Principled estimation of regression discontinuity designs with covariates: a machine learning approach

10/14/2019
by   Jason Anastasopoulos, et al.
0

The regression discontinuity design (RDD) has become the "gold standard" for causal inference with observational data. Local average treatment effects (LATE) for RDDs are often estimated using local linear regressions with pre-treatment covariates typically added to increase the efficiency of treatment effect estimates, but their inclusion can have large impacts on LATE point estimates and standard errors, particularly in small samples. In this paper, I propose a principled, efficiency-maximizing approach for covariate adjustment of LATE in RDDs. This approach allows researchers to combine context-specific, substantive insights with automated model selection via a novel adaptive lasso algorithm. When combined with currently existing robust estimation methods, this approach improves the efficiency of LATE RDD with pre-treatment covariates. The approach will be implemented in a forthcoming R package, AdaptiveRDD which can be used to estimate and compare treatment effects generated by this approach with extant approaches.

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1 Covariate adjusted LATE in regression discontinuity designs

Regression discontinuity designs provide a framework for the causal estimation of treatment effects with observational data. This is accomplished using a running variable which assigns treatment on the basis of some threshold value such that if , a unit (individual, geographic unit etc) is assigned to treatment and is not assigned to treatment otherwise . Assuming continuity of the forcing variable, the sharp RDD leverages this mechanism by allowing for the causal estimation of LATE around a narrow window of the threshold by making the assumption that, in the limit of this window, units are as “as if” randomly assigned to a treatment (Hahn, Todd, and Van der Klaauw 2001).

Under the potential outcomes framework (Rubin 2005), define as the outcome, as the outcome, had unit received treatment, and as the outcome had unit not received treatment, RDDs allow us to estimate the local average treatment effects (LATE) at the threshold . For purposes of illustration, we assume that :

(1)

Estimation of is typically accomplished through a local linear regression (LLR) in a neighborhood of the cutpoint which is determined through optimal bandwidth selection procedures designed to minimize cross-validated MSE (Imbens and Kalyanaraman 2012).

(2)

In Equation 2, is the estimated local average treatment effect, is a binary treatment indicator function which equals 1 when and is a function of the forcing variable which often takes the form of a non-parametric kernel or order polynomial. A common LLR model estimated in the literature is the model shown in Equation 3:

(3)

In Equation 3, a set of covariates X are added to increase the precision of LATE.  Calonico et al. (2016) derive the covariate adjusted estimator of and demonstrate that covariate adjustment typically leads to more efficient estimates of but there is little guidance regarding which covariates one should include to maximize the efficiency of LATE. Table 1 which lists the types of covariates chosen for similar close-election RDD designs highlights this problem. This is particularly problematic in small N estimation contexts and when covariates are correlated with the running variable, cases in which covariate selection can have a much greater impact on LATE efficiency and point estimates.

INSERT TABLE 1 HERE
Table 1: Table 1: Covariate types chosen for RDD estimation in top 3 political science journals. 2000-2019

As a solution to a similar problem in the context of randomized experiments Bloniarz et al. (2016) propose selecting covariates using a shrinkage and variable selection method known as the LASSO, a practice which I modify and extend to LATE estimation in the regression discontinuity design here by employing the adaptive lasso, the only version of the lasso which has oracle (correct model selection) properties (Zou 2006).

Covariate selection using the adaptive lasso has a number of benefits. First, given any initial set of covariates chosen by the researcher, subsequent covariate selection using this method can improve optimal bandwidth choice via model MSE minimization independent of the bandwidth estimation algorithm; second, this method can maximize LATE efficiency and; third, the method constrains the extent which a treatment effect estimate can be “p-hacked” through the practice of adding covariates. Each of these properties are demonstrated below.

