We present a hierarchical architecture based on Recurrent Neural Networks (RNNs) for predicting disaggregated inflation components of the Consumer Price Index (CPI). While the majority of existing research is focused mainly on predicting the inflation headline, many economic and financial entities are more interested in its partial disaggregated components. To this end, we developed the novel Hierarchical Recurrent Neural Network (HRNN) model that utilizes information from higher levels in the CPI hierarchy to improve predictions at the more volatile lower levels. Our evaluations, based on a large data-set from the US CPI-U index, indicate that the HRNN model significantly outperforms a vast array of well-known inflation prediction baselines.