptions.(i(a b c) ii(a b)) ol an tat E(Y) <. uppose furter tat (a) (Z W) causally isolates D an U or (b) tat P stocastically isolates D an U given (Z W). Ten (i) For all ( x) upp (D ) ρ ( x) is structurally ientifie as ρ ( x) = µ ( x). 6

17 (ii) uppose furter tat D jrzu ( ) is ominate on a compact neigboroo of by an integrable function (see assumption.3 of W). Ten ξ j ( x) is structurally ientifie as ξ j ( x) = D j ρ ( x) = D j µ ( x). Wen.(ii(c)) ols we say tat µ ( x) is stocastically ientifie. If furter (i) an (ii) in te conclusion of Teorem 3. ol tat is structural ientification ols we say tat ρ ( x) an ξ j ( x) are fully ientifie. Next we buil on te results of ection to stuy conitions sufficient for (Z W) to causally isolate D an U or for P to stocastically isolate D an U given (Z W). Recall tat for all = we write D U B an (Z W ) = ( ) =. In particular orollary 3.3 emonstrates tat knowlege of te ~ -causality matrices associate wit can permit te ientification of causal effects of interest. orollary 3.3 tructural Ientification wit tructural Proxies (I) uppose tat assumptions.(i(a b c) ii(a b)) ols tat is finite an tat E(Y) <. For = H let ~ enote te ~ -causality matrices associate wit. uppose tat for all ~ = H eiter c ~ 0( j) = 0 for all ( j) or c 0( l ) = 0 for all ( l) B ten (i) an (ii) in te conclusion of Teorem 3. ol. Furtermore knowlege of te irect causality matrix associate wit alone is sufficient for ensuring te structural ientification of certain effects of interest. orollary 3.4 states tis formally. orollary 3.4 tructural Ientification wit tructural Proxies (II) uppose tat assumptions.(i(a b c) ii(a b)) ols tat is finite an tat E(Y) <. For = H let P be te pat conitional matrix given associate wit. uppose tat for all = H eiter p 0( j) = 0 for all ( j) or p 0( l ) = 0 for all ( l) B ten (i) an (ii) in te conclusion of Teorem 3. ol. orollary 3.4 as a convenient grapical representation. In particular te covariateconitione average effect on Y of te intervention to D given = x is structurally ientifie if for all = H (a) tere oes not exist a ( 0 j )-pat tat oes not contain elements of for all ( j) or (b) tere oes not exist a ( 0 l )- pat tat oes not contain elements of for all ( l) B. Uner te regularity conitions in Teorem 3. (ii) conitions (a) an (b) also are sufficient for te covariateconitione average marginal ceteris paribus effect on Y of D j given (D ) = ( x) to be 7

18 structurally ientifie. Wen furter assumptions are impose on te probability measure P (see Pearl 000 p ; an W section 5) orollary 3.4 contains te back oor criterion (Pearl ) as a special case. In aition an application of orollary 3.4 elivers te front oor criterion (Pearl ) were te total effect of interest is ecompose into multiple effects eac ientifie via orollary 3.4. alak an Wite (007a sections 4.. an 4..3) iscuss suc ientification scemes for te special case of linear structural equations systems. We now turn our attention to te structural ientification of causal effects via preictive proxies. First we accommoate in our structure settable variables V for =. ssumption.(i) () For = let Π \ an let V b k [0: ]( :( )) variables. Put k = ( ) an {( m ) ( m ) } enote te corresponing vector of settable = ( ) = (Z V ). Next ssumption.(ii()) accommoates ranom variables V i associate wit V in our sample. ssumption.(ii) Let a canonical settable system generate a sample {(Y i D i Z i U i W i V i )} involving settable variables (Y D Z U W V). () uppose tat te joint istribution of (D i i ) (D i Z i V i ) is H an te conitional istribution of U i given (D i i ) = ( x ) is G ( x ) for all i = were D i Z i U i are as in.(ii.a) an k4 V i is -value k 4 N. (e) Te realizations of V i are observe an tose of Y i D i Z i an W i are as in.(ii(c)). We now state a structural ientification result via preictive proxies. Observe tat now we ave two vectors of potential covariates: (Z W) an (Z V). onsequently we let µ ( x ) rzugu ( ) ( x ) enote te average response given (D ) = ( x ) an we put µ ( x ) µ ( x ) µ ( x ). imilarly we let ρ ( x ) rzugu ( ) ( x ) enote te average counterfactual response at given = x lso we let ρ ( x ) ρ ( x ) ρ ( x ) enote te covariate-conitione average effect on Y of te intervention to D given = x an ξ j ( x ) D rzugu ( ) j ( x ) enote te covariate-conitione average marginal ceteris paribus effect on Y of D j given (D ) = ( x ). orollary 3.5 tructural Ientification wit Preictive Proxies uppose tat assumptions.(i(a b c ) ii(a b )) ol an tat E(Y) <. Let D U (Z W) (Z V) generate σ-algebras D U W an V F. uppose tat (a.i) (Z V) causally isolates D 8

