Selection in car insurance when claims are heterogeneous

Econometric studies of insurance markets have analyzed the Positive Correlation Property to test for the presence of asymmetric information. Car-insurance studies frequently compare policies purchasing Mandatory Third-Party Liability alone with policies that purchase additional coverage and use the presence of a liability claim as a measure of risk. Using data from the Italian market, we show that this approach can yield conflicting results from the same dataset. First, different types of additional coverage can yield different results; second, the presence of a claim is a poor or even misleading indicator of the cost to the insurance company. A test using an unused observable is similarly problematic. Our findings highlight that selection may operate along dimensions of risk not captured by claim probabilities, complicating the interpretation of standard empirical tests and the distinction between adverse and advantageous selection.