Household income requirements and financial conditions

Abstract

Understanding the income requirements of households is important for examining why households become financially stressed and liquidity constrained. Our econometric approach relies on actual incidences of household-specific financial stress to determine household income requirements. Using an extensive longitudinal dataset of Australian households, we find significant lifecycle effects in income requirements and identify the household types which benefit or are disadvantaged by the typical 30% debt to income measure of financial stress. We also find that, in general, households are locked into tight spending patterns such that, with the exception of households in the top income quintile, financial stress occurs when negative income shocks exceed 30%.

Appendix: Approach to model estimation

Model estimation broadly follows the approach used in Buera et al. (2011). The relevant steps for the model proposed in this paper are:

1. Formulate a binary decision rule that involves unobserved information, subject to the restriction that the decision or outcome is observed. In our case, the decision rule is given by

$$\begin{aligned} m_{it}=1\left( h_{it}^{y^{*}}>h_{it}^{y}\right) , \end{aligned}$$

where $m_{it}$ is an observed instance of financial stress and $h_{it} ^{y^{*}}$ is not observed.

2. Specify a stochastic functional form for $h_{it}^{y^{*}}$. In our case,

$$\begin{aligned} h_{it}^{y^{*}}=f_{it}^{h}+\gamma _{0i}+x_{it}^{\prime }\gamma +z_{it}^{\prime }\beta +u_{it}, \end{aligned}$$

where $u_{it}\sim N\left( 0,\eta _{i}^{2}\right)$.

3. Given that $u_{it}$ is Gaussian and the decision rule for $m_{it}$ is binary, the conditional density of $m_{it}$ is given by

$$\begin{aligned}&L_{i}\left( m_{i}|\beta ,\gamma ,\gamma _{0i},\eta _{i},X_{i}\right) \\&\quad = {\displaystyle \prod \limits _{t=t_{0i}}^{T_{i}}} \Phi \left( \frac{\gamma _{0i}+x_{it}^{\prime }\gamma +z_{it}^{\prime } \beta +f_{it}^{h}-h_{it}^{y}}{\eta _{i}}\right) ^{m_{it}}\\&\qquad \times \left( 1-\Phi \left( \frac{\gamma _{0i}+x_{it}^{\prime }\gamma +z_{it}^{\prime }\beta +f_{it}^{h} -h_{it}^{y}}{\eta _{i}}\right) \right) ^{1-m_{it}}. \end{aligned}$$

4. Specify priors for the model parameters $\beta ,\gamma ,\gamma _{0i},\eta _{i}$. It is natural to specify a Gaussian prior for $\gamma _{0i}$ and an inverse-gamma prior for $\eta _{i}$. We change this slightly, specifying a Gaussian prior for $\widehat{k}_{it}$ rather than $\gamma _{0i}$ since it is easier for the econometrician to consider the prior distribution of non-housing income requirements $\widehat{k}_{it}$ rather than the individual parameters $\gamma _{0i}$. In our case, we specify

$$\begin{aligned} \widehat{k}_{it}&\sim N\left( \mu _{0},\sigma _{0}^{2}\right) , \\ \eta _{i}^{2}&\sim IG\left( n_{0},s_{0}\right) {.} \end{aligned}$$

5. Formulate the posterior density given by the product of the model’s likelihood function and the prior densities and choose the model parameters that maximise the resulting posterior density.