Assessing Hidden Bias in the Estimation of Causal Effect in Longitudinal Data by. Using a Matching Estimator with Rosenbaum Bounds.

Transcription

1 Aeing Hidden Bia in the Etimation of Caual Effect in Longitudinal Data by Uing a Matching Etimator with Roenbaum Bound Tim Futing Liao Univerity of Illinoi Paper Prepared for the BHPS Conference June 30-July 2, 2005 Pleae treat a copyrighted material

2 Aeing Hidden Bia in the Etimation of Caual Effect in Longitudinal Data by Uing a Matching Etimator with Roenbaum Bound Tim Futing Liao Univerity of Illinoi Urbana, IL 61801, USA tfliao@uiuc.edu; tfliao@eex.ac.uk (voice) (fax) Abtract Some ocial, economic, and behavioral outcome are affected by natural experiment, or treatment that take place not by deign but by natural occurrence. Treatment effect in natural experimental etting or obervational tudie are difficult to ae becaue there may be potential election bia generated by hidden confounder and becaue counterfactual are unobervable, regardle of how many control variable the analyt include in the model. Whilt panel data in general allow the reearcher to better gauge caual effect and factor into individual heterogeneity, they are not immune to poible hidden bia. Thi paper attempt to ae uch potential hidden bia of teenage birth, a natural treatment event, on an outcome variable uch a the mother potnatal mental wellbeing. When there i no hidden bia, propenity core matching provide an etimate of the treatment effect on an outcome variable. However, matching method are not robut againt hidden bia ariing from unoberved variable imultaneouly affecting aignment to treatment and the outcome variable. A viable trategy to deal with the problem i the Roenbaum bound approach, which provide a wort-cae cenario. The bound convey important information about the level of uncertainty contained in matching etimator by howing jut how large the influence of a hidden confounder mut be to undermine the concluion of an analyi baed on matched data. The firt 10 wave of the Britih Houehold Panel Survey data are ued to etimate the bound, thu the enitivity, of the treatment effect of teenage motherhood on potnatal mental wellbeing to ee how robut thee effect are in variou year from birth.

3 Aeing Hidden Bia in the Etimation of Caual Effect in Longitudinal Data by Uing a Matching Etimator with Roenbaum Bound Tim Futing Liao 1. Introduction Recent economic, ociological, and tatitical reearch ha paid increaing attention to the proper aement of treatment effect, notably the o-called counterfactual approach to caual analyi (e.g., Angrit, Imben and Rubin 1996; Heckman, LaLonde and Smith 2000; Roenbaum 2002; Sobel 1995, 1996; DiPrete and Gangl 2004; Winhip and Morgan 1999). Matching method, including propenity core matching which ha gained popularity of late, potentially improve on regreion analyi by nonparametrically eliminating potential bia from oberved explanatory variable (e.g., Imai and van Dyk 2004; Smith 1997). However, thee method are not robut if there i endogeneity or hidden bia from unoberved variable related to both aignment to treatment and the outcome variable. Alo popular in recent year i bound analyi, which can ae the potential impact of hidden bia ariing from unoberved confounding variable (e.g., Heckman and Vytlacil 2001; Manki 1990). Some bound analyi, uch a the Roenbaum bound trategy, combine both method by firt conduct a matching of the data before building bound for a enitivity analyi (Roenbaum 2002). The goal of thi paper i to extend the application of Roenbaum bound on matching etimator to panel data analyi. The interet in Roenbaum bound i quite recent, and i confined to analyi that i not longitudinal in nature. However, 1

4 longitudinal or panel data, uch a the Britih Houehold Panel Survey (BHPS), have become increaingly available and have been widely ued in a great number of economic and ocial tudie. The paper repreent a firt attempt to ue Roenbaum bound in a panel data etting. Section 2 dicue the aement of cauality when there i hidden bia. Section 3 introduce the Roenbaum bound trategy on matching etimator. Section 4 propoe a imple trategy to implement the Roenbaum bound trategy for panel data analyi. Section 5 preent an empirical applying the trategy to the firt 10 wave of BHPS, a high quality longitudinal dataet, and Section 6 conclude. 2. Caual Aement in the Preence of Hidden Bia While the iue of heterogeneity and confounding i familiar to mot ocial cientit, their typical approach to deal with it i to include the o-called control variable. Thi i all fine if all confounding variable can be oberved, but bia in caual analyi may be hidden in unoberved variable. In thi regard, the tandard regreion i uele in dealing with hidden bia in analyzing treatment effect. Following the preentation of Heckman (1997), a nonparametric model for the treatment effect can be given a: Y Y 0i 1i = µ ( X ) U 0 = µ ( X ) U 1 i i 0i 1i (1) where Y 0 and Y 1 are outcome for peron i according to the treatment tatu of D=0 or D=1, and U 0 and U 1 are the effect of hidden variable depending on the treatment tatu. Each individual i may have only one of thee two outcome obervable but not both. In obervational tudie, the reearch interet i to ae whether µ X i ) µ ( ), the treatment effect. 0( 1 X 1 2

