Numerous studies have been devoted to the estimation and inference problems
for functional linear models (FLM). However, few works focus on model checking
problem that ensures the reliability of results. Limited tests in this area do
not have tractable null distributions or asymptotic analysis under
alternatives. Also, the functional predictor is usually assumed to be fully
observed, which is impractical. To address these problems, we propose an
adaptive model checking test for FLM. It combines regular moment-based and
conditional moment-based tests, and achieves model adaptivity via the dimension
of a residual-based subspace. The advantages of our test are manifold. First,
it has a tractable chi-squared null distribution and higher powers under the
alternatives than its components. Second, asymptotic properties under different
underlying models are developed, including the unvisited local alternatives.
Third, the test statistic is constructed upon finite grid points, which
incorporates the discrete nature of collected data. We develop the desirable
relationship between sample size and number of grid points to maintain the
asymptotic properties. Besides, we provide a data-driven approach to estimate
the dimension leading to model adaptivity, which is promising in sufficient
dimension reduction. We conduct comprehensive numerical experiments to
demonstrate the advantages the test inherits from its two simple components