Multi-layered rational inattention and time-varying volatility

Standard rational inattention models suppose that agents process noisy signals about otherwise fully revealing data. I show that introducing imperfect data quality yields new insights in settings in which volatility is time-varying. I impose a two-layered signal structure in which agents learn imperfectly about noisy sources. Treating data as only partially revealing of the true fundamental amplifies impulse responses to a second moment shock and, if data quality is sufficiently poor, can change the qualitative direction of the response. I apply my findings to the price-setting problem of firms and find that higher data quality enhances the transmission of monetary policy and reduces macroeconomic volatility. I also show how the empirically documented procyclicality of data quality has non-trivial implications for the Phillips curve.