Abadie s Semiparametric Difference-in-Difference Estimator

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1 The Stata Journal (yyyy) vv, Number ii, pp. 1 9 Abadie s Semiparametric Difference-in-Difference Estimator Kenneth Houngbedji, PhD Paris School of Economics Paris, France kenneth.houngbedji [at] ps .eu Abstract. The difference-in-differences (DID) estimator measures the effect of a treatment or policy intervention by comparing change over time of the outcome variable across treatment groups. To interpret the estimate as a causal effect, this strategy requires that, in the absence of the treatment, the outcome variable would have followed the same trend in treated and untreated groups. This assumption may be implausible if selection for treatment is correlated with characteristics that affect the dynamic of the outcome variable. This paper describes the command asdid which implements the semi-parametric differences-in-differences (SDID) estimator of Abadie (2005). The SDID is a reweighing technique that addresses the imbalance of characteristics between treated and untreated groups. Hence, it makes the parallel trend assumption more credible. In addition, the SDID estimator allows the use of covariates to describe how the average effect of the treatment varies for different groups of the treated population. Keywords: st0001, semi-parametric estimations, difference-in-difference, propensity score 1 The semiparametric difference-in-difference estimator Let s consider the general setting of studies of causal effects used by Rosenbaum and Rubin (1983). We want to estimate the causal effect of a treatment on a variable of interest y at some time t. Each subject has two potential outcomes (y 1t, y 0t ). y 1t is the value of y if the subject receives the treatment by time t. y 0t is the value of y had the participant not received the treatment at time t. d is an indicator of whether or not a participant is treated by time t. At time t = 0 the baseline b no one is treated. At the time t 0, d is equal to 1 for treated participant and 0 otherwise. We want to estimate the average effect of the treatment on the treated (): ) E (y 1t y 0t d = 1. (1) Since y 0t is never observed for a treated participant, the cannot be directly estimated. Assume y 0b is the value of y at time t = 0 i.e. the baseline. Let s x b be a set of pre-treatment characteristics; y t y t y b is the change of y between time t and the baseline b and π (x b ) P ( d = 1 ) xb is the conditional probability to be in the treatment group also called the propensity score. Abadie (2005) shows that the c yyyy StataCorp LP st0001

2 2 Semiparametric Difference-in-Difference Estimator sample analog of ( yt E P (d = 1) d π (x ) ) b (2) 1 π (x b ) gives an unbiased estimate of the if Equation (3) and Equation (4) hold. ) ) E (y 0t y 0b d = 1, xb = E (y 0t y 0b d = 0, xb. (3) P (d = 1) > 0 and π (x b ) < 1. (4) The estimator is a weighted average of the difference of trend y t across treatment groups. It proceeds by reweighing the trend for the untreated participants based on π(x their propensity score π (x b ). As b ) 1 π(x b ) is an increasing function of π (x ), untreated b participants with higher propensity score are given higher weight. Abadie (2005) suggests to approximate the propensity score π (x b ) semiparametrically using a polynomial series of the predictors. Thereafter, the values predicted are plugged into the sample analogue of Equation (2). Even though the approximation improves for higher polynomial order, the estimation become less precise. It is also possible to estimate π (x b ) with the series logit estimator (SLE) (see Hirano et al. 2003). This method uses a logit specification to constrain the estimated propensity score to vary between 0 and 1. Consider for instance that ˆπ (x b ) is the approximated propensity score; k is the order the polynomial function used to approximate π (x b ). The approximation of π (x b ) produced by the linear probability model can be written as follows: k ˆπ (x b ) = ˆγ 0 + ˆγ 1 x 1 + ˆγ 2i x i 2, (5) where x 1 is a binary variable; x 2 is a continuous variable and x i 2 = i j=1 x 2. The coefficients ˆγ 0, ˆγ 1, ˆγ 21,..., ˆγ 2i,..., ˆγ 2k are estimated using an ordinary least square estimator. With a SLE estimator approach, the propensity score π (x b ) is estimated as follows: ( ˆπ (x b ) = Λ ˆγ 0 + ˆγ 1 x 1 + i=1 K ) ˆγ 2k x k 2 exp (x) where Λ (x) = is the logistic function. Higher order the binary variables 1 + exp (x) like x 1 are not considered because for any value k > 1, x k 1 = x 1. Independently of the approximation method used, the errors related to the estimation of the propensity scores are taken into account when estimating the standard error of the as described in Abadie (2005). Other estimators use the propensity score to estimate the. The kernel matching and nearest neighbor matching estimators are among the most widely used estimators for quasi experimental identification. However, both estimators assume that the propensity score is given and not estimated and produce on average estimates with smaller standard errors than the estimator of Abadie (2005). k=1 (6)