19 an U or (a.ii) tat P stocastically isolates D an U given (Z V); an (b.i) σ(d V) W (U V) or (b.ii) σ(u V) W (D V). Ten (i) For all ( x) upp (D ) ρ ( x ) is structurally ientifie as ρ ( x ) = µ ( x ) (ii) uppose furter tat D jrzu ( ) is ominate on a compact neigboroo of by an integrable function ξ j ( x ) is structurally ientifie as ξ j ( x ) = D j ρ ( x ) = D j µ ( x ) (iii) onclusions (i) an (ii) in Teorem 3. ol. onsier te case were.(ii(e)) fails because te realizations of V i are not observe but were.(i.c) ols so tat te realizations of W i are observe. In particular suppose tat (Z V) causally isolates D an U but tat (Z W) oes not causally isolate D an U. Ten it suffices tat σ(d V) W (U V) or σ(u V) W (D V) to ensure tat P stocastically isolates D an U given (Z W) an tus tat ρ ( x) an ξ j ( x) are structurally ientifie. Heuristically it suffices tat te information in W relates to te information in D U an V in te sense iscusse in ection.. Observe tat it is a preictive relationsip between Y i D i Z i an W i tat ensures conitional exogeneity ere. In particular since (Z W) oes not causally isolate D an U we ave tat te conitions for Pearl s back oor criterion o not ol ere. One tus cannot conclue tat structural ientification ols on tis basis. Neverteless orollary 3.5 ensures tat te covariate-conitione average effect an marginal average effect on Y of te intervention to D given = x are inee structurally ientifie. lternatively suppose tat.(ii(e)) also ols so tat te realizations of bot W i an V i are observe. orollary 3.5 gives tat ρ ( x ) an ξ j ( x ) are structurally ientifie an so are ρ ( x) an ξ j ( x). Tis over-ientification provies a basis for constructing tests for conitional exogeneity. It is also of irect interest to construct te optimal set of covariates tat ensures conitional exogeneity an elivers an efficient estimator of te conitional average an marginal effects. We leave tese questions to oter work. Finally we note tat altoug Teorem 3. an orollary 3.5 are concerne wit average effects our results generalize to accommoate effects on oter features of te istribution of te response suc as its variance quantiles or its probability istribution (see W section 4). 9

