We use a multinomial probit model to analyze LTC use. Although alternative specifications can allow for correlated error terms and correct for unobserved heterogeneity over time for respondents with multiple observations, we chose not to exploit the panel structure of the data because it would unnecessarily5 complicate the decomposition (e.g., Van de Poel et al., 2009). Instead, standard errors are adjusted for correlation of choices over time by clustering observations at the individual level.
Institutional differences are expected to contribute to between-country differences in LTC use as described in Section 3. They will do so because they lead to differences in the relationship between LTC use and the covariates rather than to differences in means of covariates themselves. As a first step, we compare coefficients and average partial effects (APEs) resulting from separate regression analyses for both countries. But differences in APEs estimated by nonlinear models may result from both between-country differences in coefficients and differences in the distribution of other independent variables included in the model. Therefore, we use a decomposition method for nonlinear models proposed by Yun (2004) to examine whether differences in LTC use between the Netherlands (NL) and Germany (DE) result from differences in means of covariates or in the functional relationship. The decomposition is
where Y is LTC use and X and β are the sets of covariates and coefficients, respectively. F denotes the multinomial probit. The first part represents the contribution of the difference in covariates to the difference in outcomes, and the second part represents the contribution of the difference in coefficients. Subsequently, both terms can be broken down further to identify the contribution of each variable. The detailed decomposition is based on a Taylor expansion at the sample averages and and results in sets of weights W that measure the contributions of between-country differences in means and coefficients:
where K is the number of independent variables in the model, and for variable i ; and hence (Yun, 2004). It is customary to decompose the conditional expectation into the relative contributions, but in a multinomial outcome model, this approach is not feasible. Because the values of the choice alternatives are arbitrary, the conditional expectation of this model cannot be interpreted (Bauer and Sinning, 2008). Therefore, we focus on decomposing the differences in predicted probabilities for informal care (IC) and formal care (FC) separately instead. That is, rather than decomposing , we decompose P(IC)NL − P(IC)DE and P(FC)NL − P(FC)DE, where P() denotes the probability of use. In other words, we treat each part of the multinomial probit as if it were a binary probit model. The interpretation of the results changes accordingly. Following Yun (2008), the contribution of differences in coefficients of dummy variables is normalized, and standard errors are calculated using the delta method.