Whenever you see risk, you think insurance. And there are different ways to insure yourself. This may be by buying some contingent claims, often bundled into an insurance policy. Or this may be through self-insurance, whereby you build some assets for eventualities beyond savings needs. Formal insurance and self-insurance sure look like substitutes. For the case of health risk, this means that people with formal insurance should have less assets, other personal characteristics being controlled for. This statement is, however, factually wrong: insured people, ceteribus paribus, have more assets. That is difficult to square with standard theory.
Minchung Hsu makes a good attempts at solving this puzzle. The major assumption here is that health insurance is provided through employment (there is also private health insurance, but it is of minor importance, as in the data because it is crowded out by social programs). This means that the loss of employment bears a larger risk for someone who formally insures: one may lose income and insurance. Then, ironically, more self-insurance is needed than for someone who self-insures, but one has also to keep in mind that a self-insurer typically has lower income and is partially covered by social programs. Thus Hsu performs the same regressions as done in the literature and still finds the fact mentioned above. Interestingly, his regression also rejects the existence of precautionary savings, while it is the central element of the model. So much for the power of this test.