5 Uing (1), we may further et up the following effect: E X ) = µ ( X ) µ ( ) (2) ( 1 0 X E X, D = 1) = µ ( X ) µ ( X ) E( U U X, D 1) (3) ( = The left-ide of (2) denote the average treatment effect (ATE), conditional on X, and the left-hand ide of (3) define the average treatment effect on the treated (ATT), conditional on X. The correponding unconditional ATE and ATT can be further given a: [ 1 ( X ) µ ( X )] d( P( X )) E ) = µ E( X ) p( X ) (4) ( 0 X X E( D = 1) = X [ µ 1 ( X ) µ 0 ( X ) E( U1 U 0 X, D = 1) ] X E( X, D = 1) E( U 1 U 0 X, D = 1) p( X ) d( P( X )) (5) Thee equation imply that there exit interperonal variation in the treatment effect becaue of interperonal variation in X and in the unoberved variable of U 0 and U 1. They alo how a incorrect the olution of taking a imple mean difference between the treatment and the control group when hidden or unoberved variable are preent. 3. Matching and Roenbaum Bound Before we introduce the Roenbaum bound trategy, let u firt examine the method of matching. The goal of matching i to adjut the data prior to ome core analyi, which can be parametric a a conventional regreion or a regreion-baed model, or nonparametric uch a ome type of rank or ign tet. By matching the data, the following objective are achieved: (1) the aociation between D and X i reduced or eliminated, and (2) no bia and little inefficiency are introduced. However, we can only match the data uing available, oberved variable. 3

6 A DiPrete and Gangl (2004) correctly commented, tandard regreion framework are unatifactory becaue they do not rigorouly define pecific average treatment effect, and alternative framework uch a random coefficient model rely on trong aumption that the effect of confounding variable ha been captured by model pecification, and that uch pecification can be extrapolated to the population ditribution. Matching method, repreented by a popular ubet known a propenity core matching, do not rely on thee aumption. The matched data will include a elected ubet of the original ample in which D and X are unrelated, or in other word, the treatment and control group have identical background characteritic uch that: p ( X D = 0) = p( X D = 1) (6) The implet way to atify (6) i to ue one-to-one exact matching, by having each treated unit matched with one control unit with identical X value. Propenity core matching begin by ummarizing all variable in X with a ingle variable known a the propenity core (Roenbaum and Rubin 1983). Thi core i the true probability of obervation i receiving treatment, given X i, or p D i = 1 X ). It i ( i uually etimated with a logit or probit model of D on X (without any involvement of the outcome variable). Theoretically, the true propenity core i a balancing core becaue, if a group of obervation have imilar propenity core ditribution, then the covariate X will be balanced in the treatment and control group in all variable of X. Furthermore, if aignment of treatment i ignorable given X, then it i alo ignorable given only the propenity core. In practice, however, the true propenity core i unknown (unle in 4

7 true experiment). Neverthele, if we have a conitent etimate of it, we can till ue ome of it theoretical propertie. Auming we have a good etimate of the true core by the logit or probit model of D on X, then our matching will be ucceful. Needle to ay, the uual caveat hould be repeated: Our etimation can only be a good a the X variable included in the matching. Matching method, however, cannot rid the aement of treatment effect of the effect of hidden bia. The Roenbaum bound method allow the analyt to get a ene of the wort-cae cenario when uch bia i preent. The preentation of hi bound method for 1-1 matched pair follow Chapter 4 of Roenbaum (2002) and Appendix A of DiPrete and Gangl (2004). Let Z record whether each of the pair i treated, and r regiter the outcome for reach cae in the et of S pair. Z j equal 1 if an obervation i treated, and 0 otherwie; c i accordingly define a follow: c c c = 1, c = 0, c = 0, c = 0 = 1 = 0 if if if r 1 r 1 r 1 > r < r 2 2 = r 2 Common tet tatitic for matched pair are igned core tatitic, including Wilcoxon igned rank tatitic and McNemar tatitic, take the form below: T S 2 d = t( Z, r) = c Z (7) = 1 j= 1 j j where d i the rank of r r with average rank ued for tie. The product of c and Z 1 2 variable indicate which pair are to be elected where the outcome for the treated cae i greater than that for the control cae. The ummed rank of the ample are compared with the ditribution of the tet tatitic under the null hypothei that treatment ha no 5

