Proposed modified probit model incorporating non-parametric density estimation: how to measure asymmetric information in the health insurance market?

Abstract

On the basis of the theory of Chiappori and Salanie (2000), this paper proposes a simple modified bivariate Probit model incorporating non-parametric kernel density estimation. The model is applied to test asymmetric information in a health insurance market, using MEPS96 data.1 Results show that asymmetric information (whether moral hazard or adverse selection) exists between the contract of insurance coverage, and some non-emergency visits services, which appear to support the conclusions of Cardon and Hendel (2001). It is also shown how this non-parametric approach plays an important role in the delicate task of correctly testing, by computing generalized residuals, the existence of asymmetric information.

Acknowledgements

The author thanks Professor Ming Li, Professor Nagesh Revankar and Professor Young Yin for their lectures.

Notes

MEPS96: Medical Expenditure Panel Survey data in 1996.

According to this definition, a notion of so-called ‘herd behavior’ may belong to AI.

For classical examples, see Ichimura (1987), or Klein and Spady (1987), both of which are referenced by Newey et al. (1990).

Chiappori and Salanie (2000) assume the following form of bivariate Probit model.

The terms, X_i β and X_i γ are considered to be a decision-making function, whose values plus disturbance term, if they exceed a critical value, influence each agent's final decision. Of course, this functional form implicitly assumes the linearity and separability between explanatory variables.

So, ε _i ∼ N(0, 1) and η _i ∼ N(0, 1).

Assuming other distributions as in Logit model (logistics distribution), we cannot derive such an explicit closed form.

Here the subscript i is used simply in the sense that the values of dependent variables and explanatory variables of interest are not limited in those which were actually taken in each observation.

A normal distribution is adopted as a kernel function, among other candidates such as Epanechinikov, biweight, triangular or rectangular. Although it is slightly asymptotically inefficient compared with Epanechinikov, it enables smoothness (differentiability) over all possible values of explanatory variables, which might seem also to enable further development of a specification test regarding semi-parametric (partially linearized) regression setting.

For the theoretical detail of a generalized residual, see Gourieroux et al. (1987).

Observations whose total income (TTLPNX) was less that $5000 were deleted because dependants, especially children, may not have decision-making power regarding insurance coverage. Also deleted were observations more than 60 years old, in order to avoid statistical bias which might arise from the Medicare/Medicaid programme.

HEALTH is calculated as the weighted sum of the variables regarding health status for round 1 and 2.

Both y_i and z_i are, for convenience, converted into a binary variable, but we can easily extend the model into a multiple discrete choice model. Here y_i is 1 if the agent (the observation) covers a private insurance, and 0 otherwise. z_i is 1 if the number of visits is more than or equal to 1, 0 otherwise.

These results are rather different from those shown in the presentation, because there were some errors in the course of computation. I would like to apologize to the participants for it.

For the regular Probit, the whole 9831 data points were used for regression.

The policy for choosing the bandwidth h was arbitrary, although one popular criterion is to choose

.

In OBTOV, the test statistic was 15.8530 for 2000 data points and 7.0909 for 1000 data points. This was a rather drastic change in test statistics, although it does not overturn the conclusion that AI exists.