Estimation of Health Care Demand and its Implication on Income Effects of Individuals

Abstract

Zero inflation and over-dispersion issues can significantly affect the predicted probabilities as well as lead to unreliable estimations in count data models. This paper investigates whether considering this issue for German Socioeconomic Panel (1984–1995), used by Riphahn et al. (2003), provides any evidence of misspecification in their estimated models for adverse selection and moral hazard effects in health demand market. The paper has the following contributions: first, it shows that estimated parameters for adverse selection and moral hazard effects are sensitive to the model choice; second, the random effects panel data as well as standard pooled data models do not provide reliable estimates for health care demand (doctor visits); third, it shows that by appropriately accounting for zero inflation and over-dispersion there is no evidence of adverse selection behaviour and that moral hazard plays a positive and significant role for visiting more doctors. These results are robust for both males and females’ subsamples as well as for the full data sample.

Appendices

Additional Tables

Table 11 Summary Statistics

Table 12 (continued)

Table 12 Standard distributions for doctor visit for males (complete table)

Table 12 (continued)

Table 12 (continued)

Table 13 Standard distributions for doctor visit for females (complete)

Table 13 (continued)

Table 14 Predicted and original numbers of doctor visits (Standard Poisson)

Table 15 Predicted and original numbers of doctor visits (ZI Poisson)

Table 16 Predicted and original numbers of doctor visits (Standard NB2)

Table 17 Predicted and original numbers of doctor visits (ZINB2)

Table 18 (NB2) Predicted and actual probabilities

Table 19 (ZINB2) Predicted and actual probabilities

Table 20 (Poisson)Predicted and actual probabilities

Model Specifications

To account for over-dispersion and deviations from E(Y _i ) = V(Y _i ) = μ _i in the Poisson distribution, a new distribution is obtained by adding an individual unobserved effect (u _i ) to the log of the mean of the Poisson model, ln(mean _i ) = ln(μ _i ) + ln(u _i ). Thus, by defining different distributions for u _i , new versions of the Poisson distribution are created. Table 21 presents a list of those distributions, known as standard distributions in this paper, with their variances. A Gamma distribution for u _i , for example, gives a Negative Binomial 2 (NB2) distribution with mean μ _i and conditional variance μ _i  + αμ _i ², with the constant parameter α controlling for heterogeneity or dispersion among individuals. The additional parameter p in the Power Negative Binomial (NB-P) distribution, introduced by Greene (2008), provides NB1 or NB2 distributions when p = 1 or p = 2 , respectively. Also, the Heterogeneous NB2 model allows the heterogeneity explained by α in the NB2 distribution to be a function of the individual’s characteristics (z _i ), α = exp(z _i γ). Thus, α can vary among individuals. In special case, where ϕ → 1, the variance of Famoye’s (1995) distribution approaches to that of the NB. The Waring Negative Binomial distribution introduce by Irwin (1968) converges to NB if $k\to \frac{1}{\alpha },\rho \to \infty$. Also, if δ = 0, the GP distribution reduces to the usual Poisson distribution with parameter θ _i . (See Hilbe (2011, 2014) for more details)

Table 21 The list of standard distributions

1.1 Zero-inflated Count Models

As Hilbe (2011) discuss, the framework of zero-inflated models are based on separating zero outcomes and positive ones. The probability of zero outcomes results from the group of individuals who are not the subject of an event (Q(0) for those who do not have physician to visit), and those who are the subject of the event but with zero outcome (P(0) for those who do not visit their physicians). The two part of the model is written as:

The probability of a zero outcome for the system is given by⁸:

$$ \Pr \left(y=0\right)=Q(0)+\left\{1-B(0)\right\}\Pr (0) $$

And the probability of a nonzero count is⁹:

$$ \Pr \left(y=k;k>0\right)=\left\{1-B(0)\right\}\Pr (k) $$

A Probit or logit model estimates Q(0) while one of the standared models in Table (1) estimates Pr(k), k = 0, 1, …n. The mixture model have more power in explaining over-dispersion in the data (see also Hilbe and Greene (2008)).

Table 22 presents different zero-inflated distributions that are used in the next sections for the purpose of estimation and comparison.

Table 22 Zero-inflated distributions