Household Resilience and Recovery

Income Dynamics in the SIPP

Kristina Bishop

Introduction

Income inequality is an economic concern
Income inequality depends on factors such as:
- Distribution of shocks
- Effect of shocks*
  - Income loss is difficult to offset
  - Long recovery from income loss widens the income inequality gap

Motivation

Policy can be designed more effectively to aid recovery:
- Who recovers slower?
- How long does it take for them to recover?
- How much income to give?
- Is income aid the right policy?

Research Questions

How long does it take for households to return to their pre-shock level of income?
What characteristics are common of households who recover quickly from income shocks?

Contribution

Provide an empirical model to estimate the speed of recovery from shocks
Account for measurement error in self-reported income (Lee et al. 2009)
Assess heterogeneity in recovery time across household types

Methods

Empirical Strategy

Model the income process with measurement error
Simulate corrected income values
Introduce a shock into the initial period
Recovery times across households and periods

Income Process

\[\begin{align*} %y^{*}_{it} &= \alpha_{i} + \gamma_{1} y^{*}_{it-1} + \gamma_{2} (y^{*}_{it-1})^{2} + X_{it} \beta + \delta_{it} + \tilde{\lambda} m_{it} + \epsilon_{it} \quad t = 2, \ldots, T \\%\label{eq:true} y^{*}_{it} &= \gamma y^{*}_{it-1}+ X_{it} \beta + \alpha_{i} + \delta_{it} + \epsilon_{it} \quad t = 2, \ldots, T \\ \delta_{it} &= \delta_{it-1} + u_{it} + d_{t} \nonumber \end{align*}\]

\(y^{*}_{it}\): measure of true income of household \(i\) at \(t\)

\(\alpha_{i}\): household fixed effect

\(X_{it}\): time-varying household characteristics

\(\delta_{it}\): household-specific stochastic time trend

\(u_{it}\): time-varying shock with permanent effect on income

\(d_{t}\): average time effect across households

\(\epsilon_{it}\): random shock

Measurement Error

Data measured imprecisely: recall, misinformation, survey methods
Income contains non-classical measurement error
Measurement error is mean-reverting and serially correlated (Bound and Krueger, 1991; Cristia and Schwabish, 2007)

Measurement Error

Model

\[\begin{align*} y_{it} &= y^{*}_{it} + e_{i} + v_{it} \end{align*}\]

\(y_{it}\): observed, actual income

\(y^{*}_{it}\): unobserved, true income

\(e_{i}\): time-invariant component

\(v_{it}\): time-varying component

Estimation

Model of observed total household income:

\[\begin{align*} y_{it} &= \alpha_{i} + \gamma y_{it-1} + X_{it} \beta + \delta_{it} + \tau_{it} \quad t = 2, \ldots, T \\ \tau_{it} &= (1 - \gamma_{1})e_{i} + v_{it} - \gamma_{1} v_{it-1} + \epsilon_{it} %\nonumber \end{align*}\]

First difference the model: \(\Delta y_{it} = y_{it} - y_{it-1}\)

Estimating equation using Two-step GMM

\[\begin{align*} \Delta y_{it} &= \gamma_{1} \Delta y_{it-1} + \Delta X_{it} \beta + d_{t} + \Delta \tau_{it} \quad {t = 3, \ldots, T} \\ %\label{eq:delta_yit} \Delta \tau_{it} &= u_{it} + \Delta v_{it} - \gamma \Delta v_{it-1} %nonumber + \Delta \epsilon_{it}, %\nonumber \end{align*}\]

Instruments (Arellano and Bond, 1991): \(y_{i,t-s}\) for \(s=3, \ldots, 8\)