8 effect. When there i no hidden bia, or in other word, aignment to treatment i random, thi nonparametric tet can be applied without complication. In the preence of nonrandom aignment to treatment, however, the above tet tatitic can be bounded. When thi happen, we may ue U to repreent the unoberved heterogeneity that affect the treatment probability. Now let π i be the probability that the ith cae receive the treatment and X be the oberved covariate vector determining the probability of receiving treatment (a well a the outcome), the treatment aignment can be defined a follow: π i log = κ X i γui π ( ) (8) 1 i where 0 U 1. Roenbaum propoe that (8) implie the following bound on the i ratio of the odd that either of two obervation matched on X (or on the propenity core p(x)) will receive treatment: 1 π 1(1 π 2 ) Γ Γ π (1 π ) 2 1 (9) where Γ = exp(γ ). Under the aumption that an endogenou variable U exit, the tet tatitic in (7) become the um of S independent random variable, where the th pair equal d with the probability, p c1 exp( γu1) c2 exp( γu2 ) = (10) exp( γu ) exp( γu ) 1 2 and the probability of the th pair equaling 0 i jut 1 p. Although the null ditribution of t(z,r) i unknown, for each fixed γ, the two known ditribution bound the null ditribution. With Γ = exp(γ ), we define p and p, 6

9 which repreent thee two ditribution a the lower and upper limit for p in the following way: p 0 if = 1 if Γ 1 Γ c c if 1 1 = c = c c = 0 = 1 c 2 and p 0 if = 1 if 1 1 Γ c c if 1 1 = c = c c = 0 = 1 c 2 It follow that p p p for = 1,, S. Let u further define T a the um of S independent random variable, where th take the value d with probability the value 0 with probability1 p, and T imilarly with (2002) put forward a propoition, tating that, for all u U p in place of, the unknown null ditribution of the tet tatitic T = t( Z, r) i bounded by the ditribution of p and take p. Roenbaum T and T : { T a m} p( T ) p( T a) p a (11) for all a and u U. For each value of γ, (11) place bound on the ignificance level that would have been appropriate had u been oberved. The ditribution of thee bounding variable and T and T have moment which can be readily defined a: S ( ) = (12) E T = 1 d p S 2 var( T ) = d p (1 p ) (13) = 1 7

10 For T the moment are given by the ame formula of (12) and (13) with p. The ditribution of T and p in place of T are approximated by normal ditribution a the number of pair S increae, provided that the number of dicordant pair increae with S and that the d are reaonably well-behaved, a they are for the McNemar and Wilcoxon tatitic. 4. Roenbaum Bound in Panel Analyi The Roenbaum bound method decribed in Section 3 can be extended traightforward in a typical cro-ectional etting. With panel data, there i additional heterogeneity in the ytem generated by the potential clutering effect of individual contributing multiple obervation over time. The claical regreion can be readily extended to handle the complex of panel data: y it = x β u ε (14) it i it where u i are individual-pecific effect and ε it are random error that vary by cae a well a with time. The pecification of u i give rie to fixed or random effect model. Another way to view the two type of pecification i to undertand how omitted variable may function. Fixed effect regreion i called for when omitted variable differ between cae but are contant over time. However, when there i reaon to believe that ome omitted variable may be fixed over time but vary between cae while other may be contant between cae but vary over time, then random effect regreion i the choice. Fixed effect regreion i tantamount to the leat quare with dummy variable etimator, and random effect can be viewed a effect reulted from a random draw from a larger population. However, hidden bia may till exit even though fixed 8

