This program offers a suite of tools in STATA for implementing difference-in-differences style analyses with staggered treatment onset using the two-way fixed effects approach proposed in Wooldridge (2021) and the high dimensional fixed effects estimators developed by Correia (2017). Features include:
Author: Thomas A. Hegland, Agency for Healthcare Research and Quality
Contact: thomas.hegland@ahrq.hhs.gov, thomashegland.com, @thomas_hegland
Citation: Hegland, Thomas A. wooldid: Estimation of Difference-in-Differences Treatment Effects with Staggered Treatment Onset Using Heterogeneity-Robust Two-Way Fixed Effects Regressions. Statistical Software Components s459238, 2023.
Disclaimer: This document reflects only the views of the author and does not necessarily represent the views of the Agency for Healthcare Research and Quality, the Department of Health and Human Services, or the United States government. The program with which this file is associated is the work of the author and does not come with any endorsement by or warranty from the Agency for Healthcare Research and Quality, the Department of Health and Human Services, or the United States government.
ssc install wooldidor
net install wooldid, from(https://raw.githubusercontent.com/thegland/wooldid/master/wooldidinstall/) replace
For more detail on syntax, please refer to wooldid's included stata help file. This is only a brief overview and does not contain as much detail as the help file. _Note: the file wooldid_simpleexample.do offers a basic example of how to run the program and set up the treatment variable.
wooldid y i t ttre [if] [aw/pw=weights], options
wooldid estimates various types of difference-in-differences treatment effects in the vein of Wooldridge (2021), with extensions to handle continuous treatments and estimation of effects by subgroup or treatment arm. The program does not currently accommodate cases where treatment is reversible -- once treated, cohorts must remain treated.
Broadly speaking, wooldid estimates difference-in-differences treatment effects using a two-step procedure. In step 1, wooldid estimates the underlying two-way fixed effects regression, which consists of (at baseline) a regression of the outcome variable on cohort fixed effects, time fixed effects, and a set of indicator variables used to flexibly capture the effect of treatment. Each indicator variable is 1 in a specific cohort-period (i-t) cell, and 0 otherwise. The slate of indicator variables spans all cohort-periods for which treatment effects are to be estimated. In step 2, wooldid completes a set of calculations using the {help margins} command that convert the large slate of coefficients on the treatment indicator variables from the underlying regression in step 1 into treatment effects of interest. These margins calculations proceed by computing, for all treated observations, the difference between the predicted values from the underlying regression and the counterfactual predicted values that would be observed given each observation was not treated. Various averages of these differences are then presented to obtain various types of user-specified average treatment effects. To flesh this out, the underlying regression used is as follows:
Step 1 (Underlying Regression): Y_jt = fe_i + fe_t + sum_over_it_in_Z(beta_it TreatedFlag_it Tz_it)
where i indexes cohorts, t indexes time periods, j indexes individual units (possibly equivalent to cohorts, possibly more numerous than cohorts), and where Z is the set of all (i,t) pairs where i is a treated cohort and t is a treated period for cohort i. In the above regression, Tz_it is a dummy variable that is 1 if, for a given z=(iz,tz) in Z, i==iz and t==tz, TreatedFlag_it is a variable that is 1 if an observation is part of an (i,t) cell in Z and 0 otherwise, fe_i is a cohort fixed effect, fe_t is a time fixed effect, and Y_jt is the outcome variable.
The output of the underlying Step 1 regression is then used for margins calculations in step 2. The exact margins calculation performed depends on the estimands specified by the user. The overall average treatment effect would be obtained by computing fitted values from the underlying regression among all treated observations (those where TreatedFlag_it is equal to 1), subtracting the counterfactual fitted values for these observations that set TreatedFlag_it to 0, and then taking the average of these differences (applying any sample weights). To target estimands other than the overall average treatment effect, this procedure can be modified by either introducing a set of weights (or modifying the pre-existing sample weights) when averaging the computed differences in the margins calculation or by restricting the sample so that the average only contains observations from a subgroup or time period of interest. The margins calculation can also be modified here to compute effects in terms of semi-elasticities.
With respect to inference, there are several approaches to standard error estimation available. The default estimates standard errors in the underlying regression and then obtains standard errors on the treatment effects produced by the margins calculation using the delta method. These errors are conditional standard errors in the sense that they are conditional on the data: the difference in fitted values potentially depends on the value of various covariates, and this approach takes these values as fixed when performing the margins calculation. Given these standard errors, p-values are then obtained using a t-distribution of same degrees of freedom as the residual number of degrees of freedom in the underlying regression model. The alternative is to have margins produce unconditional standard errors, which account for sampling variation in covariates.
This overall approach can be extended to cover many additional use cases with little or no modification. The user can switch from standard regression to poisson regression without requiring any essential modification to the regression equation or margins procedure. Additional controls and fixed effects can be added to the underlying regression without requiring adjustments to the margins calculation. Effects specific to particular subgroups may be estimated simply by performing the margins calculation within a specified subgroup. Where multiple subgroups are present, wooldid can use margins to calculate pairwise comparisons of the subgroup-specific effects and compute joint tests of the hypothesis that all subgroups have the same effect. This can be useful for tests of differences in treatment efficacy across treatment arms and can be used to replicate functionality that interaction terms in a traditional difference-in-differences would have provided.