3 K. Houngbedji 3 2 The absdid command The command absdid is the Stata equivalent of a Matlab code written by Abadie in an empirical application of the semiparametric difference in difference estimator. 1 absdid estimates the by comparing change over time of the outcome of interest across treatment groups while adjusting for difference between treatment groups on the observable characteristics at baseline which are correlated to the propensity score. The general syntax for the command absdid is: absdid depvar [ if ] [ in ] [, tvar(varname) xvar(varlist) yxvar(varlist) order(#) csinf(#) csup(#) sle ] depvar is a variable that represents the change of the outcome of interest between baseline and post treatment for each observation. tvar(varname) is the treatment variable. It takes the value 1 when the observation is treated and 0 otherwise. xvar(varlist) are the control variables. They can be either continuous or binary variables and are used to estimate the propensity score. yxvar(varlist) list of variables that can modify the treatment effect. order(#) takes integer values and represents the order of the polynomial function used to estimate the propensity score. The default is order(1). sle uses a series logit estimator of Hirano et al. (2003) to estimate the propensity score instead of simple polynomial series (the default). csinf(#) drops the observations of which the propensity score is less than the value provided as csinf. The default is csinf(0). csup(#), drops the observations of which the propensity score is greater than the value provided as csup. The default is csup(1). It is mandatory to declare depvar, tvar(varname) and xvar(varlist). 3 Example To illustrate how absdid works, we reproduce the application exercise available on Abadie s website and estimate the effect of participation to a worker union on wages of unionized female workers. The data used is an excerpt of the current population survey (CPS) a US Government monthly survey of unemployment and labor force participation. It consists of female workers observed in 1996 who were resurveyed in The workers were not unionized in 1996 and we can identify the union-wage effect on the workers that joined a worker union between 1996 and The original code is tailored to measure the effect of union membership on wages for workers. It is available at union.m.

4 4 Semiparametric Difference-in-Difference Estimator Table 1: Characteristics of female workers across treatment groups. Variables Entire sample Unionized Non- Unionized Diff. Union coverage in [0.22] Wage variables: Log wage in *** [0.52] [0.49] [0.53] (0.02) Log wage in ** [0.54] [0.52] [0.54] (0.02) Covariates in 1996: Age (years) *** [11.01] [10.55] [11.03] (0.37) High school [0.26] [0.27] [0.26] (0.01) College *** [0.43] [0.48] [0.43] (0.01) African American *** [0.29] [0.39] [0.29] (0.01) Hispanic [0.24] [0.26] [0.24] (0.01) Married [0.48] [0.48] [0.48] (0.02) Number of workers 18, ,512 18,470 Note: Standard deviations are in brackets and standard errors are in parentheses and significance levels are denoted as follows: * p<0.10, ** p<0.05, *** p<0.01. Let s note w 1,97 the wage of a worker in 1997 if she joins a worker union and w 0,97 the wage hasn t she joined the union. As wage variations are traditionally modeled through a log-normal distribution, the parameter of interest is: ( ) (log(w)) E log(w 1,97 ) log(w 0,97 ) union 97 = 1. (7) For simplicity, we report estimates of (log(w)) and interpret the results as the percentage effect of worker union on wage. 2 If female workers were randomly selected to join a union in 1997, one could estimate (log(w)) by comparing the log of wages of unionized and non-unionized workers in To account for the fact that the female workers who joined a union in 1997 differ 2. Actually, a more accurate estimate of the percentage effect of worker union on wage can be obtained using the transformation suggested by Kennedy (1981).

