Testing Parallelism of Nonparametric Regression Curves

This paper considers the inference of regression functions in the context of multiple time series. For an arbitrary number of time series observed at a large number of time points, we test the hypothesis that the regression curves are parallel to each other. A central limit theorem is obtained for a parallelism index based on the distances between the estimates of the regression curves and their average. To implement the testing procedure, we propose a simulation-based approach that significantly improves upon the normal approximation to the test statistic. Our method is applied to a data set of daily download volumes from Motorola cell phones over 129 area codes in the United States.