Resilience and Recovery

Household Income Dynamics in the Survey of Income and Program Participation

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

  1. How long does it take for households to return to their pre-shock level of income?
  2. 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
  • Assess heterogeneity in recovery time across household types
  • Account for measurement error in self-reported income (Lee et al. 2009)

Data

Survey of Income and Program Participation (SIPP)

  • Multistage-stratified sample of US population

  • Rotating panels of 14,000 - 37,000 households last 2.5-4 years

  • Explore changes across economic environment with cohorts: 2004 and 2008

    • 2004 cohort: Feb 2004 - Jan 2008
    • 2008 cohort: May 2008 - Nov 2013

Sample composition

  • Households with head age 25-55, not currently enrolled full-time in school nor on active duty

  • Aggregate data to every four months due to seam bias

Variables

  • Outcome: Total real household income

  • Time-varying covariate: Household size

  • Demographic characteristics:

    • Race

    • Education

    • Marital status

    • Metro status

Methods

Empirical Strategy

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

Income Process

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

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

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

\(X_{it}\): household size

\(\delta_{it}\): 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 \(s=3, \ldots, 8\)

Estimation

Generalized Method of Moments (GMM)

  • Coefficient estimates: \(\gamma\), \(\beta\), and \(d_t\)
  • Estimated residuals \(\Delta \tau_{it}\) form moment conditions to estimate \(\sigma^{2}_{v}, \sigma^{2}_{u}, \text{ and } \sigma^{2}_{\epsilon}\)

\[\sigma^{2}_{v} = \gamma^{-1} E[\Delta \tau_{it} \Delta \tau_{it-2}] \] \[\sigma^{2}_{\epsilon} = -(E[\Delta \tau_{it} \Delta \tau_{it-1}] + \sigma^{2}_{v}(1 + 2\gamma + \gamma^{2}))\]

\[ \sigma^{2}_{u} = E[(\Delta \tau_{it})^{2}] - 2\sigma^{2}_{v}(1 + \gamma + \gamma^{2}) - 2\sigma^{2}_{\epsilon}\]

Simulate Measurement-Error-Free Income

  • True income path: \[y^{*}_{it} = y^{*}_{it-1} + \Delta y^{*}_{it}, \quad t \geq 3\]
  • Initial values require a linear projection:
\[\begin{align*} \Delta y_{i2} &= \delta_{0} + \beta'_{0} X_{i1} + \beta'_{1} X_{i2} + \Delta v_{i2} + \zeta_{i1} \\ y_{i2} &= \delta_{1} + \beta'_{2} X_{i1} + \beta'_{3} X_{i2} + e_{i} + v_{i2} + \zeta_{i2} \end{align*}\]

  • Define estimating equation errors: \(\psi_{1} = \Delta v_{i2} + \zeta_{i1}\); \(\psi_{2} = e_{i} + v_{i2} + \zeta_{i2}\)

  • Projection error variances:

\[\begin{align*} \sigma^{2}_{1 } &=var(\psi_{1}) - 2 \sigma^2_{v} \\ \sigma_{12} &= cov(\psi_{1},\psi_{2}) - \sigma^{2}_{v} \\ \sigma^{2}_{2} &= var(\psi_{2}) - \sigma^{2}_{e} - \sigma^{2}_{v} \end{align*}\]

Projection Error Bounds

  • Upper bound on \(\sigma^{2}_{2}\) is when \(\sigma^{2}_{e}=0\) \[\sigma^{2}_{2} \leq Var(\psi_{2}) - \sigma^{2}_{v}.\]

  • Lower bound on \(\sigma^{2}_{2}\): Variance matrix \(\Sigma\) of simulated errors must be positive semi-definite.

  • Observed income: set \(\sigma^{2}_{v}=0\).

Shock Simulation

  • Simulate paths of \(y^{*}\) using parameter and variance distributions

  • Introduce a negative income shock at t=2 and trace out the income path

  • Recovery time: how long to return to \(y_{i1}\) after shock in t = 2

  • Repeat the process by splitting the sample by demographic characteristics

  • Compare estimates across different data periods

Results

Main Sample

Figure 1: Corrected income with and without the shock for one replication of one household using the upper bound of the projection error variance for a 10% shock. Each wave of the sample is four months.

