In this master thesis we conduct a simulated out-of-sample experiment to compare the one-month-ahead inflation density forecasting performance of competing econometric models on US data. We pay particular attention to the ability to forecast inflations rates that are generally considered as harmful. In our comparison GARCH models consistently deliver the best density forecasts, outperforming the Markov switching model proposed by Amisano and Giacomini (2007). This is in line with the finding of Clark and Ravazzolo (2014), that time varying is an important feature for density forecasts of macroeconomic variables. With regards to the forecasts of excessively high and low inflation rates, we find that models based on the Phillips curve outperform a univariate model at this task, while the models perform similarly overall. This is consistent with earlier findings that Phillips curve generally prove to be useful during certain times, such as recessions, but not during normal times .