Eqns (1) and (2) assume that conditional on the propensity score, participation in the insurance program is independent of the _change_ in outcome between the year before the program was introduced and the date of the second wave of the survey, five years later. Eqns (1) and (2) are compatible with individual-specific time-invariant unobservables, so that our estimates will not be contaminated by any selection on time-invariant unobservables. We have also allowed for a common unobservable growth component in the outcome growth equations. Under our assumptions, the only driver of differential change between the insured and matched uninsureds is the fact that the former are insured and the latter are not. The risk we run is that the unobserved time trends are not—contrary to what we assume—independent of insurance status. If the insured have a more favorable time trend than the uninsured, we will end up attributing differences in changes in _y_ to insurance when they are, in fact, attributable—at least in part—to different unobserved time trends.

The link with standard exogeneity assumptions and the attractions of the matching approach can be seen as follows (cf. Imbens 2004). Suppose one were prepared to assume that the treatment effect is constant, and that the outcome among the uninsured is linear in _Xi_ . With cross-section data one could try to recover the treatment effect from a regression of the outcome on an insurance status dummy ( _I_ ) and the vector _Xi_ : Estimation might proceed on the assumption that _I_ is exogenous in the sense that conditional on _X_ , _I_ and ε are uncorrelated, which is the regression equivalent of the ‘unconfoundedness’ or ‘ignorable treatment assignment’ assumption made in the treatment effects literature when crosssection data are used. Or one might use a Heckman selection model and rely either on functional form or exclusion restrictions to identify the insurance effect. Both have been used in the insurance literature, but neither is altogether satisfactory.<sup>7</sup> In the present context, with a two-wave panel conducted before and after the insurance scheme was introduced, we are able to make a much weaker exogeniety assumption (cf. Blundell and Costa Dias 2000). Instead of (5), the regression analogue of what we have is: and so that if ∆ denotes the difference between period _t_ =1 and _t_ =2, we get

> 7  Studies relying on the non-linearity of the Heckman model to secure identification include the studies of Waters (1999) and Jowett et al. (2003).  A number of studies explore exclusion restrictions to identify the effects of insurance in a cross-section context.  Gertler and Strum (1997) consider various possibilities, but in the end conclude they have no plausible instruments. Harmon and Nolan (2001) exclude education from the utilization equation, which they acknowledge is problematic, and resort to non-linearity as the alternative identification strategy.  Guariglia and Rossi (2004) in their analysis of the impact of health insurance on savings in the UK use as identifying instruments occupation, workplace size, whether the individual has an occupational pension, and whether he or she works in the private sector.  It is not at all obvious why any of these ought to be excluded from the savings equation.

6