2 Regularization, machine learning and variable selection

Regularization methods are tools used primarily for prediction problems and machine learning applications as a means of reducing the dimensionality of a feature space to avoid over fitting of a prediction model. In the context of linear models, ridge regression and lasso regression are the primary regularization methods used for linear prediction problems 

(Tibshirani 1996). Each method applies a term which penalizes each additional variable added to an OLS model in a different way. For instance, in all OLS problems our goal is to find coefficient estimates which minimize the squared error loss:

OLS under mild assumptions is guaranteed by Gauss-Markov to be the best linear unbiased estimator (BLUE) of the coefficient values. However, if our ultimate goal is

prediction

using a linear model, as is typically the case in the machine learning context, the bias–variance trade-off allows us to exchange unbiasedness of coefficient estimates for a model that makes better out-of-sample predictions (lower MSE) 

(Tibshirani, Wainwright, and Hastie 2015). This was first demonstrated by statistician and mathematician Charles Stein in 1956 and improved upon by statistician Williard James and Stein in 1961 and came to be known as James-Stein shrinkage estimation of linear models (Stein 1956; James and Stein 1961).

2.1 Shrinkage and ridge regularization

As its name suggests, shrinkage estimation is a means of optimizing the predictive abilities of linear models through shrinking coefficient estimates toward zero. One of the first shrinkage methods developed for linear models was ridge regression which added a penalty to the OLS minimization problem (Tihonov 1963):

In ridge regression equation above, the original OLS loss function is estimated with a penalty which penalizes the addition of more variables and is determined by the tuning parameter

which is typically estimated using cross-validation (Tibshirani 1996).

2.2 Shrinkage and selection with lasso regularization

This ridge regression estimator ends up introducing biased (shrunken) coefficient estimates, but through the introduction of this bias, minimizes MSE and improves ability of the model to make better predictions in out of sample data. Unfortunately, ridge regression cannot be used as a variable selection tool because it will never shrink coefficients to zero (Tibshirani, Wainwright, and Hastie 2015), however, the LASSO, an acronym for “least absolute shrinkage and selection operator,” which slightly modifies the penalty term above to an norm allows the model to serve as both a shrinkage and selection method:

Due to the nature of the constrained optimization problem presented by the objective function above, some coefficients will be shrunk toward zero, thus allowing for the lasso to be model selection and shrinkage tool. Additional versions of the lasso which involved tweaks of the penalty for specific high dimensional problems include the elastic net, which can be thought of as a middle ground between ridge regression and the lasso, and the “group lasso”, which is used to select out large groups of covariates.

2.3 Variable selection and oracle properties of the adaptive lasso

Most variations of the LASSO applicable to high dimensional () data often do a good job of minimizing MSE, but fare poorly in simulations in which the ultimate goal is to retrieve the correct subset of covariates from a relatively large pool (Zou 2006). As such, the usefulness of the ordinary lasso for LATE adjustment in RDDs, which do not typically involve high dimensional problems with covariates, is somewhat questionable. Fortunately, the adaptive lasso first introduced by Zou (2006) was developed with the goal of maximizing “correct” variable selection for both low and high-dimensional estimation problems, making it an ideal candidate for covariate adjustment of LATE in RDDs and other causal inference contexts which call for covariate adjustments. As with other flavors of the LASSO the adaptive LASSO involves some alterations of the regularization term:

The inclusion of a set of weights , differentiates the adaptive LASSO from other LASSO varieties. In the adaptive LASSO as developed by Zou (2006), weights are chosen from the OLS estimates of the coefficients such that:

where the are the coefficients estimated from an OLS model:

and is a tuning parameter that can take on positive values. In his original adaptive lasso study, Zou (2006) tuned the adaptive lasso with values of 0.5,1 and 2 or cross-validation. In his simulations, the best results were achieved with followed by selected by cross validation. The tuning parameter is estimated in the ordinary way via k-fold cross-validation222In most software packages k is set to 10 but this should be adjusted depending upon sample size..

