Phillips, starting with the assumption that deterrence should be strongest in the weeks immediately following an execution, analyzed the weekly homicide statistics for London, England, before and after 22 highly publicized executions from the years 1858-1921. He found that the homicide rate declined by about 35 percent for each of the 2 weeks following these executions. He also found that the more extensive the newspaper accounts of an execution, the greater the drop in homicides. In the third through the fifth weeks following the executions, however, the homicide rates rebounded above the original baselines, virtually cancelling the deterrent effects of the 2 previous weeks. Following the fifth week, the homicide rates returned to their baselines. The study reported here examined the short-term deterrent impact of the executions of Gary Gilmore (1977 in Utah), John Spenkelink (1979 in Florida), Jesse Bishop (1979 in Nevada), and Steven Judy (1981 in Indiana). Each of the executions received extensive news coverage, although the amount of coverage decreased substantially with each successive execution. Since weekly homicide rates are available from 1972 to the present, it was possible to use Box-Jenkins ARIMA modeling for the data analyses. The advantages of this procedure over the nonparametric procedures used by Phillips are that one can examine time-series data for nonstationarity, autoregressive processes, and moving-average processes. Seasonal or annual changes in the data due to nonstationarity, autoregression, and moving-average processes can also be identified. In a sequential process, ARIMA modeling permits identification of which of these sources of systematic variance are contained in the time-series data set, to calculate a parameter which estimates the strength of each source of variance, and to use these parameters in predicting particular data-points within the series. While the study failed to find any evidence that capital punishment has either a short-term deterrent or a rebound effect on homicides, the findings do not mean that such effects may not occur at other times and in other social circumstances. Graphic data and 52 footnotes are provided.