11 or random effect are taken into account becaue mot often we cannot be ure of the pecific pattern of unoberved variable; in addition, thee variable may vary between cae and over time imultaneouly. For application purpoe, we conider both type of effect in the next ection demontrating a BHPS data example. For a fixed effect or random effect model teting treatment effect, (14) can be modified lightly to the following: y = x β δ D u it it it i ε (15) it where δ capture the average treatment effect, D it i a dummy variable coded 1 if treated and 0 otherwie, and the x it vector contain all the explanatory variable erving a control. To extend the Roenbaum bound method to panel data, we mut omehow clean the outcome variable of the influence of fixed or random effect. To do o, the following procedure i appropriate: (1): Etimate a model in the form of (15) minu the D it variable (becaue we want to keep the treatment effect in the outcome variable for the bound analyi), (2): Conduct matching (uch a propenity core matching) a uual. However, the outcome variable to ue for pot-matching analyi i no longer y it but ε it, which can be conider a the outcome variable with meaurable confounding effect a well a individual (fixed or random) effect partialled out. (3): Ue ε it a outcome core in contructing Roenbaum bound, which give the limit for thee tet tatitic of (7) with d = r r where r. now record the new 1 2 outcome with ε it in place of y it. Note that while the parameter for ATT may be unidentified in the fixed effect model, it bound can actually be aeed becaue of the 9

12 election due to matching and becaue rank i ued intead of real difference in value in the bound analyi. Thi mean that both the fixed effect and the random effect model can be ued for contructing bound. (4): Finally, Roenbaum bound can be contructed eparately for each level of certain longitudinal characteritic in the analyi. For example, we may etimate Roenbaum bound for each year ince the completion of econdary education, the onet of unemployment, firt marriage, or firt birth. Thi ugget a modification of (9) a follow: 1 π k1(1 π k2 ) Γ π (1 π ) k kt 2 k1 Γ k (16) Note that the ubcript k in (16) here doe not repreent the wave of panel data even though it varie with it. The ubcript refer to individual-pecific time meaure uch a the number of year ince a firt birth. Two conideration upport uing the full vector of X variable in calculating the adjuted outcome for tep 3. Matching method, propenity core or not, reduce bia but will not produce a perfect match and cannot replace randomization in eliminating confounding factor. Moreover, it make ubtantive a well a tatitical ene that the X vector in (8) and in (15) may not be identical (i.e., there are different mechanim reponible for aignment to treatment and for generation of outcome) though all ignificant X variable in (15) hould alo be in (8). That i, the all the important covariate ued for modelling the outcome variable hould be included in matching, thought not necearily the other way around. The procedure decribed here will guide the application in the next ection. 10

13 5. A BHPS Example: The Effect of Teenage Birth on Mental Health Thi ection give an empirical example that applie Roenbaum bound for aeing the treatment effect of having a teenage birth on the mother mental health, uing the BHPS data. The example ue the firt 10 wave (1991 to 2000) of the BHPS, an annual urvey of each adult aged 16 and over member of a nationally repreentative ample of more than 5,000 houehold (approximately 10,000 individual interview). The ame individual are re-interviewed in ucceive wave and, if they plit off from original houehold, all adult member of their new houehold will alo be interviewed. The main objective of the urvey i to invetigate ocial and economic change at the individual and houehold level in Britain, and to identify, model and forecat uch change, their caue and conequence in relation to a range of ocioeconomic variable. The analyi in thi paper i retricted to women of aged in the year of birth for mother and in the reference year for nonmother from the 10 wave. Becaue the BHPS doe not pecifically record event of birth, the age at firt birth variable i contructed uing the following four ource of information repreented by ix variable: (1) whether the hopital (or clinic) tay in the pat year wa due to childbirth, (2) the year and month of birth of the repondent, (3) the year and month of the interview, and (4) the number of own children in houehold. A median value of childbirth date i aigned to the time period between the previou and the current interview, and an age at firt birth i calculated accordingly uing the repondent birth date information. While the approach i not perfect a home birth are mied, the mie hould not be too numerou ince home birth rate are a low a about 2% in Britain. 11

