Justin McCrary presents a test for manipulation in regression discontinuity designs by examining the continuity of the running variable density function. The test involves estimating the discontinuity in the density function at the cutoff point and conducting a Wald test of the null hypothesis that the discontinuity is zero. The methodology uses a local linear density estimator, which involves two steps: first, creating a finely-gridded histogram, and second, smoothing the histogram using local linear regression on either side of the cutoff. The test is implemented as a Wald test and is useful in applications where pre-determined characteristics are not available or not relevant to the outcome. The test complements existing specification checks in regression discontinuity applications and is particularly useful in settings where the density function is the object of interest. The paper applies the test to popular elections to the House of Representatives and roll call voting in the House, where manipulation is both expected and found. The results show that the density test provides strong evidence of manipulation in the latter case. The paper also discusses the theoretical implications of manipulation in regression discontinuity designs and the conditions under which identification problems may arise. The test is shown to be effective in detecting manipulation when it is monotonic, and the paper provides simulation evidence supporting the test's performance. The empirical analysis demonstrates the test's utility in real-world applications, highlighting the importance of considering manipulation in regression discontinuity designs.Justin McCrary presents a test for manipulation in regression discontinuity designs by examining the continuity of the running variable density function. The test involves estimating the discontinuity in the density function at the cutoff point and conducting a Wald test of the null hypothesis that the discontinuity is zero. The methodology uses a local linear density estimator, which involves two steps: first, creating a finely-gridded histogram, and second, smoothing the histogram using local linear regression on either side of the cutoff. The test is implemented as a Wald test and is useful in applications where pre-determined characteristics are not available or not relevant to the outcome. The test complements existing specification checks in regression discontinuity applications and is particularly useful in settings where the density function is the object of interest. The paper applies the test to popular elections to the House of Representatives and roll call voting in the House, where manipulation is both expected and found. The results show that the density test provides strong evidence of manipulation in the latter case. The paper also discusses the theoretical implications of manipulation in regression discontinuity designs and the conditions under which identification problems may arise. The test is shown to be effective in detecting manipulation when it is monotonic, and the paper provides simulation evidence supporting the test's performance. The empirical analysis demonstrates the test's utility in real-world applications, highlighting the importance of considering manipulation in regression discontinuity designs.