Peaks in the spectrum of a stationary process are indicative of periodic phenomena, such as seasonality or business cycles. Hence one important aspect of developing parametric models for periodic processes is proper characterization of spectral peaks. By using diagnostics constructed from the average ratio of two spectral densities this work proposes to test whether a hypothesized model is locally supported in the frequency domain by the data. The local fit of a model is assessed by considering a subset of the whole frequency band, focused on the locality of the spectral peak. This technique can therefore be used to test the goodness-of-fit of a model with respect to local frequency domain properties of the data. For example, one can test for the appropriateness of a hypothesized seasonal or cyclical spectral peak in the model for the data. In the development of these diagnostics we provide a result of independent interest, the asymptotic distribution of general polynomial functionals of the periodogram. We present theoretical properties for several new diagnostics, and also explore their finite-sample properties through simulation and application to several economic time series.