Regression discontinuity versus matching with spatial discontinuity I am interested in using a spatial research design.  Imagine a line, like a time zone line.  For example, in the United States, the line that makes between Eastern Standard Time and Central Standard Time runs North to South through the U.S. (and other places), more or less. 
Suppose the United States implemented a policy where, with respect to county boundaries, all counties west of the 75th longitude were to get a tax deduction.  No counties to the right of it would.  I want to evaluate the impacts of the this tax deduction on crime.  
For the sake of argument, can we please assume that the discontinuity is valid.  I am trying to understand this idea for spatial discontinuities, in general?  
I can imagine proceeding one of two ways.  One way would a regression discontinuity.  Exploiting the distance of say a county from this line.  The other way would be a matching of counties that straddle this line on opposite sides.  Note that a county might appear more than once if it borders more than one county on the opposite side of the time zone line.  
The regression discontinuity includes a larger sample because it may include counties that are not exclusively along the border.  The matching hones in on counties that touch each other, whereas the regression discontinuity, at best, can partition the time zone into smaller segmented lines where segments likely include multiple pairs of counties along the time zone.  
What are the trade-offs of the research design?  Which is preferred?  Does one offer something that the other doesn't?  
 A: I would say matching is more appropriate (and easier), but the logic to each has some comparable aspects worth expounding upon.
Regression discontinuity designs are predicated on the fact that there is some observable relationship between some variable, $X$, and the outcome, $Y$. Then in RDD there is some other exogenous impact that occurs at some threshold of $X$. Note, implicit in the design is that cases are comparable on each side of the threshold (that is, no other differences between the cases exist on each side of the threshold), and so any discontinuity in the effect of $X$ and $Y$ before and after that threshold can be considered the treatment effect.
One of the things that makes RDD in this circumstance difficult is that it is unclear what $X$ is in your circumstance (you could think of many in addition to the distance one you mentioned) and for the social science variables listed, it is unlikely $X$ has a clear/obvious/strong relationship to $Y$. Also I would be skeptical that cases on either side of the threshold are entirely comparable, and so one would want to include other socio-demographic indicators. This can be done, but makes such a quasi-experiment markedly less appealing.
Thus I would suggest matching or estimating propensity score models. You can certainly find a history of examples of matching across the border (see for instance Card & Kreuger, 1994). Also I have seen matching spatial units extended to propensity score models, for instance Ridgeway (2006) uses a flexible set of generalized boosted models to estimate propensity scores for post traffic stop outcomes (e.g. searches, arrests). Such flexible models are attractive because spatial trends can be hard to characterize with such social science data and may take many parameters to effectively model. Also such models are readily capable of including other sets of socio-demographic covariates one would be expected to include in such research designs (for at least the outcomes you mention).

Citations

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*Card, David & Alan Krueger. 1994. Minimum wages and employment: A case study of the fast-food industry in New Jersey and Pennsylvania. The American Economic Review 84(4):772-793. PDF Here.

*Ridgeway, Greg. 2006. Assessing the effect of race bias in post-traffic stop outcomes using propensity scores. Journal of Quantitative Criminology 22(1): 1-29. PDF Here
A: I think the main tradeoff is that RD has high internal validity, but low external, and the opposite is true for matching. Since we don't have a great deal of detail about what you are actually doing, you will have to decide which one is better in your case.
The regression discontinuity design comes closest to an ideal experiment, so it has high internal validity, but unfortunately low external validity. The parameter it estimates is a very localized sort of local average treatment effect since you're mainly considering units right around the discontinuity cutoff. If you have reasons to think there are heaps of heterogeneity in the effect, you're stuck up the proverbial creek with no paddle.  
Propensity score matching/reweighting methods can only eliminate bias due to selection on observables, and will yield efficient estimates of many types of treatment effects (average treatment effect or the treatment effect on the treated), so they often have higher external validity compared to RD, but the internal validity that is often questionable since it is rare that selection depends only on observables. If there's selection on unobservables, matching can actually exacerbate bias.
Finally, you might also want to consider an RD/Difference-in-Differences hybrid since you may have panel data on your counties:
\begin{equation}
\hat\beta=\frac{(y_{1}^{+}-y_{1}^{-})-(y_{0}^{+}-y_{0}^{-})}{x_{1}^{+}-x_{1}^{-}}
\end{equation}
where $y$ is the mean outcome, the signs denote position relative to the cutoff and the numbers denote before and after, and $x$ is the mean treatment (may be 1 in the case of the deterministic assignment or sharp RD design). 
Update:
I just remembered a paper by Keele and Titiunik about geographic RD. 
A: The matching procedure you described can be thought of as a special case of regression discontinuity where you set the estimation bandwidth to zero.
Speaking somewhat informally to develop intuition - the problem of causal inference is that there may be some unobserved variables that are related to both your treatment variable and your outcome. In a regression discontinuity framework, the key assumption needed to establish causality is that none of these unobserved variables change discontinuously at the treatment threshold. The treatment effect is then the difference between the limit of Y as it approaches the threshold from the left, and the limit approaching from the right.
Calculating these limits in the real world requires us to make some choices in modelling how Y changes with respect to the running variable. In your case the running variable could be Euclidian distance to the time zone boundary. It's unlikely that this distance causally effects your outcome variable, however it could very well be correlated with some unobservables. In this case it makes sense to think about this in a geographic regression discontinuity framework. It is perfectly reasonable to include other covariates in the regression. In this sense, geographic RD can control for the same observables as a propensity score matching type technique, but it will also control for unobservables that change with distance to the border. Propensity score matching requires the somewhat stronger assumption that there are no unobservable confounders in order to establish causality.
The question then becomes how far from the border should we include observations. There is a tradeoff choosing this 'bandwidth' between bias and variance, and optimal methods for choosing the bandwidth have been developed (see Imbens and Kalyanaram 2011). If you restrict observations to the border counties, then you are forcing the bandwidth to be zero.
A: I may be mistaken so please feel free to correct me if this is not the correct answer.  However, I think the correct answer is that both approaches would be acceptable but rather that the assumptions for RD to hold are more plausible, making it preferable.  
Consider a research design in which both hold.  Basically, RD only requires that the unobservable characteristics are continuous.  The matching effectively requires that there are no selection on unobservables and matching by proximity would be adequate to take care of all unobeservables.  While the matching probably does account for some unobservables--unobservables that RD would have to address--that RD only requires them to be continuous is probably more plausible.  At the same time, RD estimates are more efficient.  
Any reason that matching would not hold would also cause RD to not hold.  Likewise, if spatial RD is plausible then so must spatial matching.  So, when spatial RD is possible, it is preferable to spatial matching.  
