Why would complete manipulation on running variable result in misidentification of treatment effect Why would complete manipulation on running variable result in misidentification of treatment effect and why the partial manipulation may not. i.e. why is density test necessary?
May someone gives an example? Thanks in advance!
 A: The intuition is that the things around the cutoff are very similar, except that those on one side are treated. For example, a product on Amazon has an average rating of 4.49 stars and another has the rating of 4.51, but the first gets rounded to 4 stars and the second gets rounded to 5 when the ratings are displayed to the customer. These two products are basically identical, but appear very differently to the consumers browsing which allows us to say something about the role of stars and reputation on the purchasing decision.
If some sellers can create a couple of fake reviews and get their products rounded up to 5, then this comparison of probabilities of purchase is conflating both the star effect and with the effect of being a cunning merchant, which may be correlated with other changing factors intended to boost sales, like adding better/deceptive photos or fixing product descriptions, which are observable to consumers and also influence their purchase decisions, but maybe not to the analyst doing the data analysis. Now the two groups of products are not comparable along other dimensions as well as the star rating and the analysis of star rating on purchase is biased.