What makes the adaptive lasso appealing for causal inference, in general, is that with the appropriate value of estimated from the data, the adaptive lasso exhibits oracle properties: it tends to consistently select a correct subset of variables out of a larger set and has asymptotic guarantees of unbiasedness and normality (Zou 2006). This is especially useful when the lasso is used as a variable selection, rather than shrinkage tool, which will be true more often in the context of covariate adjustments of LATE in RDDs and other causal inference contexts more generally.

Indeed, as with ridge regression and other varieties of lasso, however, raw parameter estimates ( can be biased in finite samples, which seemingly limits the utility of this method for causal inference more generally. Fortunately, however, as  Bloniarz et al. (2016) and others point out, estimation through a two-step procedure in which the lasso is used as a model selection tool and final parameter values are estimated using OLS allows us to obtain BLUE coefficient estimates with appropriate standard errors in an easily interpretable model.

Accordingly, this is the approach that we employ here discussed in more detail below. Furthermore, here, as in Bloniarz et al. (2016), we argue that adaptive lasso covariate adjustment of LATE can improve the precision of estimates and also function as a means of “principled” model selection that can avoid some of the pitfalls of model manipulation to recover statistically significant treatment effects (ie “p-hacking”) for RDDs. Based on a series of simulations and on the basis of the theoretical results discussed here and previously in Bloniarz et al. (2016), we recommend a four–step process for RDD treatment effect estimation when covariates are included. This process is outlined in Table 2 and each step is described in more detail below.

Step 1 Researcher pre-treatment Covariates are selected by the researcher
covariate selection on the basis of substantive concerns.
and data issues.
Step 2 Adaptive lasso regularization The model from Step 1 is estimated using an adaptive lasso
as described below.
Step 3 Covariate adjustment Covariates and higher-order terms whose coefficients are
shrunk to 0 are excluded from the final model.
The adaptive lasso is tailored in this case
such that the treatment effect, forcing variable
and variables included in the kernel chosen are
NOT penalized.
Step 4 CCT robust estimation The modified model from Step 3 is estimated
of final model via the CCT robust procedure
(Calonico, Cattaneo, and Titiunik 2014).
Table 2: Overview of an RDD estimation algorithm with the adaptive lasso.

Briefly, the four steps involve researcher model selection based on substantive or theoretically motivated concerns, the application of a adaptive lasso regularization with tuning parameter cross validation; variable selection based on the results of adaptive lasso estimation in the previous step and finally CCT robust estimation of the model selected from Step 3. Each of these steps along with treatment effect estimates produced by this method in the context of RDDs with local linear regression and covariates are derived below.

3 Adaptive lasso estimation of LATE for RDDs

3.1 Step 1: Researcher Pre-Treatment Covariate Selection

The purpose of including pre-treatment covariates in RDD estimation, as in randomized experiments, is to increase the precision of treatment effect estimates (Bloniarz et al. 2016; Calonico et al. 2018). This increase in precision can be the result of improved bandwidth selection, reduced model variance or a combination of the two. Some questions that researchers may struggle with, however, are: (1) which pre-treatment covariates to include in the model and; (2) whether pre-treatment covariates should be included before or after optimal bandwidth selection.

This is a thorny issue because all of these decisions can have significant downstream consequences for LATE estimation and efficiency, particularly when covariates included are highly correlated with the forcing variable and in small N local linear regression contexts which tend to be common in RDD estimation within the political science literature. As a result, temptations to manipulate covariate selection to maximize the statistical significance of LATE estimates is high, particularly in cases where LATE estimated without covariates are marginally significant (i.e. ).

While the automated model selection algorithm proposed here in Table 2 cannot eliminate “p-hacking”, it is a procedure that can at the very least attenuate the ability of researchers to engage in this practice while simultaneously providing LATE estimates when covariates are introduced than researcher model selection alone. That being said, initial decisions regarding which pre-treatment covariates to include should always be made on the basis of expert judgment/the researcher’s expectation of which are the most relevant to the problem at hand. Since RDDs in the political science literature are typically conducted with close election vote share as the forcing variable and the treatment of interest is an election win where if and otherwise we focus on this type of RDD to illustrate the method.

