Simultaneous Inference for Time Series Functional Linear Regression

We consider the problem of joint simultaneous confidence band (JSCB) construction for regression coefficient functions of time series scalar-on-function linear regression when the regression model is estimated by roughness penalization approach with flexible choices of orthonormal basis functions. A simple and unified multiplier bootstrap methodology is proposed for the JSCB construction which is shown to achieve the correct coverage probability asymptotically. Furthermore, the JSCB is asymptotically robust to inconsistently estimated standard deviations of the model. The proposed methodology is applied to a time series data set of electricity market to visually investigate and formally test the overall regression relationship as well as perform model validation. A uniform Gaussian approximation and comparison result over all Euclidean convex sets for normalized sums of a class of moderately high-dimensional stationary time series is established. Finally, the proposed methodology can be applied to simultaneous inference for scalar-on-function linear regression of independent cross-sectional data.