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We analyze the optimal Taylor rule in the standard New Keynesian model when output and inflation are imperfectly observed. When the central bank observes inflation and the output gap with error, the optimal Taylor rule features tempered responses so as not to impart unnecessary volatility to the economy. If the Taylor rule is expressed in terms of estimated output and inflation, it is optimal to respond infinitely strongly to estimated deviations from the targets. Because filtered estimates are based on current and past observations, such Taylor rules appear to exhibit interest rate smoothing even though the monetary authority has no explicit preference for smooth interest rates. Under such a Taylor rule, the estimates of inflation and the output gap are perfectly negatively correlated. In the data, these gaps are slightly positively correlated, suggesting that the central bank is systematically underreacting to estimated inflation and the output gap.