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 on predicting headline inflation, many economic and financial institutions are interested in its partial disaggregated components. To this end, we developed the novel Hierarchical Recurrent Neural Network (HRNN) model, which utilizes information from higher levels in the CPI hierarchy to improve predictions at the more volatile lower levels. Based on a large dataset from the US CPI-U index, our evaluations indicate that the HRNN model significantly outperforms a vast array of well-known inflation prediction baselines. Our methodology and results provide additional forecasting measures and possibilities to policy and market makers on sectoral and component-specific prices.