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About text below the content, author: Wu Aoba, Australian National University Business School and School of Economics, communication mail: [email protected]
Author's previous article:
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Pedro H.C.Sant, AnnaJunZhao. (2020). Doubly robust difference-in-differences estimators. Journal of Econometrics.
This article proposes doubly robust estimators for the average treatment effect on the treated (ATT) in difference-in-differences (DID) research designs. In contrast to alternative DID estimators, the proposed estimators are consistent if either (but not necessarily both) a propensity score or outcome regression working models are correctly specified. We also derive the semiparametric efficiency bound for the ATT in DID designs when either panel or repeated cross-section data are available, and show that our proposed estimators attain the semiparametric efficiency bound when the working models are correctly specified. Furthermore, we quantify the potential efficiency gains of having access to panel data instead of repeated cross-section data. Finally, by paying particular attention to the estimation method used to estimate the nuisance parameters, we show that one can sometimes construct doubly robust DID estimators for the ATT that are also doubly robust for inference. Simulation studies and an empirical application illustrate the desirable finite-sample performance of the proposed estimators. Open-source software for implementing the proposed policy evaluation tools is available.
Abstract:
This article proposes the average effect (ATT) of the experimental group using the DR estimation method in the DID model. Compared with the DID estimator, if a propensity score or outcome regression model is correctly set, then the estimator is consistent. In addition, the author also derives the semi-parametric effective bounds of ATT in the DID model in the case of panel data or repeated cross-sectional data, and proves that the semi-parametric effective bounds of the estimator proposed by the author can be obtained when the model is correctly set. In addition, the authors quantified the potential efficiency gains of using panel data instead of repeated cross-sectional data. Finally, the author proves that sometimes the double-robust estimator constructed for ATT is also double-robust for statistical inference.
1. Introduction
DID is the most popular method used by researchers to evaluate policies using observational data. In its classic model, DID determines the average effect of the experimental group by comparing the results of the two groups before and after the experiment: one group was tested, and the other group was used as a control. In order to facilitate causal explanations, researchers usually quote the Parallel Trend Hypothesis (PTA): Before the experiment, the average value of the experimental group and the control group are parallel for a period of time. Although PTA is fundamentally untestable, its credibility is often questioned: if the observed characteristics are considered to be related to the evolution of the results, and the evolution of the results is not balanced between the two groups. In this case, the researcher usually deviates from the standard DID model settings and incorporates the pre-experimental covariates into the DID model analysis, and assumes that PTA is only met under these covariate conditions.
In the article, the author studies the robustness and validity of the ATT estimator in the DID model when the PTA assumption is satisfied based on the condition of the covariate. The author considered both the case of panel data and the case of only repeated cross-sectional data. The author has made contributions to the research of DID in different aspects. First, the author introduced the estimation of ATT by the DR method under the DID model setting, and proved that when the propensity score and the result model are correctly set, the estimated results of the DR method are consistent. The author proposes two different estimations of ATT using the DR method. They are in the same experimental group, but one is based on the regression of the results before the treatment, and the other is based on the regression of the results after the treatment. Nevertheless, the authors found that the DR method estimator does not depend on such a choice.
Secondly, the author derives the semi-parametric effective bounds of ATT under the DID design. The semi-parametric effective bounds derived by the author are non-parametric, because the author does not assume that the researcher will have a certain understanding of the propensity score function and the result regression function. Therefore, these boundaries provide a standard based on which researchers can compare any semi-parametric DID estimates of ATT. It is also worth emphasizing that these semi-parametric effective bounds explicitly include all the constraints implied by the recognition hypothesis. It is important that these limits differ depending on whether it is panel data or repeated cross-sectional data. In both cases, they both involve the constraints implied by the conditional PTA. However, when there are repeated cross-sectional data, they also include the constraints implied by the identification hypothesis, that is, the joint distribution of the covariate and the processing state is not Affected by the sampling period. It should be emphasized that if these implicit constraints are not considered, it will lead to differences in the efficiency boundaries derived, which in turn may indicate that some estimators are semi-parametrically valid, when in fact they are not.
With the semi-parametric effective bounds, some questions can be answered. For example, one may wonder whether using panel data instead of repeated cross-sectional data will improve efficiency. By directly comparing the effective bounds under these two settings, it can be seen that the answer to the above question is not only affirmative, but also that when the sample size of the repeated cross-sectional data before and after the experiment is more unbalanced, the gain tends to be larger. .
Another issue that the author discusses is whether the estimator proposed in the article using the DR method to estimate the DID model can reach the semi-parametric effective bound. The author shows that when the propensity scoring work model and the control group result regression model are correctly set, the DR DID estimator is partially valid under the panel data setting, but not under the cross-sectional data setting. In fact, when there are only repeated cross-sectional data, the author found that as long as the regression model is set correctly depending on the propensity scoring model and the results of the experimental group and the control group, the DR DID estimator can reach the semi-parametric effective limit. The author quantified the efficiency loss of using the inefficient DR DID estimator instead of the local effective estimator, and showed that the loss is really large through Monte Carlo simulation.
