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Join the community! Visit our GitHub or LinkedIn page to join the Tilburg Science Hub community, or check out our contributors' Hall of Fame! Want to change something or add new content? Click the Contribute button!--> IntroductionRegression discontinuity (RD) designs are a popular method for causal inference in non-experimental contexts in economics and other fields, such as political science and healthcare. They rely on the existence of a threshold defined as a value of some running variable, to one side of which subjects are treated, and to the other not. Regression discontinuity in time (RDiT) designs are those RD applications where time is the running variable. The cutoff consequently is the treatment date: after it, subjects are treated, and before it, they are not. The RDiT design is thus useful in cases where there is no cross-sectional variation in treatment status; that is, on a certain date, treatment is applied to all subjects, meaning designs such as difference-in-differences (DiD) are not applicable. Like in canonical (continuity-based) RD designs, in RDiT we identify the treatment effect as a discontinuity in the outcome variable at the cutoff, assuming that any potential time-varying confounders change smoothly around the threshold. However, there are also important differences between standard RD and RDiT designs which have implications for causal inference. Differences between standard RDD and RDiTFirst, in RDiT it is often the case that we only have one observation per unit of time, unlike in many other applications of RDD with a discrete running variable. This happens, for example, with hourly pollution data in environmental economics, a field where RDiT designs are frequently used. In these cases, in order to attain sufficient sample size, researchers often use very wide bandwidths, of up to 8-10 years. In such large time periods, many changes unrelated to treatment are

July 2007. In this case, we use a second-order polynomial (a quadratic), but you could also experiment with other polynomials by changing p in the function below. We also select a 36-month (3-year) bandwidth, and you could also try experimenting with that (in the paper, effect estimates are robust to bandwidth choice). R#use the with function to select a bandwidth#set polynomial order with p=...#set number of bins with nbins=...#y.lim is used to define the height of the y-axiswith(subset(fertility, date >= 540 & date OutputWe see a clear discontinuity at the treatment date. Now, we can test for this formally by running a standard RDiT regression:R#estimation of treatment effect with a second-order polynomial#use natural log of abortions as in the paperoptions(scipen = 999) #remove scientific notationfertility$month_qdr = 540)) summary(rdit)OutputWe see a statistically significant negative effect of the treatment on abortions, which is also what Gonzalez (2013) finds in her paper. Now, let's run some of the robustness checks mentioned above. We start by testing for serial correlation in the residuals using the Durbin-Watson test.R#Durbin-Watson test for serial correlation in the residualsdwtest(rdit, alternative = "two.sided")OutputWe see that the p-value is well below 0.05, meaning we can reject the null hypothesis of no serial correlation. Given that residuals exhibit this serial correlation, we compute HAC standard errors:R#use vcovHAC in coeftest to re-estimate the model with HAC standard errorscoeftest(rdit, vcovHAC(rdit))OutputThe HAC standard errors do differ from the conventional ones, but do not change the inference we made from the original estimates. The treatment coefficient remains strongly statistically significant. Now, we estimate a donut RDD, allowing for the possibility that individuals anticipate the policy and change their behaviour accordingly. We assume that they can do so within 1 month of the treatment date (before and after).R#donut RDD removing 1 month before and 1 month after

Likely to affect the outcome variable. This could be caused by time-varying confounders (in the pollution example, think of weather or fuel prices) or simply time trends in the outcome. Using a long time period after the treatment date can also result in us capturing treatment effects that vary over time, a circumstance RD designs are unsuited to. In addition, if our time variable is measured sparsely, i.e. at low-frequency units like months or quarters, the usual assumptions of smoothness and continuity of the conditional expectation function relating the running variable to the outcome (and potential confounders) are a priori less likely to hold. Intuitively, almost any variable is less likely to vary smoothly from month to month than from week to week, for example. Secondly, as time is the running variable, the data will have time series properties. This can translate into two issues: autoregression in the outcome variable (i.e. the outcome depends on its previous values) and serial correlation in the residuals of our fitted RD models (i.e. residuals from observations next to each other are not independent of each other, violating the Gauss-Markov assumptions). The third difference is that we cannot interpret an RDiT design with the local randomisation approach, because we cannot perceive time as being assigned randomly in a bandwidth around a threshold. Instead, we can only use the continuity-based approach to RD designs in an RDiT setting, using discontinuities in the outcome at the cutoff to identify causal treatment effects. Finally, as time has a uniform density, it is impossible to conduct standard density tests (such as the McCrary test) for manipulation of the running variable in RDiT designs. This makes it impossible to test for selection into treatment in the form of anticipation, sorting, or avoidance, all of which could bias the treatment