5 K. Houngbedji 5 from those that remained non-unionized with respect to age, education level and race see Table 1 we use a semiparametric difference-in-difference approach. Assume that, in absence of worker unions, wage dynamics of unionized workers would have been similar to that of non-unionized workers with the same age, education level, race, state of residence, and sector of activity. If that assumption holds, we can use the absdid command to compute the semiparametric difference-in-difference estimator of the union-wage effect for female workers. First, we need a variable which, as suggested in (2), measures the change of log wage between baseline and follow-up: dlwage. Second, we need a binary variable which indicates treated and untreated observations: union97. Third, we need a list of control variables along which unionized and non-unionized workers differ from one another. Let s consider the variables age, black, hispanic and grade which report the age, the ethnic background, and the education level of workers in With these inputs we show below the Stata command for estimating the semiparametric difference-indifference estimator of the union-wage effect for female workers.. absdid dlwage, tvar(union97) xvar(age black hispanic married i.grade) Abadie s semi-parametric diff-in-diff Number of obs = _cons Number of obs shows the number of observations used for the estimation which satisfy (4), i.e. those for which the estimated propensity score is bigger than 0 and smaller than 1. Though the sample has 18, 470 observations, only 18, 466 observations are used to estimate the. This suggests that 4 observations have an estimated propensity score which is either smaller or equal than 0 or bigger or equal than 1. This is not surprising since, by default, absdid uses a liner regression to estimate the propensity score. Hence there the predicted values can often be eitheir negative of bigger than 1. To avoid any loss of information we can add the option sle. 3 See the example below.. absdid dlwage, tvar(union97) xvar(age black hispanic married i.grade) sle Abadie s semi-parametric diff-in-diff Number of obs = _cons To discard the observations with very small or high propensity score, one can use the 3. When the option sle is chosen, some observations can still be left out from the estimation of the propensity score when there is perfect prediction. This is for instance the case when all workers of a given industry are either unionized or non-unionized. In those cases the is estimated only for the observations for which the treatment status is not perfectly predicted by observed characteristics.

6 6 Semiparametric Difference-in-Difference Estimator options csinf and csup to indicate the lowest and highest acceptable values of the propensity score. In the example below we restrict the estimation of the to female workers whose propensity score is between 0.01 and absdid dlwage, tvar(union97) xvar(age black hispanic married i.grade) csinf(0. > 01) csup(0.99) Abadie s semi-parametric diff-in-diff Number of obs = _cons Independently of the method used to estimate the propensity score, the outputs of absdid show a point estimate of the when the union-wage premium is constant and does not vary with worker characteristics. Overall, the results suggest that joining a worker union increased wage of female workers by 3.5 percent in The effect is estimated at 3.6 with the option sle. Similarly, we can also consider that the effect of union on wage varies with worker characteristics. For instance, the union-wage premium may vary with the age of the worker. Experience workers proxied by their age are often scarce in the economy. As such they have more bargaining power and may not need to join a worker union to negotiate their wage. Hence, we may expect the union-wage premium to decrease with the age of the worker. Likewise, the union-wage premium may also vary with the education level. Workers who have not completed high school should expect a higher premium compared to similar workers who have completed either high school or college. We show below the Stata command for estimating how of the union premium for female workers varies with age and education level.. absdid dlwage, tvar(union97) xvar(age black hispanic married i.grade) yxvar(ag > e hschool college) sle Abadie s semi-parametric diff-in-diff Number of obs = age hschool college _cons As expected, the results indicate that union premium decreases with age and education. Considering that the average female worker of the sample was 39 years old in 1996, joining a worker union should increase wage of the average female worker by 20.2 percent = In contrast, the premium is estimated at 16.1 percent for a worker who was 50 years old in Likewise, compared to workers with no diploma in 1996, the union premium decreases by 18.9 percentage points for workers whose highest diploma is high school. Surprisingly, there is no statistically significant

