  This paper focuses on testing for the presence of alpha in time-varying
factor pricing models, specifically when the number of securities N is larger
than the time dimension of the return series T. We introduce a maximum-type
test that performs well in scenarios where the alternative hypothesis is
sparse. We establish the limit null distribution of the proposed maximum-type
test statistic and demonstrate its asymptotic independence from the sum-type
test statistics proposed by Ma et al.(2020).Additionally, we propose an
adaptive test by combining the maximum-type test and sum-type test, and we show
its advantages under various alternative hypotheses through simulation studies
and two real data applications.