In a similar vein, event study style estimates that present treatment effects specific to various periods relative to treatment can be obtained by performing the margins calculation restricted to the subsample of observations corresponding with particular periods relative to treatment. If effects in periods prior to treatment are of interest, this requires modifying the underlying regression with additional treatment indicator variables for cohort-periods that correspond with the relative time period of interest in treated cohorts. The regression equation itself remains the same: only the set of (i,t) pairs appearing in Z changes for this approach. The margins calculations involved also remain similar, though may be performed on varying subsets of the data by relative period. When computing event study estimates, wooldid can supplement the standard treatment effect estimates with joint tests of the hypothesis that all pre-treatment relative time period effects are zero, as well as of the hypothesis that all relative time period effects fall on a single line (namely, the line of best fit through the estimated relative time period effects).
In terms of implementing event study-style estimates, wooldid offers two approaches. The default pooled reference period approach is conceptually similar to estimating results like normal, but with the treatment onset date for all treated cohorts pulled earlier in time by just enough periods to estimate all requested relative time period effects. With this approach, we add no more (i,t) pairs to Z than is necessary and we allow the reference period to potentially pool multiple periods together for each cohort. Event study estimates derived from this approach always have a normalized effect of 0 in the earliest period reported (the pooled reference period). The alternative approach is to use a fixed reference period, in which case Z is modified to include all (i,t) pairs for treated cohorts, except the pair corresponding with the period immediately prior to treatment (or corresponding with some other user specified period). In this case, the regression is saturated with i-t indicator variables and all treatment effects are estimated relative to the single specified reference period.
With greater modification, the approach above can be extended to handle continuous treatment variables and treatments that come with an associated measure of treatment intensity or dosage. When cohorts are coarse -- i.e., there are multiple units within each cohort-period cell -- and when a continuous treatment variable varies within i-t cells, estimates of the effect of the continuous treatment variable can be computed within each treated i-t cell. To ensure treatment effect heterogeneity is adequately captured, a flexible function of the continuous treatment variable can be used to measure its effect within each i-t cell if needed. Note that this approach is a bit speculative and is not covered in Wooldridge (2021), though Wooldridge has informally suggested handling continuous treatments using interactions between treated cell indicators and the continuoust reatment variable.
The particular modification to the underlying regression used for estimation in the continuous treatment context is as follows: Step 1 (Continuous): Y_jt = fe_i + fe_t + sum_over_it_in_Z(beta_it TreatedFlag_it Tz_it + TreatedFlag_it Tz_it f(Contreat_jt)) + g(f(Contreat_jt))
where Contreat_jt is a continuous treatment variable that potentially varies arbitrarily across observations, where f() is a flexible function of Contreat_jt, and where g() consists of interactions between various fixed effects-like indicator variables and f(Contreat_jt). Average treatment effects can be computed using the same margins calculation as before. The average marginal effect of the continuous treatment variable can be computed similar to the main margins calculation, but comparing differences in fitted values after modifying the variable Contreat_jt rather than modifying the variable TreatedFlag_it. In addition to overall and event study style estimates, wooldid can also compute average treatment effect estimates at varying points along the distribution of the continuous treatment variable. It can also compute various types of elasticity estimates in lieu of standard estimates of the effect of the continuous treatment variable. For more detail on how f(Contreat_jt) and g(Contreat_jt) are constructed, please refer to the help file's discussion of this in its section continuous treatments.
A final feature available in wooldid is its implementation of Ibragimov and Muller (2010) style cluster robust inference. This approach estimates the same underlying regression as normal, but then modifies the margins calculation significantly. In particular, for any treatment effect requested (be it an average treatment effect or an effect of a continuous treatment variable), this approach has the margins calculation compute that effect separately within each cluster. The final estimate presented is then the simple average of these cluster-specific estimates, with inference being conducted by t-testing the cluster-level estimates.
Note: greater detail on the program's implementation can be found in wooldid's help file, including comparisons with other estimators, greater detail on how constinuous treatments are handled, tips for working with wooldid output, etc.
Borusyak, Kirill, Xavier Jaravel, and Jann Spiess. Revisiting Event Study Designs: Robust and Efficient Estimation. Working Paper, 2023.
Correia, Sergio. reghdfe: Stata module for linear and instrumental-variable/gmm regression absorbing multiple levels of fixed effects. Statistical Software Components s457874, 2017. Boston College Department of Economics. https://ideas.repec.org/c/boc/bocode/s457874.html
Ibragimov, Rustam and Muller, Ulrich K. t-Statistic Based Correlation and Heterogeneity Robust Inference. Journal of Business & Economic Statistics, 2010.
Wooldridge, Jeffrey M. Two-Way Fixed Effects, the Two-Way Mundlak Regression, and Difference-in-differences Estimators. Working Paper, 2021. https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3906345