From Isaac Ehrlich and Yong Yin (NBER, 2012):
Self-insurance is intrinsically a substitute for market insurance whereas self-protection could in principle be a substitute or a complement, depending on whether insurance companies monitor individual efforts at self-protection and reward such behaviors with lower premiums – a rather unlikely prospect in the case of typical, menu-based health insurance policies where premiums are based on overall community rating. As we show,  both alternatives, when sufficiently effective, increase the likelihood of a “corner solution” in which the purchase of insurance is eschewed altogether. This possibility  has been entirely missing from the debate about the rationale for mandating uniform health coverage, as well as from the micro-simulation models offered by the CBO and Rand’s COMPARE, which project the take-up rate of the Patient Protection and Affordable Care Act (ACA) and assess some of its welfare implications. [...]
Our calibrated simulations  indicate that self-insurance and self-protection account for 31.3% of the uninsured by the baseline model, or 28.2% by the extended model. Jointly with the safety net system, these alternatives account for 50.3% and 45.5% of the uninsured, respectively. [...]
As our analysis  illustrates, estimates of compliance rates with the mandated provision of the ACA of by the previously uninsured could be significantly overstated if no account is given to the role that [self-insurance/self-protection] and the safety net have played in motivating the original decision of individuals to be uninsured. The precise estimates depend on the magnitude of the penalty actually imposed on non-compliers, but our illustrated results indicate that it might be significantly overstated, perhaps by over 50%.