The method proposed in the article is suitable for the linear and nonlinear working models of the interference function. When the general parameter working model is used for the interference function, the author establishes the picture consistency and asymptotic normality of the DR DID estimator. The correct form of the asymptotic variance of the estimator proposed by the author depends on whether the propensity score and/or outcome regression model is correctly set. In view of the fact that in the actual application process, it is not known which models are correctly set, so when estimating the asymptotic variance, the researcher should consider the estimation effect of all the first estimators. Failure to do so may lead to invalid statistical inferences. .
The third contribution of the article is that by observing the estimation methods used to estimate redundant parameters, sometimes it may be possible to establish a simple computable ATT estimator in the DID model. Not only is DR consistent and local semi-parametric effective, but also double Robust.
2.
The theoretical framework of related literature articles is mainly based on the two branches of causal inference research. First, the methodology of the article is inherently related to other DID literature, such as Section 6.5 of Imbens and Wooldridge (2009) and its references. The two main contributions of this branch related to the article are the regression estimator of the DID model based on the kernel function proposed by Heckman et al. in 1997, and the (parametric and non-parametric) DID inverse probability weighting (IPW) estimator proposed by Abadie (2005). . The authors note that when the dimensionality of the covariate is very high or even just moderate, completely non-parametric methods usually do not give useful inferences. In this case, researchers usually use parametric methods. The DR DID proposed in the article belongs to the latter kind.
Second, the content of the article is also closely related to the classic literature on bistable estimation, for example, Robins et al. (1994), Scharfstein et al. (1999), Bang and Robins (2005), Wooldridge (2007), Chen et al .(2008), Cattaneo(2010), Graham et al.(2012,2016), Vermeulen&Vansteelandt(2015), Lee et al.(2017), Sloczynski & Wooldridge(2018) and Seaman&Vansteelandt(2018). Recently, DR estimation methods have also played an important role in using adaptive data and machine learning to estimate interference functions. For example, Belloni et al. (2014), Farrell (2015), Chernozhukov et al. (2017), Belloni et al. (2017) ) And Tan (2019). On the other hand, the author noticed that the above-mentioned papers focused on the assumptions of the “selection model of observations” or “IV/LATE” type, and the author specially looked at the conditional DID model design, so the results of the article were inconsistent with the existing An addition to research.
In order to derive the semi-parametric effective bounds of the ATT estimator under the DID framework, the article is based on the research of Hahn (1998) and Chen et al. (2008). Although the authors follow the structure of the semi-parametric effective bounds derived from the above paper, the semi-parametric effective bounds derived by the authors supplement their research because their results depend on the assumption of the “selection model” type under the cross-sectional data setting. Focus on the model design of DID.
The research on further improvement of DR DID estimator is based on the research of Vermeulen & Vansteelandt (2015). They proposed the DR estimator for statistical inference in the cross-sectional data setting under the assumption of observable type selection. The content of the article is also based on Graham et al. (2012), because their propensity score estimator is an important part of the article.
Finally, related to the article but independent of the content of the article is the research work of Zimmert (2019), who proposed that under high-level conditions, researchers can use the first estimator of machine learning to estimate the ATT statistics in the DID model. His research results supplement the author's research results, but the author noticed that his estimation of repeated cross-sections did not reach the semi-parametric effective bounds derived by the author, and the efficiency loss is also first-order important. At the same time, the author also noticed that Zimmert (2019) did not provide a detailed comparison between panel data and repeated cross-sectional data applications, nor did it discuss the inference process of DR, which is very relevant when the model is incorrectly set.
3. DID model
1.
The model symbols that will be used in the background article are as follows:
In order to assess the sensitivity of the conclusion and the model representation, the author considered three different model representations: (i) linear model, in which all covariates are linear; (ii) based on DW, adding the square of age on the basis of the linear model, The cube of age divided by 1000, the square of years of education, the dummy variable of zero income in 1974, the product variable of years of education and actual income in 1974; (iii) In the expression of the DW model, added married and 1974 The product variable of real income, the product variable of married and zero income in 1974.
Table 3 summarizes the results. As ST pointed out, these relative deviations are useful for comparing the DID estimates within each sample, but since the experimental benchmark of ATT is very different among the three experimental samples, it should not be used for comparison between samples. Table 3 also reflects some other phenomena. First, the estimator based on the two-way fixed effects regression model is relatively stable in different model representations, but they all show a significant positive evaluation bias; second, the DID estimator based on the regression method will lead to the most accurate estimate. However, in the Lalonde sample, the point estimate is seriously underestimated, leading to significant evaluation bias. Abadie’s IPW estimator has the largest standard error among all estimators, but its evaluation bias is relatively small. Like the Monte Carlo simulation results, the standardized IPW estimator can improve the stability of the estimation. Finally, the author found that the DR DID estimator not only has the advantage of the small evaluation error of Adadie's IPW estimator, but also has a smaller standard deviation. At the same time, through comparison, the authors found that the DR DID estimate they proposed is a valuable alternative to the existing DID framework.
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7. Summary
Under the DID model, when the assumption of parallel trends based on the pre-experimental covariates is satisfied, the author proposes a dual robust estimation of ATT. When either the propensity scoring model or the outcome regression model is correctly specified, the estimator proposed by the author is consistent. When the interference model of the working model is also correctly specified, the semi-parametric effective bound can be obtained. At the same time, the author also confirms that this estimator can be used as a causal inference tool through Monte Carlo simulation and empirical application.
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