In this article, we present a copula regression model for testing asymmetric information as well as for predictive modeling applications in automobile insurance market. We use the Frank copula to jointly model the type of coverage and the number of accidents, with the dependence parameter providing for evidence of the relationship between the choice of coverage and the frequency of accidents. This dependence therefore provides an indication of the presence (or absence) of asymmetric information. The type of coverage is in some sense ordered so that coverage with higher ordinals indicate the most comprehensive coverage. Henceforth, a positive relationship would indicate that more coverage is chosen by high risk policyholders, and vice versa. This presence of asymmetric information could be due to either adverse selection or moral hazard, a distinction often made in the economics or insurance literature, or both. We calibrated our copula model using a one-year cross-sectional observation of claims arising from a major automobile insurer in Singapore. Our estimation results indicate a significant positive coverage-risk relationship. However, when we correct for the bias resulting from possible underreporting of accidents, we find that the positive association vanishes. We further used our estimated model for other possible actuarial applications. In particular, we are able to demonstrate the effect of coverage choice on the incidence of accidents, and based on which, the pure premium is derived. In general, a positive margin is observed when compared with the gross premium available in our empirical database.