Alternatives to polynomial trend-corrected differences-in-differences models

ABSTRACT

A common problem with differences-in-differences (DD) estimates is the failure of the parallel-trend assumption. To cope with this, most authors include polynomial (linear, quadratic…) trends among the regressors, and estimate the treatment effect as a once-in-a-time trend shift. In practice, that strategy does not work very well, because inter alia the estimation of the trend uses post-treatment data. An extreme case is when sample covers only one period before treatment and many after. Then the trend’s estimate relies almost completely on post-treatment developments, and absorbs most of the treatment effect. What is needed is a method that i) uses pretreatment observations to capture linear or nonlinear trend differences, and ii) extrapolates these to compute the treatment effect. This article shows how this can be achieved using a fully flexible version of the canonical DD equation. It also contains an illustration using data on a 1994–2000 EU programme that was implemented in the Belgian province of Hainaut.

KEYWORDS:

• Treatment-effect analysis
• differences-in-differences (DD) models
• correction for trend differences
• parallelism

JEL CLASSIFICATION:

• C20
• C31
• C52

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

¹ See Vandenberghe (2018) for more details about EU-Objective 1-Hainaut.

² Defining time = year-1993.

³ If outcome level change by unit of time (i.e. 1st derivate) is ‘speed’, then Parallel_[1] means stable level differences due to identical speeds.

⁴ If outcome growth rate change by unit of time (2nd derivative) is ‘acceleration’, then Parallel_[2] means stable growth rate differences due to same accelerations.

⁵ If outcome acceleration change by unit of time (3rd derivative) is ‘surge’, then Parallel_[3] corresponds to a situation where acceleration differences remain stable due to identical surges.

⁶ The pattern of lagged effects is usually of substantive interest (e.g. if treatment effect should grow or fade as time passes).

⁷ When estimating Equation (2) with only two periods, γ^D_t* is subsumed into the constant γ^Dand DD_[1] is directly captured by the time X treatment coefficient.

⁸ Treatment effect’s SE must account for the fact that it consists of a linear combination of estimated coefficients, and thus of the covariance between variables. That is automatically done by STATA test or lincom commands used hereafter, that exploit the variance-covariance matrix of the estimated coefficients.

⁹ Net of the initial handicap in t*-1: γ^D