The PCDID Approach: Difference-in-Differences When Trends Are Potentially Unparallel and Stochastic

Abstract

We develop a class of regression-based estimators, called Principal Components Difference-in-Differences (PCDID) estimators, for treatment effect estimation. Analogous to a control function approach, PCDID uses factor proxies constructed from control units to control for unobserved trends, assuming that the unobservables follow an interactive effects structure. We clarify the conditions under which the estimands in this regression-based approach represent useful causal parameters of interest. We establish consistency and asymptotic normality results of PCDID estimators under minimal assumptions on the specification of time trends. The PCDID approach is illustrated in an empirical exercise that examines the effects of welfare waiver programs on welfare caseloads in the United States.

Keywords:

• Difference-in-differences
• Factor-augmented regressions
• Interactive fixed effects
• Parallel trends
• Principal components
• Treatment effects

Acknowledgments

We would like to thank Manuel Arellano, Mario Fiorini, John Ham, Rosa Matzkin, Tim Moore, Morten Nielsen, Tim Robinson, Chris Skeels, Yongcheol Shin, Katrien Stevens, Chris Taber, seminar participants at Adelaide, ANU, Deakin, HKU, Melbourne, Monash, NYUAD, Oxford, Sydney, UTS, and participants of the Econometrics Colloquium (UQ), IAAE, AASLE and IWEEE conferences for their valuable comments. The article has greatly benefited from suggestions by the editor, associate editor and the five referees. Special thanks to James Ziliak who provided the key years of welfare caseload data to us.

Notes

Notes

1 Similar to PCDID estimators, the CCE estimators are robust to different factor dynamics including stationary (Pesaran 2006) and unit-root processes (Kapetanios, Pesaran, and Yamagata 2011). Weighted cross-sectional averages are allowed for CCE estimators; see Westerlund and Urbain (2015) and Greenaway-McGrevy, Han, and Sul (2012).

2 Quasi-differencing methods have been developed for fixed-T models assuming that the factors are unknown (e.g., Holtz-Eakin, Newey, and Rosen 1988; Ahn, Lee, and Schmidt 2001, 2013). These methods require stronger assumptions on idiosyncratic errors (e.g., iid) or require exclusion restrictions.

3 In a nonparametric framework, parallel trend holds when, given each $t>T_0$ , the potential outcome $y_{it}(0)$ satisfies conditions akin to $E(y_{it}(0)-y_{i,t-1}(0)|i\in E)=E(y_{it}(0)-y_{i,t-1}(0)|i\in C)$ ; see Callaway and Sant’Anna (2018) for details. Given t and t – 1, the factor structure implies $E(y_{it}(0)-y_{i,t-1}(0)|i\in E)=E({\mu }_i^{\prime }|i\in E)(f_t-f_{t-1})$ and $E(y_{it}(0)-y_{i,t-1}(0)|i\in C)=E({\mu }_i^{\prime }|i\in C)(f_t-f_{t-1})$ , hence the interest in PTW.

4 We do not impose the time-homogeneity condition on period-specific disturbances ( ${\mu }_i^{\prime }{f}_t+{ϵ}_{it}$ in our context) as in the semi-/non-parametric panel data models of Chernozhukov et al. (2013). This also highlights the importance of imposing a (factor) structure on the disturbances. Nonetheless, the unobserved μ_i and f_t involve minimal assumptions; see Assumptions F and FL_C .

5 See also Wooldridge (2005) and Pesaran (2006), where x_it is correlated with other regressors as well as unit-specific β_i . While Wooldridge shows that a fixed-effects estimator may consistently estimate the population average of β_i , he does not consider a factor structure, a key feature in Pesaran (2006) and our model. As in the 2wfe literature, time-invariant covariates are subsumed into the fixed effect ${\varsigma }_i$.

6 Moon and Weidner (2015) established consistency of the least-square estimator of the regression coefficient when the number of factors used in estimation is at least $\mathcal{l}$.

7 The asymptotic normality result hinges on the pivotal nature of the studentized statistic and a standard conditioning argument that applies to the case with nonstationary regressors (Park and Phillips 1988). This is a nontrivial result, for example, Chernozhukov, Wüthrich, and Zhu (2018) requires stationarity as the key assumption underlying their inference procedure.

8 Although we are unable to provide formal proofs, the asymptotic normality result provides support for bootstrapping the t-statistic. By contrast, the limiting distribution of the nonstudentized statistic may vary discontinuously with the serial dependence properties of the factors (e.g., stationary, near unit-root and unit-root processes). Hence we do not recommend bootstrapping the PCDID estimate directly.

9 Although we only focus on the ATET, other estimands are available, such as the conditional moments and quantiles of ITET among $i\in E$. These estimands can help unveil the distributional features of ITET.

10 Suppose α = 1 and $E(v_j|j\in E)=0$. It follows that $E({\mu }_j^{\prime }|j\in E)f_t=E({\mu }_i^{\prime }|i\in C)f_t$ for all t, or in vector form: $F[E({\mu }_j|j\in E)-E({\mu }_i|i\in C)]=0$ (see online appendix). That F has full column rank $\mathcal{l}$ (Assumption F(ii)) implies $E({\mu }_j|j\in E)=E({\mu }_i|i\in C)={\mu }_0$ where ${\mu }_0\ne 0$ , and hence PTW is satisfied.

11 Existing studies have typically considered factors in simpler forms, for example, iid factor or deterministic sinusoid functions (Bai 2009; Gobillon and Magnac 2016). For space reasons, we do not report scenarios that involve deterministic trends; such cases are trivial and are encompassed by PCDID as a special case. We also do not report scenarios that involve a mix of nonstationary and stationary factors because the results look similar to the scenario with nonstationary factors only.

12 Doubly weighted methods such as Arkhangelsky et al. (2019) numerically solves for vertical and horizontal weights with nonnegativity constraints. Hence it can be viewed as an extension of Abadie, Diamond, and Hainmueller (2010) where vertical nonnegative weights are involved. Nonnegative weights require strong support conditions as derived in Gobillon and Magnac (2016).

13 As mentioned above, the DGP for ITET inference sets ${\rho }_{ϵ}=0$ ; see Gonçalves and Perron (2014) for a similar setup. The formula can be replaced by the Newey-West HAC estimator if ${\rho }_{ϵ}\ne 0$.

14 We follow the definitions of variables in Ziliak et al. (2000). Work requirements and time limits are the key components of welfare reform and they unambiguously reduce welfare participation. For simplicity, we do not consider earnings disregards and parental responsibility waivers, which yield mixed evidence in the literature, see, for example, Chan (2013) and Chan and Moffitt (2018) for details.

15 We include some covariates that are potentially endogenous to the factor structure, for example, unemployment rate. PCDID is robust to such covariates (see Section 4.1 and online appendix). We perform fixed effects estimation of y_it on x_it in the control panel (2340 observations) to obtain residuals, which are then used for constructing factor proxies ${\widehat{f}}_t$ by PCA.