Depending on how many control units and different treatment periods you have this would be a good application for difference-in-differences (DiD) with pre-treatment matching on the outcome or the synthetic control method.
DiD with matching on the pre-treatment outcome
The most important thing in DiD is that the pre-treatment trends are the same in the treatment and control group. For this you can do matching in an initial step where you match treatment and control locations based on the pre-treatment outcome via nearest neighbor, propensity score, or caliper matching methods. This ensures that you select the most similar pre-treatment locations for the analysis. Here is an explanation for how to do this in Stata. The logic would be the same for R with the corresponding matching package. In any case, if you want to convince your audience with a DiD design you need to plot these pre-treatment trends. A nice graph would show the average in the outcome for the treatment and control groups over time with the two lines having the same trend before the treatment and a divergence after the treatment.
Once you have done the matching you can use the matched sample for the following regression,
$$y_{lt} = \beta_1 \text{location}_l + \beta_2 \text{time}_t + \beta_3 \text{treated}_{lt} + \beta_4 X'_{lt} + \epsilon_{lt}$$
where $y_{lt}$ is the outcome for location $l$ at time $t$, $\text{location}_l$ is a dummy for whether location $l$ is in the treatment group, $\text{time}_t$ is a dummy for each time period, $\text{treated}_{lt}$ is a dummy which equals one if a location is in the treatment group AND the time period is in the treatment period. $X'_{lt}$ are location specific controls (if you have any), and $\epsilon_{lt}$ is a random error. The treatment effect is captured by the coefficient $\beta_3$. This is the most general representation of a DiD model which allows for multiple time periods and treatments at different points in time. This is useful if location $A$ is treated at $t=5$, location $B$ is treated at $t=7$, etc. so you don't need to drop other treated observations.
If you have several treated locations then it is a good idea to keep those and not just keep one. Discarding other treatment locations will result in a loss of statistical power for your hypothesis tests and you may end up finding no significant effect even though you should have. Regarding testing you should also definitely cluster the standard errors on the location identifier. Bertrand et al. (2004) have shown that without an adjustment for autocorrelation the standard errors in DiD designs will be too small and may lead to false inference. Clustering on the location identifier is the easiest way to get around it and it deals with both heteroscedasticity and autocorrelation.
Synthetic control
If you only have a single treated location then the synthetic control method can be useful. Basically what it does is that it compares your treatment location with all the control locations in the sample. From the control locations it then builds a synthetic location from the control locations that are most similar to the treated location. It does so by assigning different weights to each control location such that the resulting synthetic location most closely fits the pre-treatment outcome of the treatment location.
The original application of this was on the question how terrorist attacks affected economic growth in the Basque region. Given that no one region in Spain is as similar as the Basque region, Abadie and his co-authors took a weighted average of the control regions. This is how this synthetic control method came along. So if your application is similar to their idea then this is not a bad choice. At least in economics this method has not yet found many followers. The main problem is that you cannot really have statistical inference in the traditional sense, i.e. there won't be any standard errors or confidence intervals and most of the "inference" is done via graphical analysis and the use of placebo tests. This lecture introduces DiD and the synthetic control method if you want to know more about the details.
As with DiD, graphical evidence is needed to show the validity of your approach. Here is a discussion on the Statalist on how to implement these placebo tests. The basic idea behind them is that you randomly re-assign the treatment to different time periods that lie before the treatment or you re-assign the treatment identifier to control units. In either case you should not find an effect, otherwise it casts doubt on your initial finding.
Advantages and Disadvantages
Both methods are easily implemented in any statistical software. The initial matching for the DiD approach can be a bit tricky, especially if you don't have a balanced panel but the answer that I linked above provides a good solution for this problem. Otherwise DiD is the easier method to implement as regression with some generated dummy variables is straight forward. You get interpretable results with standard errors and confidence intervals right away, something you won't have in the synthetic control approach. This is probably the main reason for why economists have not used synthetic control much. You rarely have just one treated case and most often what you can do with synthetic control you can also do with DiD.
Synthetic control has the advantage that it is something relatively new. New ways of doing things are always more interesting than old ones, plus it works very well if you only have one treated case. The construction of the placebo graphs can be a bit annoying but some software packages do it for you. In general, this approach is slightly more difficult to implement but canned packages are available for R, Matlab and Stata (link). The R package has a nice documentation. People are also currently working on new ways to have more solid inference methods for synthetic control (see this working paper, for example) though most of them you will have to program yourself.