7 K. Houngbedji 7 difference between the union premium of workers with college diploma and those with no diploma. This is likely due to the sample size as very few 7.3 percent female workers with college diploma joined a union between 1996 and To reproduce the same results as those from Table II in the empirical illustration available from Abadie s website we need to consider other control variables which may affect the propensity score. We also need to increase the order of the polynomial function used to estimate the propensity score. First, Abadie considers a larger list of control variables which includes age, education level, ethnic group, state of residence, sector of activity and date of the interview. Let s call this list cvars and save it in a macro as below.. local cvars age black hispanic married i.grade i.state i.dind i.month Second, Abadie uses a polynomial function of order 4 to estimate the propensity score. Using the control variables listed above and 4 as the order of the polynomial function, we reproduce below the same results as those of Abadie s website for female workers.. absdid dlwage, tvar(union97) xvar(`cvars ) order(4) Abadie s semi-parametric diff-in-diff Number of obs = _cons absdid dlwage, tvar(union97) xvar(`cvars ) yxvar(age hschool college) order(4) Abadie s semi-parametric diff-in-diff Number of obs = age hschool college _cons Those results are presented below in columns (1) and (2) of Table 2. They are similar to the union-wage premium for female workers found by Abadie in his empirical exercise. 4 Discussion For a given set of control variables and predictors, the semiparametric difference-indifference estimates vary with the type of approximation used sle or simple linear probability model (the default) and the order of the polynomial approximation used order(#). To reduce the margin for arbitrage, one could use a cross validation technique to decide the combination of method which suits best the semiparametric approximation of the propensity score. It can also help to consider that the LPM is likely

8 8 Semiparametric Difference-in-Difference Estimator to produce estimates of the propensity score which are either negative or greater than 1. This is not the case of the SLE approximation. Conversely, when the SLE approximation is used, the observation for which the treatment status is perfectly predicted by a control variable are discarded from the estimation. In most cases, however, the sample size used to estimate the is larger when the propensity score is approximated with the sle option. Table 2: Effects of worker union on log of wage of female workers. Union premium () LPM SLE (1) (2) (3) (4) Constant ** *** ** *** (0.0160) (0.0956) (0.0168) (0.1082) Age (years) ** ** (0.0016) (0.0017) High school ** *** (0.0648) (0.0724) College (0.0349) (0.0361) Number of workers 16,374 16,374 18,273 18,273 Note: The table reports estimates of the effects of worker union on log of wage of female workers using absdid. Models (1) and (3) report estimates of the average union premium for unionized workers. Models (2) and (4) show how the union premium varies with worker age and eductaion level. The average union premium reported in (1) and (2) are estimated using a linear polynomial function of degree 4 to approximate the propensity score. The premiums reported in (3) and (4) are estimated using a logit specification of degree 4 to estimate the propensity score. Standard errors are in parentheses and significance levels are denoted as follows: * p<0.10, ** p<0.05, *** p<0.01. Using our latest example as benchmark, Table 2 shows how our estimates of the union premium for unionized workers vary the type of approximation used. To conclude, the semiparametric difference-in-difference approach is mostly suited for longitudinal surveys with a baseline and follow-up rounds. To use absdid, the user needs to have a measure of the change of the main outcome variable over time for each observation along with their treatment status and their baseline characteristics.

9 K. Houngbedji 9 5 References Abadie, A Semiparametric Difference-in-Differences Estimators. Review of Economic Studies 72(1): Hirano, K., G. W. Imbens, and G. Ridder Efficient Estimation of Average Treatment Effects Using the Estimated Propensity Score. Econometrica 71(4): Kennedy, P. E Estimation with Correctly Interpreted Dummy Variables in Semilogarithmic Equations. American Economic Review 71(4): 801. Rosenbaum, P. R., and D. B. Rubin The Central Role of the Propensity Score in Observational Studies for Causal Effects. Biometrika 70(1): About the authors Kenneth Houngbedji is a researcher at the Paris School of Economics. His main research interests are in studies of economic behavior and decision-making processes of households in developing countries to help design better public policies.

Web Appendix Figure 1. Operational Steps of Experiment

Web Appendix Figure 1. Operational Steps of Experiment 57,533 direct mail solicitations with randomly different offer interest rates sent out to former clients. 5,028 clients go to branch and apply for