14 Becaue a major concern i to follow the mental health trajectory of mother in their potpartum year, it i deirable to have a minimum of five year potpartum if not longer becaue it ha been hown that mother potpartum depreion alleviate greatly after three year (Liao 2003). The total number of obervation with no miing value from the 10 wave of the BHPS i 2,366, and 494 of thee obervation were contributed by teenage mother. To capture any potential differential effect of motherhood, epecially teenage motherhood (i.e., mother are by themelve different from nonmother), the pychological wellbeing during the year leading to the firt birth are alo included a long a the information i available, in addition to the time-invariant, unit-variant term of u i in equation (2), which typically capture unoberved individual characteritic in panel analyi. The variable year from birth record the year a mother wa in with regard to the firt birth (i.e., with a maximum range from -4 to 10 and 0 indicating the birth year. Becaue nonmother have no birth a a reference point, a randomly choen reference point wa aigned to them by auming a uniform ditribution throughout the five year o that they became comparable to the mother. The aumption i valid a long a birth i a random event in term of time. That i, there i no reaon to doubt that birth i independent of time. Similarly, the nonmother reference birth year are independent of time, thereby ditributed evenly over the five wave. Uing the aignment under the uniform ditribution, the year from birth variable reflect the departure in number of year from thi arbitrarily defined yet randomly choen reference point for nonmother. 12

15 The mot often ued intrument for meauring peronal well-being and mental health in the UK i the General Health Quetionnaire (GHQ), which ha been widely and uccefully teted and ued. The full-length verion of the GHQ include everal dozen of quetion. The BHPS adminiter a hortet verion (12 item) of the GHQ (GHQ-12), which ha been found a conitent and reliable intrument when ued in general population ample with relatively long interval between application a in the BHPS. The quetion ak the repondent how they have been feeling over the pat few week concerning the 12 area uch a whether a repondent could concentrate, had lo of leep, or felt under train. The four-point anwer follow the pattern of more/le than uual, ame a uual, le/more than uual, and much le/more than uual. To reduce the potential problem of regreion toward the mean and to avoid the unreaonable equal pacing aumption when treating the four point a numerical value, dummy variable were created for the 12 item with feeling wore coded 1 regardle of the degree of how much wore. The 12 variable were then aggregated into a ingle count variable with a value ranging from 0 to 12. For the ake of pace, the detail about the covariate are omitted from thi paper, and intereted reader are referred to Liao (2003). To begin with, let u compare the group mean of teenage mother and teenage nonmother (Table 1). On overage, the mother have a GHQ-12 core of 0.9 point higher than the nonmother, with a ignificant t-tet. Comparing the mean GHQ-12 value acro the level of the year from birth variable reveal an intereting pattern: there i no ignificant difference between the two group until the birth year. The difference get maller in the potpartum year, and become inignificant by the fifth year potpartum. 13

16 ---Table 1 about here--- Next, two imple model of depreion were fit, one uing the OLS and the other, the random effect etimator. Obviouly the OLS regreion ignore the panel tructure of the data while the random effect model doe not. (In a fixed effect model, the parenthood parameter i not identified.) Two remark can be made about thee reult. For mot X variable, either the OLS or the random effect etimator appear to lead to imilar concluion. According to the OLS regreion, the treatment effect among thee women i about 0.50 while according to the random effect model it i about 0.55, after other oberved confounding factor having been taken into account. ---Table 2 about here--- In thi example, we ue a one-to-one nearet-neighbor caliper matching algorithm without replacement that matche treatment and control cae with imilar propenity core within the tolerance level for acceptable matche defined in term of the empirical variance of the propenity core. (The Stata macro pmatch2 wa ued in thi part of the analyi.) The propenity core i defined a the predicted probability of a teenage mother obervation from a probit model of teenage motherhood on a et of explanatory variable. Table 3 contain the etimate from thi model. ---Table 3 about here--- Wherea mot of the variable are the ame one a thoe in Table 2, their ignificance level in predicting teenage motherhood are different from thoe predicting depreion, epecially in two of the chronic train variable a well a ome of the tructural and life coure variable. 14

17 Before matching, many pair of the covariate value appear to be quite different. Matching ha brought them much cloer (Table 4). In other word, matching ha reduced the dicrepancie exiting in the raw data. (The Stata macro rbound wa ued in thi part of the analyi.) Therefore, the proce of matching create a high degree of covariate balance between the treatment and control cae ued in the etimation. How would one quantify the amount of bia and it reduction? Here we ue the tandardized mean difference between treatment and control ample (DiPrete and Gangl 2004): where x T and and 100( xt xc ) Bia = (17) 2 2 ( T C ) 2 2 T are the ample mean and variance for the treatment ubample, and x C 2 C are the ame tatitic for the control ubample. ---Table 4 bout here--- For majority of the variable the abolute bia ha reduced, a much a 100%. Three variable how an increae in bia. However, the important finding here i that before matching, jut about all the variable were different enough between the two ubample (ignificant at the 0.05 level) while afterward, none of the t-tet are ignificant at the ame level anymore. The reult in Table 5 how that robutne to hidden bia varie coniderably acro the level of year from birth variable. The overall treatment effect appear to be fairly robut to the poible preence of hidden bia. The critical level of Γ at which we would have to quetion the concluion of a poitive effect i jut below That i, if unoberved heterogeneity caued the odd ratio of treatment aignment to differ between 15

