How Much Does Household Collateral Constrain Regional Risk Sharing? Separate Appendix for Non-Separable Preferences

In the main text, we focus on the case of preferences that are separable between non-housing and housing consumption streams {ct(s)} and {ht(s)}. Here we extend the analysis to the case of non-separable preferences. We use two approaches to derive testable implications from the model. The first approach links our model to the traditional risk-sharing tests based on linear consumption growth regressions, the workhorse of the consumption insurance literature. That is, we make assumptions on the regional consumption share processes needed to derive a linear consumption growth equation from the model. This equation linearly relates regional consumption share growth and regional income share growth, conditional on the housing collateral ratio. We refer to this as the linear model (section 4). We estimate the linear risk-sharing regressions for the case of non-separable preferences, both using aggregate collateral measures and regional collateral measures. We confirm the findings of the main text: even with non-separable preferences do we find evidence for a time-varying degree of risk-sharing between U.S. metropolitan areas. In the second approach (section 5) we fully incorporate the model’s non-linear dynamics in the estimation, and we use a simulated method of moments estimation to estimate the non-linear law of motion for consumption shares . We refer to this as the non-linear model. This is new approach to structural estimation of non-linear risk-sharing models.