(4)

Equation 4 is a typical local linear regression model estimated to obtain the treatment effect estimate where the observations are in some neighborhood, of the forcing variable around the cutpoint , i.e. and is a matrix of covariates. In these circumstances, the covariates included are often characteristics of the candidate (age, sex, etc) and characteristics of an electoral unit that they represent (pre-treatment demographics etc).  Szakonyi (2018), for instance includes candidate controls such as age, gender, incumbency, ruling party membership, state ownership, foreign ownership, and logged total assets in the pre-election year in his estimates. As I mentioned above, selection of this initial set of covariates should always be dictated by a substantive understanding of the problem at hand.

3.2 Step 2: Adaptive lasso regularization

Once the model in Equation 4 has been selected, a question that remains is whether this is the best possible model that can be fit which invariably raises the question of what “best” means in this context. Here we define “best” as a model in which a set of covariates are chosen out of the original set of covariates which minimizes the variance of LATE, . All things equal, it can be shown that minimizing can be accomplished by minimizing the mean squared error (MSE) of the local linear regression.

Formally, if is a subset of covariates from , , we seek to choose an that minimizes the mean squared error (MSE) of Equation 4

w.r.t to the LLR parameters which we describe as the vector

for convenience. Thus:

(5)

While many methods exist for choosing , LASSO regularization is suited directly to the estimation of linear models and has been found to outperform other automated variable selection methods (Tibshirani, Wainwright, and Hastie 2015). Also, since we are primarily concerned with an optimal variable selection and minimizing the bias of the LATE in a low-dimensional context, the adaptive lasso discovered by Zou (2006) is a natural choice since it is the only lasso variety which possesses the oracle property, as mentioned above. This is important because it guarantees that it will be consistent in both estimation of and in variable selection. Formally this implies asymptotic unbiasedness of in the ordinary sense:

while simultaneously identifying the correct set of non-zero coefficients. These properties ensure that adaptive lasso estimates of are asymptotically at least as good, in terms of efficiency and bias, as LLR without adaptive lasso variable selection.

Learning about which covariates to exclude in RDDs, however, requires modifying the adaptive lasso to the RDD context. In particular, we do not want to penalize the treatment effect, forcing variable or kernel, but do want to penalize any additional covariates. This can be accomplished by simply estimating a modified version of the adaptive lasso in which the weights for these coefficients are set to 0 while the weights of the added covariates are identical to those of the adaptive lasso. The full initial model to be estimated is thus:

(6)

Where are obtained through OLS estimation of and is determined through cross-validation as described above. Again, the tuning parameter is estimated with k-fold cross validation.

3.3 Step 3: Automated Model Selection

Once parameters from the adaptive lasso model in Equation 7 are estimated using the optimal penalty value and optimal weights, those covariates which are shrunk to zero are excluded from the model prior to calculating the optimal bandwidth. The resulting model used to estimate optimal bandwidth and subsequently, robust treatment effects, is will thus be:

(7)

where is the truncated set of covariates selected out by the adaptive lasso described above. Since optimal bandwidth selection algorithms such as Imbens–Kalyanaraman use cross-validated MSE as criteria for selecting the “best” possible bandwidth, MSE for bandwidth values estimated using covariates pre-processed by the adaptive lasso method described should be equal to or less than model MSE for bandwidth values estimated using the full model from Step 1.

As I demonstrate below, this method can be incorporated into RDD estimation with covariates before bandwidth selection, which will alter the optimal bandwidth chosen, or after bandwidth selection if the bandwidth is set to a predetermined value (eg. 1%, 5% etc for close election RDDs).

3.4 Step 4: Regularized CCT Robust Estimation

Steps 1-3 involve selecting an optimal LLR conditional expectation function (CEF), , and estimating an optimal bandwidth based on the CEF. Once this has been accomplished, final treatment effect estimates are produced using CCT robust estimation (Calonico et al. 2018). The final LATE estimated using this procedure, will be at least as efficient as the un-adjusted treatment effect estimated from the original model, :

A proof of this is provided in the Appendix.