18 treatment and control cae by a factor of about 1.20, then we would have to quetion the credibility of a poitive effect of teenage birth on women mental health. ---Table 4about here--- The dynamic variation over time i what make the bound analyi mot worthwhile. In the year leading to the birth, the treatment effect are not robut at all. During the birth year, teenage mother level of depreion become ignificantly higher than the teenage nonmother. Uing the outcome variable with confounding effect partialled out with the fixed effect model, we would have to quetion the concluion of ignificant difference between the two group of women at the level of change in the odd ratio of treatment aignment by a factor of between 1.4 and 1.5; uing the outcome variable with confounding effected partialled out with the random effect model, thi critical level of change in the odd ratio i by a factor of between 1.3 and 1.4. For one year potpartum, we would have to quetion the concluion of treatment effect if hidden bia caued the odd ratio of treatment aignment to differ by a factor between 1.10 and 1.15 according to the fixed effect model, and a factor between 1.25 and 1.30 according to the random effect model. The ubequent year appear to be le robut to potential hidden bia, epecially in the year beyond three. However, the treatment effect i only mildly robut for three year potpartum (but not for two year potpartum). It i important to note that thee reult are wort-cae cenario. A value of 1.20 doe not ugget that there i no true poitive effect of teenage birth on depreion; it only implie that the confidence interval for the treatment effect would include zero if an unoberved endogenou variable caued the odd of treatment aignment to differ between treatment and control group by a factor of 1.20 and if the effect of thi 16

19 unmeaured variable on mental health wa o trong a to almot perfectly determine whether the depreion level would be higher for the teenage mother or the nonmother in each pair of the matched cae. In other word, if thi unoberved variable had a trong effect on treatment aignment but only a weak effect on the outcome variable, the confidence interval for the treatment effect would not contain zero. A wore-cae cenario, Roenbaum bound neverthele carry ome ueful information about the level of uncertainty: They demontrate the extent to which the influence of hidden bia would undermine the concluion of a matching analyi. 6. Concluion Thi paper preented Roenbaum bound for longitudinal data analyi. In obervational tudie, hidden bia can be the wort nightmare for data analyt. The Roenbaum bound method can provide reaonable confidence that a caual relationhip between a treatment and an outcome variable exit even when there i unmeaured confounding variable. The BHPS example howed well that Roenbaum bound can be traightforwardly contructed for panel data analyi, and that they may hed light on the iue of robutne that a conventional analyi cannot. Thi i epecially true if the data analyt exploit the dynamic tructure of panel data by demontrating how robutne may vary by ome time variable(). The propoed method of incorporating the Roenbaum bound method into panel data analyi i rather conervative in the ene that the fixed or random model firt clean the outcome variable of confounding effect a much a poible before an 17

20 application of Roenbaum bound on matched data. Coupled with nonparametric matching, the Roenbaum bound method provide reearcher uing obervational data a feaible way to ae robutne of treatment effect in the preence of potential hidden bia, thereby giving them a mean that i more informative and more uperior when a conventional method i alo ued to give ome ubtantive inight into the data than uing the conventional method alone. 18

21 Reference Angrit, J., G.W. Imben and D. Rubin, (1996). Identification of Caual Effect Uing Intrumental Variable, (with dicuion). Journal of the American Statitical Aociation vol. 91, no 434, DiPrete, Thoma A. and Marku Gangl Aeing Bia in the Etimation of Caual Effect: Roenbaum Bound on Matching Etimator and Intrumental Variable Etimation with Imperfect Intrument. Pp in Sociological Methodology, Vol. 34, edited by Ro M. Stolzenberg. Wahington, DC: Blackwell. Heckman, Jame J., Robert J. LaLonde, and Jeffrey A. Smith The Economic and Econometric of Active Labor Market Program. Pp in Handbook of Labor Economic, Vol. III, edited by Orley Ahenfelter and David Card. Amterdam: North Holland. Heckman Jame J. and Edward J. Vytlacil Intrumental Variable, Selection Model, and Tight Bound on the Average Treatment Effect. Pp in Econometric Evaluation of Active Labor Market Policie in Europe, edited by Michael Lechner and Friedhelm Pfeiffer. New York: Springer. Imai, Kouke and David A. van Dyk Caual Inference with General Treatment Regime: Generalizing the Propenity Score. Journal of the American Statitical Aociation 99: Liao, Tim F Mental Health, Teenage Motherhood, and Age at Firt Birth among Britih Women in the Working Paper of the Intitute of 19

