A narrow bandwidth will have a lower bias because more observations are near the cutoff, but will have a larger variance because of the smaller number of observations. Bias increases as one moves away from the cut-off while variance increases with a smaller number of observations as one moves closer to the cut-off and vice-versa.
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How to select the appropriate bandwidth, h, from which to estimate tau? The determination of the bandwidth is a tradeoff between bias and variance. || lfit y_post x_obs if inrange(x_obs, 0. Tw scatter y_post x_obs, msize(small) mcolor(gs10) yl(-4(2)4) /// Tw scatter y_pre x_obs, msize(small) mcolor(gs10) /// Drawing the graphs above: // Before program participation
In the right panel, the discontinuous jump, tau, at the cutoff is the estimated program impact. The figures below graphically illustrate a local linear regression RDD before and after program participation on simulated data within a specified bandwidth, h. The magnitude of the discontinuous jump at the cutoff may be estimated using a local regression that limits the observations to a specified bandwidth around the cutoff where the functional form is most likely linear. RDD can be characterized as an estimation of whether an outcome variable exhibits a discontinuous jump precisely at the cutoff of the running variable. While the evaluation results using RDD has strong internal validity properties considered by many as next only to RCT, it needs to be recognized that its external validity is limited to observation units near the eligibility threshold. The main caveat in RDD is that because program impact is estimated locally, or using observations very close to the cutoff, the generalizability of RDD estimated effect is limited.
For example, Hahn, Todd, and van der Klaauw (2001) showed that RDD requires milder assumptions relative to those needed for other non-experimental methods. Among the advantages of RDD are the weaker assumptions required for its validity compared to other non-experimental impact evaluation methods. Thus, a large jump in the outcome variable, observed precisely at the threshold value of the running variable, after program intervention can be attributed to the program itself. In the absence of the program, one would expect that any shifts in outcome variables would happen smoothly alongside minor changes in the running variable. Observations just below the cutoff are deemed similar to, and therefore, compare well to those just above the cutoff. RDD is a quasi-experimental method for evaluating program impact when observation units (example, households) can be sorted using some continuous metric (example, income) and program assignment is based on a pre-determined threshold or cutoff point of the sorting metric. In Part 3, validation or falsification tests are discussed. In Part 2, a comparison of user-written Stata estimation packages is provided. Lee and Lemieux (2010), Imbens and Lemieux (2007), and Cook (2008) provide comprehensive reviews of regression discontinuity design and its applications in the social sciences. There has been a growing use of regression discontinuity design (RDD), introduced by Thistlewaite and Campbell (1960), in evaluating the impacts of development programs.
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