20 4. onclusion We buil on results of W an W to stuy te specification of proxies tat can act as conitioning instruments to support te structural ientification of causal effects of interest. In particular we provie causal an preictive conitions sufficient for conitional exogeneity to ol tus permitting structural ientification of effects of interest. We buil on results in W to provie two proceures for inferring conitional causal isolation among vectors of settable variables in canonical settable systems. Our first proceure employs te ~-causality matrix associate wit to state necessary an sufficient conitions for conitional causal isolation. Our secon proceure employs te irect causality matrix associate wit to infer conitional inepenence relationsips. Te secon proceure contains as a special case te notion of -separation introuce in te artificial intelligence literature (see Pearl ). It follows from W Teorem 4.6 tat tese two proceures permit verifying conitional inepenence relationsips among certain vectors of ranom variables in canonical settable systems. We also buil on results of van Putten an Van cuppen (985) to provie necessary an sufficient conitions tat preserve conitional inepenence relationsips wen enlarging or reucing te σ-algebras generate by te conitioning vector of variables. corollary of tis result elivers a preictive relationsip among variables of interest sufficient to ensure tat P is conitionally stocastically isolating. We buil on tese results to istinguis between two categories of conitioning instruments. Te first reners te causes of interest an te unobserve irect causes of te response of interest conitionally causally isolate. Hence we refer to tese as causally isolating conitioning instruments or structural proxies. We provie two proceures tat permit verifying structural ientification from te ~-causality matrix associate wit te structural proxies as well as from te irect causality matrix. Te -base proceure contains as a special case te back oor criterion introuce in te artificial intelligence literature (see Pearl ). Te secon category of conitioning instruments ensures tat P stocastically isolates te causes of interest an te unobserve irect causes given te vector of conitioning instruments. We refer to tis secon category of conitioning instruments as stocastically isolating conitioning instruments or preictive proxies. We are unaware of any treatment of tis category in te literature oter tan in W an alak an Wite (007a). Here we focus on measuring total causal effects in canonical systems. We leave to oter work analysis of te measurement of irect inirect an te more refine notions of causality via an exclusive of sets of variables. We provie formal causal an preictive conitions sufficient to ensure tat conitional exogeneity ols. We leave to oter work te stuy of testing for conitional exogeneity a key conition for structural ientification. We also leave asie iscussion of ow to construct an optimal set of 0

21 conitioning instruments tat support efficient estimation of causal effects of interest. Te proceures of ection soul prove elpful in suggesting an testing for causal moels. We also leave a formal treatment of tis topic to oter work. 5. Matematical ppenix Proof of Proposition. Let n be a positive integer suc tat n H J an let {( j ) = ( n j n )} be a set of n istinct elements of {0 () (H J H )} suc tat ( ) ( ) ( ) c c... c =. From Definition. it follows tat tere exist a ( j)( j) ( j)( 3 j3) ( n jn)( j) set of n istinct settable variables { j j j n j n } suc tat j j tat is { n n j n jn yiels a contraiction wit te efinition of recursiveness (W efinition.5) tus completing te proof. } belongs to a ()-cycle. However tis Proof of Proposition.3 (a) Te proof follows from W (efinition 4.3). First we let a = min{b: tere exists ( j) Π b } an b = min{b: tere exists (i k) Π b B}. ~ (i) uppose tat 0 it follows tat a = 0 an b 0. uppose furter tat c = 0 0( ) for all (i k) B. Let in( I {0}:B ) ten Definition. gives tat 0 B an tus an B are conitionally causally isolate given. (ii) uppose tat 0 B; it ~ follows tat a 0 b = 0. uppose furter tat c = 0 for all ( j). Let 0( j) in( I {0}: ); ten Definition. gives tat 0. Tus an B are conitionally causally isolate given. (iii) uppose tat 0 B; it follows tat a 0 b 0. ~ uppose furter tat c ~ = 0 for all ( j) or c = 0 for all (i k) B; ten 0( j) Definition. gives tat 0 ~ or 0 ~ ~ 0( ) ~ B. Tus an B are conitionally causally isolate given. (b) Let P be any probability measure on (Ω F) an suppose tat eiter (a.i) (a.ii) or (a.iii) ol. It ten follows from W (teorem 4.6) tat Y Y B Y. Proof of Lemma.5 For given ( j) an (i k) B let ( j):( i k) = :B in( I ) an let ( j ):( i k be te corresponing vector of settable variables. Let ) ( j ):( i k ) I enote te ( j ik ) intercessors for pats troug an let ( j ):( i k ) enote te ( j ik ) intercessors not belonging to pats troug tat ( j)( i k) ( j ):( i k ) ( j ):( i k ). uppose p = 0. ince \ ( B) we ave ( j) (i k). From :[max( ab ) ] Definition. it follows tat c ( )( ) = 0 an tere oes not exist a subset {( j ) j i k