4 Empirical Illustration: Do Firms Profit from Having Elected Board Members?

Knowledge of whether politicians benefit financially from holding office-holding is essential is essential for ensuring the legitimacy of democratic institutions. Earlier work using RDDs to estimate the returns to office found large lifetime earnings effects by barely (initially) elected members of the British Parilament (Eggers and Hainmueller 2009). Subsequent work in different national contexts has found similar results as well (see eg Fisman, Schulz, and Vig (2014) (India), Truex (2014) (China), etc). Recent innovative work published in the American Political Science Review by Szakonyi (2018) adds an interesting dimension to this literature by using a close election RDD to explore whether office-holding affects the profits of firms whose board members held political office in Russia. Using a close election RDD,  Szakonyi (2018) finds that office holding positively affects both firm profitability and firm revenue. In the empirical illustration below, I apply both the adaptive lasso algorithim described above to optimize covariate adjusted treatment effects.

In the following analysis, I replicate the local linear regression in Table 2 of Szakonyi (2018). In this table, the author uses a close election RDD to estimate the causal effect of holding political office on firm profitability with and without covariates using a recommended 5% bandwidth and the I-K optimal bandwidth estimated without covariates. The general form of the local linear regression estimated is:

(8)

In Equation 8, the outcome variable is firm profit margins and the treatment indicator is whether the businessperson won election in their district and the running variable is the vote margin. In these analyses are also included a set of covariates X and year, sector and region fixed effects (Y, S, R). This regression is estimated around a threshold of the cutpoint where is determined through cross-validation. Define the original treatment effect of the full model (i.e. the model entered in Step 1 above), .

After selecting covariates via either Bayesian SS or LASSO through Steps 2-4, we are left with the model:

(9)

Note that the primary difference between the two equations above is the new set of covariates which satisfies the condition through the removal of covariates and a new optimal bandwidth as a result of the addition of new covariates. We thus define the new adjusted treatment effect as . It can be shown that , a result that is expanded upon in the Appendix. While coverage properties of this new estimator is less clear theoretically, results from CCT and others suggest that the regularization adjusted estimator will have superior coverage properties as well under a variety of circumstances. This is confirmed in a series of simulations below.

One important thing to note is that all fixed effects in the model were not regularized. This was done deliberately through setting the penalty parameters to 0 when estimating the LASSO model. While efficiency gains can theoretically be made through the exclusion of some fixed effects, this estimation strategy does not make substantive sense in any context.

Original Adaptive Original Adaptive Adaptive
(APSR) 5% (APSR) 5% CCT Robust
District Win 0.146 0.102 0.198 0.097 0.140
(0.065) (0.060) (0.090) (0.038) (0.052)
Bandwidth 0.113 0.120 0.050 0.050 0.120
Covariates Dropped * 4 * 2 4
Firm and Cand Full Select Full Select Select
Covariates
Region,Sector Full Full No No No
Year FE
Observations 481 520 201 201 520
Table 3: Replication of Political Connections and Firm Profitability Analysis in Szakonyi (2018) with Adaptive LASSO Adjusted Treatment Effects.

Table 3 contains original and regularization adjusted treatment effects and standard errors. One thing of note is that the standard errors of all adaptive lasso treatment effects are smaller than those of the original covariate adjusted treatment effects published. As simulations below demonstrate, this is due to the oracle property enjoyed by the adaptive Lasso, which has been demonstrated produce “correct” model specification under a wide variety of conditions.

5 Simulations

In order to explore the bias and coverage properties of the adaptive lasso method in as realistic a scenario as possible, I perform a series of simulations using the same election and profit data from  Szakonyi (2018) described above but with a true simulated treatment effect. To accomplish this, I use the correlation matrix of the covariates and vote margin used by  Szakonyi (2018) to construct 2,000 simulated data sets which have the same covariance structure and mean of the original dataset and set the true treatment effect to 0.30.

