Extensive studies have concluded that the GNSS observations are heteroscedastic and physically correlated. Typically, the observation precisions are elevation dependent and between-frequency cross-correlations and time correlations exist. The influence of these stochastic characteristics on the GNSS positioning has been numerically well understood. However, their influence on the statistic tests of reliability has been rarely studied. We will systematically study the influence of GNSS stochastic characteristics on the statistic tests involved in reliability. With BeiDou as an example, the realistic elevation-dependent model, cross-correlations and time correlations are estimated. Then their impacts on the reliability are numerically analyzed by comparing with the empirical stochastic model where the stochastic characteristics, i.e., elevation-dependent precisions, cross-correlations and time correlations, are not adequately specified. Besides the overall test and w-test, the minimal detectable bias (MDB) and the separability of two w-test statistics are examined. The results show that the realistic elevation-dependent model will reduce probabilities of both false alarm and wrong detection for both overall test and w-test. Introducing the cross-correlations and time correlations properly can obtain the realistic MDBs together with reasonable separability measures, which all are helpful for users to make objective decisions in quality control of real GNSS applications.