Economic theories are often encoded in equilibrium models that cannot be directly estimated because they lack features that, while inessential to the theoretical mechanism that is central to the specific theory, would be essential to fit the data well. We propose an econometric approach that confronts such theories with data through the lens of a time series model that is a good description of macroeconomic reality. Our approach explicitly acknowledges misspecificationas well as measurement error. We highlight in two applications that household heterogeneity greatly helps to fit aggregate data, independently of whether or not nominal rigidities are considered.