This paper stresses that estimated policy rules are reduced form equations that are silent on many important policy questions. To obtain a structural understanding of monetary policy it is necessary to estimate the policymaker's objective function, rather than its policy reaction function. With these issues in mind, this paper proposes a system-based estimation approach that uses the solution to the policymaker's optimization problem to infer the underlying policy regime from the economy's evolution over time. The paper derives conditions under which the parameters in a policymaker's policy objective function can be identified and estimated. These identification conditions apply to forward-looking rational-expectations models as well as to backward-looking models, extending existing results. We apply these conditions to a New Keynesian sticky-price model of the US economy, estimating jointly all of the model's behavioral parameters and the policy regime parameters. The results show that the implicit inflation target and the relative weight placed on interest rate smoothing both declined with Volcker's appointment to Federal Reserve chairman. However, the estimates reveal that other - non-monetary-policy - parameters have changed over time also.