22 ( n j n )} of Π \ wit n j i k H J suc tat = c c... c =. ( j)( j) ( j)( j) ( n jn)( i k) ( ):( ) We tus ave I = I an ( j ):( i k =. Furter using W s notation tere ) ( j) exist measurable functions r an ( j r ) suc tat: :( i k) ( j ) r ( z z j [0: b ]( j) y ( j): y ( j ):( i k ) y ( j ):( i k ) y ( j ):( i k ) y ) :( i k ) = ( j ) r ( z [0: b ]( j) y r ( z ( j ):( i k ) ( j) :( i k) [0: b ]( j) y ( j ):( i k )) ) oterwise j H J suc tat j = or tere exists { j j } I ( ):( ) \ wit n j j n n n. n j i k Tus for all (a) z ; an (b) z an [0: b ]( j) j z z j wit j z j we ave: ( j ) [0: b ]( j) ( j) r ( z z j y y y y y ) ( j): ( j ):( i k ) r ( z [0: b ]( j) z j y ( j) :( i k) ( j): ( j ):( i k ) y ( j ):( i k ) r ( z [0: b ]( j) z j y ( j): ( j ):( i k ) y ( j ):( i k ) y ( j ):( i k ) :( i k) y ( j ):( i k ) y ( j ):( i k ) y ( j ):( i k )) ) = ( j ) [0: b ]( j) ( j ):( i k ) ( j) :( i k) r ( z y r ( z y )) ( ):( )( j) [0: b ]( j) ( j ):( i k ) [0: b ]( j) ( j ):( i k ) j i k r ( z y r ( z y )) ( j) :( i k) [0: b ]( j) ( j ):( i k ) = 0. It follows from efinition.8(ii) in W tat j ~ ():() j ik ik. ince an (i k) B j ik are arbitrary we ave ():() j ik for all ( j) an (i k) B i.e. ~ B : B. Proof of orollary.6 uppose tat eiter (i) (ii) or (iii) ol. From Lemma.5 an Definition. we ten ave tat conitions (i) (ii) or (iii) of Proposition.3 ol. Te proof ten follows from Proposition.3(b). Proof of Teorem.8 Teorem 4.6 in W gives tat (a) causally isolates an B or (b) P stocastically isolates an B given if an only if Y Y B Y tat is is conitionally inepenent of B given. Te result is ten immeiate from teorem 3.3 in Van Putten an Van cuppen (985).

