Two questions arising in the analysis of functional magnetic resonance imaging (fMRI) data acquired during periodic sensory stimulation are: i) how to measure the experimentally determined effect in fMRI time series; and ii) how to decide whether an apparent effect is significant. Our approach is first to fit a time series regression model, including sine and cosine terms at the (fundamental) frequency of experimental stimulation, by pseudogeneralized least squares (PGLS) at each pixel of an image. Sinusoidal modeling takes account of locally variable hemodynamic delay and dispersion, and PGLS fitting corrects for residual or endogenous autocorrelation in fMRI time series, to yield best unbiased estimates of the amplitudes of the sine and cosine terms at fundamental frequency; from these parameters the authors derive estimates of experimentally determined power and its standard error. Randomization testing is then used to create inferential brain activation maps (BAMs) of pixels significantly activated by the experimental stimulus. The methods are illustrated by application to data acquired from normal human subjects during periodic visual and auditory stimulation.