Critical Discussion of the Results

Overall this paper certainly contributes to the existing literature in the field and adds evidence to the relationship between maternal labor supply and public childcare. Especially for Germany where not much evidence exists yet, the paper delivers a valuable contribution. The paper gives a concrete estimation of an important policy change in Germany. In fact in appears reasonable that the introduction of the legal claim for a kindergarten was an rather unexpected event and could therefore be described as exogenous variation. So indeed the paper uses an appropriate policy change to apply estimation strategies which are widely used in the context of quasi-experimental methods.
Indeed all robustness checks validiate the assumptions which have to be made to justify the IV approach. The relevance criterion is clearly fulfilled and also the Placebo tests further strenghtens the arguments made in favor of the IV approach. Most difficult in the IV case is to assess the validity of the exclusion restriction. The authors argue here that just around the cut-off age (36 months) the existence of the cut-off rule is the only factor pushing mothers back to work. Indeed in the close range around the cut-off it appears rather unlikely that the age matters through a different channel than the elgibility for the kindergarten. Potential threats to the identification approach such as other caring opportunities (i.e. grandparents, private daycare) driven by age as well can rejected by the fact that private daycare is so small in size in Germany (less than 1 (!) percent) and that more care through grandparents will not hang on a two month age differential, so the exclusion restriction seems reasonable in this context. 
More criticism deserves the cut-off rule itself. It is by definition very fuzzy (remember municipalities apply it as they want) and the subgroup to which the it applies is very small. In fact Bauernschuster & Schlotter only identify effects for municipalities which applied the cut-off rule and inside these municipalities mothers which complied with the treatment. This excludes all mothers which sent their children to kindergarten at a younger age or sent them to kindergarten not in August/September when the cut-off rule was potentially applied. According to their own data Bauernschuster & Schlotter report that only 70 percent of the children in the sample started with childcare in August or September. All this reduces the group for which the authors estimate their LATE.  Since the authors can't identify the municipalities they don't know for which group they finally identify the treatment effect. This is slightly unfortunate since generally it should be possible for Bauernschuster & Schlotter to obtain data for the municipalities and the application of the cut-off rule. This would have increased the knowledge which population is actually targeted by the instrumental variable.
With respect to the DiD approach it has to be stated that this approach relies on the common trends assumption. In order to check the validity Bauernschuster & Schlotter fortunately perform some placebo tests and yield insignificant results which support the common trends assumption. Some concerns could be raised about the choice of control groups as already mentioned earlier which in fact might differ in some aspects and therefore might not be optimal. The coefficient estimates vary considerably over the three different control groups which supports the notion that at least some of the control groups are far away from satisfying the common trends assumption. 
Even worse than that is the blunt way in which Bauernschuster and Schlotter try to argue that both identification strategies yield rather similar point estimates. Whereas the IV point estimate is obtained in a transparent manner, the computation for the point estimate for the DiD (remember this is an ATT)  is calculated intransparently and even worse arithmetically wrong. Whereas the authors first only consider the estimate which fits best for their purposes they then use the wrong values for the increase in kindergarten attendance in order to obtain a point estimate which is almost the same as for the IV. What was intented as argument to underline the robustness of both estimates, turns to be a serious flaw which reduces the credibility of the paper.
Furthermore it is questionable whether the assumption that the costs don't matter and can be neglected since 80 percent of the costs are subsidized by the government . First of all the fees for childcare vary dramatically over Germany since they are (usually) set by the municipality. Furthermore surveys like the SOEP show that prices actually matter when parents decide whether they should bring their child to kindergarten or not. So childcare fees appear to be crucial determinant of the final decision whether children attend a kindergarten and whether the resprective mother workds. This might overlap with the effect from the expansion and therefore the effects estimated by Bauernschuster & Schlotter could be biased.

Research Outlook

Based on that criticism issued before a study which examines the effect that childcare fees would have seems a valuable target. In order to achieve this, the IZA - Institute of Labor Economics has started to collect data on childcare fees for the largest municipalities in Germany and for municipalities in North Rhine-Westphalia. Although it is tedious to obtain the data since their is no central register for them. The most current data set on this so far has been collected by the Initiative Neue Soziale Marktwirtschaft (INSM) comparing childcare fees for almost a hundret cities in Germany for selected "prototype families" therefore the data set which is also rather old (2010) can not be used for attachment to the any other data sets. In contrast, by now the IZA has collected the largest data set containing the childcare fees for all 400 municipalities in North Rhine-Westphalia and for all urban districts in Germany. In municipalities where this was possible the IZA has obtained that data for multiple years reaching back until 2008. This data set linked with data from the SOEP or the Microcensus might be able to figure out what the effect of different childcare fees might be. Since then for each mother both the actual and the hypothetical costs for childcare could be calculated causal effects of childcare fees could be identified. 
Furthermore the particular setting in North Rhine-Westphalia allows for the application of quasi-experimental methods. In 2006 the state government transfered the legislative power from the state level to the level of the municipalities. Before the reform the childcare fees were therefore similar all municipalities of North Rhine-Westphalia. Afterwards the municipalities could set the fees individually. Whereas some (richer municipalities) reduced the fees afterwards others left the fees unchanged and some even increased them. This provides us with a set-up which could be exploited for further research (i.e. the Master thesis of the author of this report). 

Conclusion

The paper contributes to the existing literature on the topics of childcare and its effect on maternal labor supply. It is part of an growing body of papers examining this topics also in German contexts. The methods applied are standard quasi-experimental methods and their application seems to be justified overall. The magnitude of the estimates appears to be reasonable and is in line with other estimates for Germany (esp. Geyer et al. (2015)). However the paper suffersfrom some considerable flaws. Firstly the use of the instrument narrows the LATE Bauernschuster & Schlotter measure to an undefined small subgroup although identifying at least the municipalities which applied the cut-off rule should have been possible. Secondly, the calculation of the effect in the DiD framework is arbitrary and simply wrong. This indeed reduces the credibility of the paper significantly. Thirdly the ignorance towards childcare fees is questionable. This criticism is the base for further research which is already planned. The strive for knowledge continues...