23 Proof of orollary.9 Teorem 4.6 in W gives tat (a) causally isolates an B or (b) P stocastically isolates an B given if an only if Y Y B Y tat is is conitionally inepenent of B given. Te result is ten immeiate from proposition 3.4(e) in Van Putten an Van cuppen (985). Proof of Lemma 3. Let B = in( I :{( )} ) we permute te arguments of r Π to write B r ( z[0: b ]( :( i k)) z z B ) r Π ( z ). ubstituting for [0: b ] z B = y B = r B B ( z[0: b ]( :( i k)) B B z ) we ave r ( z z z B ) = r ( z z r B ( z z )) = [0: b ]( :( i k)) [0: b ]( :( i k)) B [0: b ]( :( i k)) r ( z[0: b ]( :( i k)) z ). Fix (g l) Π [0: b ]( :( i k)) suc tat gl ik. Let z enote values for settings corresponing to all elements of [0: b ]( :( i k))( g l) Π [0: b ]( :( i k))( g l) = Π [0: b ]( :( i k)) \ {(g l)}. Ten te function r ( z[0: b ]( :( i k)) z ) is constant in z gl for every ( z z ). Tus tere exists a measurable function [0: b ]( :( i k))( g l) ( gl ) ( gl ) r suc tat r ( z[0: b ]( :( i k))( g l) z ) = r ( z[0: b ]( :( i k)) z ) for all z [0: b ]( :( i k))( g l) z gl an z. ince (g l) Π [0: b ]( :( i k)) is arbitrary it follows tat tere exists a measurable function r suc tat r ( z ) = r ( z z ) for all z [0: b ]( :( i k))( ) z z [0: b ]( :( i k)) an z using te obvious notation for z. [0: b ]( :( i k))( ) Proof of Teorem 3. Te proof follows from teorem 3.3 in W. (i) uppose tat.(i(a b c) ii(a b)) ol; ten assumptions.(a b c.i) in W ol. In particular uner assumptions.(i(b c)) we ave tat D Z an D W. lso from Teorem 4.6 in W we ave tat (a) (Z W) causally isolates D an U or (b) P stocastically isolates D an U given (Z W) if an only if U D.Hence assumption. in W ols. ince E(Y) < te result follows from W (teorem 3.3 (i)). (ii) uppose furter tat D jrzu ( ) is ominate on a compact neigboroo of by an integrable function ten assumption.3 in W ols an te result follows from W (teorem 3.3 (ii)). Proof of orollary 3.3 ince 0 B for all = H Proposition.3 gives tat D an U are conitionally causally isolate given (Z W ) an tus D U. Te result ten follows from Teorem 3.. Proof of orollary 3.4 ince 0 B for all = H orollary.6 gives tat D an U are conitionally causally isolate given (Z W ) an tus D U. Te result ten follows from Teorem 3.. 3

24 Proof of orollary 3.5 uppose tat (a.i) or (a.ii); an (b.i) or (b.ii) ol. Ten from orollary.9 we ave tat (Z W) causally isolates D an U or tat P stocastically isolates D an U given (Z W). Te result ten follows from Teorem 3.. References Bang-Jensen J. an G. Gutin (00). Digraps: Teory lgoritms an pplications. Lonon: pringer-verlag. Blunell R. an J. Powell (003) Enogeneity in Nonparametric an emiparametric Regression Moels in M. Dewatripoint L. Hansen an. Turnovsky (es.) vances in Economics an Econometrics: Teory an pplications Eigt Worl ongress Vol II. New York: ambrige University Press alak K. an H. Wite (007a) n Extene lass of Instrumental Variables for te Estimation of ausal Effects UD Department of Economics Discussion Paper. alak K. an H. Wite (007b) Inepenence an onitional Inepenence in ausal ystems UD Department of Economics Discussion Paper. Heckman J. an R. Robb (985) lternative Metos for Evaluating te Impact of Interventions in J. Heckman an B. inger (es.) Longituinal nalysis of Labor Market Data. ambrige: ambrige University Press Heckman J. an B. Honore (990) Te Empirical ontent of te Roy Moel Econometrica Heckman J. an E. Vytlacil (005) tructural Equations Treatment Effects an Econometric Policy Evaluation Econometrica Matzkin R. (003) Nonparametric Estimation of Nonaitive Ranom Functions Econometrica Pearl J. (995) ausal Diagrams for Empirical Researc Biometrika (wit Discussion). Pearl J. (000). ausality: Moels Reasoning an Inference. New York: ambrige University Press. Rosenbaum P. R. (00). Observational tuies. n e. Berlin: pringer-verlag. Rosenbaum P. R. an D. Rubin (983) Te entral Role of te Propensity core in Observational tuies for ausal Effects Biometrika

25 Rubin D. (974) Estimating ausal Effects of Treatments in Ranomize an Nonranomize tuies Journal of Eucational Psycology Van Putten. an J.H. Van cuppen (985) Invariance Properties of te onitional Inepenence Relation Te nnals of Probability Wite H. an K. alak (006) Unifie Framework for Defining an Ientifying ausal Effects UD Department of Economics Discussion Paper. 5