22 Social and Economic Reearch, paper Colcheter: Univerity of Eex. Manki, Charle F Nonparametric Bound on Treatment Outcome. American Economic Review 80: Roenbaum, Paul R Obervational Studie, Second Edition. New York: Springer. Roenbaum, Paul R. and Donald B. Rubin The Central Role of the Propenity Score in Obervational Studie for Caual Effect. Biometrika 70: Smith, Herbert Matching with Multiple Control to Etimate Treatment Effect in Obervational Studie. Pp in in Sociological Methodology, Vol. 34, edited by Adrian E. Raftery.Boton, MA: Blackwell. Sobel, Michael E Caual Inference in the Social and Behavioral Science. Pp in Handbook of Statitical Modeling for the Social and Behavioral Science, edited by Gerhard Arminger, Clifford C. Clogg, and Michael E. Sobel. New York: Plenum. Sobel, Michael E An Introduction to Caual Inference. Sociological Method and Reearch 24: Winhip, Chritopher and Stephen L. Morgan The Etimation of Caual Effect from Obervational Data. Annual Review of Sociology. 25:

23 Table 1: A Comparion of GHQ12 Score between Teen Mother and Teen Nonmother Year from Birth Overall Nonmother Mother t-value p N 1872/494 44/13 87/29 148/44 283/59 253/63 244/58 226/61 179/50 Note: The t-tet i baed on the two-ample, unequal-variance tet with the null hypothei being the teen nonmother GHQ12 core lower than the teen mother. The p value are for one-tailed tet of the null hypothei that teen mother would have a higher level of depreion. The N before the / ign are for nonmother, and the one after it, mother. 21

24 Table 2: Conventional Regreion Approache to the Aement of Treatment Effect OLS Regreion Random-effect Model X Variable Etimate t Etimate Z Structural Context Tory year? Univerity degree? Left chool<16? Adjut. income cale Permanent job? Divorced/eparated? Partner together? Chronic Strain Poor health? Diabilitie? Health problem? Still in chool? Mediator Low ocial upport Job hour/week Life Coure Dynamic Age Year from birth Treatment Parenthood Contant ρ Model F or χ (p=0.000) (p=0.000) N 2,366 2,366 (366) Note: Variable with a quetion mark are dummy variable, coded 1 if ye and 0 otherwie. 22

25 Table 3: Probit Regreion Model of Treatment (Teenage Motherhood) X Variable Etimate Z Univerity degree? Left chool<16? Adjut. income cale Permanent job? Divorced/eparated? Partner together? Poor health? Diabilitie? Health problem? Still in chool? Low ocial upport Job hour/week Age Year from birth Contant Model χ (p=0.000) N 2,366 Note: Variable with a quetion mark are dummy variable, coded 1 if ye and 0 otherwie. 23

26 Table 4: Decriptive Statitic Comparing Unmatched and Matched Data Before Matching After Matching % (Un)matched Variable Control Treated Control Treated in bia t-tet GHQ ; 1.82 Still in chool? ; Univerity degree? ; Left chool<16? ; Adjut. inc. cale ; 0.53 Poor health? ; Diabilitie? ; 1.22 Health problem? ; 0.82 Permanent job? ; Job hour/week ; 0.15 Divorced/eparated? ; 1.86 Partner together? ; Low ocial upport ; 0.20 Age ; Year from birth ; Note: The firt number in the t-tet column are for unmatched comparion and the econd, the matched comparion. 24

27 Table 5: Roenbaum Bound for Teenage Birth Treatment Effect p-critical Year from birth Γ Fixed Random N of pair Overall 1 <0.001 < < year year year year year year year year Note: The third and the fourth column contain the p value for Roenbaum bound with the ε it from the fixed effect and the random effect model, repectively, in place of the outcome variable in contructing bound for the matched pair. 25