Define as a matrix which contains the set of covariates plus the vote margin used in Szakonyi (2018) discussed above. Furthermore, assume that the data generating process of

is that of a multivariate normal distribution defined by some mean parameters

and a covariance matrix . Thus:

Using this data generating process along with empirically defined parameters and covariance structure , I generate simulated data sets such that the d.g.p of each simulated dataset adheres to:

Through generating the data in this manner, we’re insuring that each simulated dataset conforms to a realistic d.g.p in the context of a close-election RDD. For each simulation the outcome , and thus the true model, is thus defined by

where the error term , the simulated vote share, is simulated as one of the variables within and is a simulated forcing variable based on . The true treatment effect that we estimate using the conventional RDD approach and adaptive lasso approach with is . Reported coefficient values, bandwidths and standard errors are CCT robust estimates using the standard and adaptive approaches.

The model estimated for each simulation is the full model including covariates:

(10)

For the simulations, the average bias of ,

and % coverage of the confidence intervals were recorded for models in which the bandwidth was allowed to vary according to the adaptive lasso procedure outlined above or was fixed at a certain value with the adaptive lasso applied afterwards.

Figure 1: Distribution of simulated treatment effects , for adaptive lasso adjusted treatment effects and conventional treatment effects across 2,000 simulated data sets with variable bandwidth select. The true is denoted by the black dotted line.

Figure 1 contains the distribution of simulated treatment effects estimated using conventional and adaptive lasso methods. Here we see that the adaptive lasso restricts the treatment effects estimated to a much narrower band around the true treatment effect.

Variable Bandwidth*
Adaptive Conventional Difference (Adaptive - Conventional)
Bias 0.274 0.397 - 0.123
% Coverage 0.944 0.699 + 0.245
Estimate 0.308 0.308 -
Bandwidth 0.38 0.292 + 0.088
Fixed Bandwidth
Adaptive Conventional Difference (Adaptive - Conventional)
Bias 0.375 0.375 - 0.001
% Coverage 0.931 0.796 + 0.135
Estimate 0.300 0.300 - 0.001
Bandwidth 0.200 0.200 -
Table 4: Performance of Adaptive Lasso v. Conventional Treatment Effect Estimates in Simulations, ,

for t-test of mean difference with

. Average simulation results across 2,000 simulations comparing “Adaptive” vs. “Conventional” treatment effect bias and coverage results. Final bias and coverage results are both estimated using CCT robust point estimates and confidence intervals. *“Variable bandwidth” results are produced through Imbens-Kalyanaraman optimal bandwidth selection based on models selected by the adaptive algorithm described above or the full model mentioned in this section. For fixed bandwidth simulations, bandwidth was set to 0.20.

Table 4 contains estimates of the bias, % coverage and other statistics from the simulation. The adaptive lasso here provides some very striking efficiency improvements which are reflected in the % coverage estimates in both variable and fixed bandwidth selection procedures. In the variable bandwidth scenario, the adaptive lasso combined with CCT robust estimation produces confidence intervals on treatment effects that achieves an average of 94% coverage versus 70% coverage under conventional estimation while under the fixed bandwidth scenario, adaptive lasso estimation achieved 93% coverage compared to about 80% coverage under conventional estimation. Each of these differences was statistically significant at the level.

6 Discussion

In this paper we have demonstrated that our algorithm which employs the adaptive LASSO can improve the efficiency of treatment effects for RDDs estimated with covariates and provide a principled framework of treatment effect adjustment for RDDs. As we emphasize above, however, this does not imply that substantive considerations in the estimation process should be abandoned and replaced by automated machine learning methods. To the contrary, substantive considerations, as reflected in the algorithm that we developed above, are and should always be at the forefront of model estimation whether in the context of RDDs or estimation strategies.

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