Figure 2: Total Real Household Income by Sample Wave for Measurement Error (ME) Correction Types Averaged across Replications for the 25th and 75th Distribution Quantiles. Note: Upper is the upper bound of the projection error variance, none is no measurement error correction, and lower is the lower bound of the projection error variance.

Table 1: Recovery Proportion by Shock Size and Projection Error Variance Bound
10% 25%
Upper 0.57 0.53
Lower 0.60 0.52
None 0.65 0.62

Recovery Speed

Measurement error

Figure 3: Main Sample Recovery Speed Distribution Varying the Measurement Error Correction by 10% shock for the 2008 Data. A recovery period is four months.

Economic Environment

Figure 4: Main Sample: Recovery Speed Distribution for 2004 and 2008 SIPP Data Release. Shock size is 10% and no measurement error correction is used. A recovery period is four months.

Half-Life Recovery

Figure 5: Main Sample Full and Half-Life Recovery Speed Distributions on Lower Bound of the Projection Error Variance. A recovery period is four months.

Recovered Income

Figure 6: Percentage of Recovered Income by the Final Period. Values greater than one represent a complete recovery.

Income Loss

(a) Household

(b) Aggregate

Figure 7: Income Loss until Recovery Occurs for a 10% Shock using the Upper Bound of the Projection Error Variance on the 2008 Data.

Heterogeneity Results

Race

Table 2: Recovery Proportion by Race for a 10% shock with the Upper Bound on the Projection Error Variance
Not recovered Recovered
Black 0.31 0.69
White 0.45 0.55

Figure 8: Recovery Speed Distribution by Race for a 10% shock with the Upper Bound on the Projection Error Variance. A recovery period is four months.

Education

Table 3: Proportion recovered by education for a 10% shock with the upper bound on the projection error variance
Not recovered Recovered
College 0.44 0.56
High School 0.42 0.58

Figure 9: Recovery Speed Distribution by Education for a 10% Shock with the Upper Bound on the Projection Error. A recovery period is four months.

Marital Status

Table 4: Proportion Recovered by Marital Status for a 10% with the Upper Bound on the Projection Error Variance
Not recovered Recovered
Married 0.45 0.55
Single 0.43 0.57

Figure 10: Recovery Speed Distribution by Marital Status for a 10% Shock and the Upper Bound on the Projection Error Variance. A recovery period is four months.

Metro

Table 5: Recovery Proportion by Metro status for a 10% shock and the Upper bound on the Projection Error Variance
Not recovered Recovered
Metro 0.47 0.53
Non-metro 0.17 0.83

Figure 11: Recovery Speed Distribution by Metro Status for a 10% shock and the Upper Bound on the Projection Error Variance. A recovery period is four months.

Decomposition

Hypotheses

What explains group differences?

  • Parameters and income persistence
  • Random Errors
  • Observable characteristics

Table 6: Demographic Decomposition by Subgroup Analysis Relative to the Baseline Reference Group in Each Demographic Category

(a) Recovery Times
Gamma Beta Time Trend Random Error Time Effect Xit yi0
Education 1.06 1.00 0.86 1.24 0.78 1.01 1
Marital 1.03 1.01 0.79 1.39 0.90 1.02 1
Metro 0.77 0.99 0.75 1.47 0.76 1.01 1
Race 1.02 1.00 0.84 1.37 1.13 1.01 1
(b) Recovery Proportion
Gamma Beta Time Trend Random Error Time Effect Xit yi0
Education 0.95 1.00 0.66 0.90 1.38 1 1
Marital 0.98 1.00 0.85 0.86 1.13 1 1
Metro 1.33 1.01 1.17 0.59 1.49 1 1
Race 0.99 1.00 0.89 0.91 0.99 1 1

Conclusion

Results summary

  • Measurement error corrections bound the observed income estimates
  • Households recover longer for 25% shock than 10% shock
  • Economic environment shows similar results

Heterogeneity Summary

  • Whites recover faster than black household heads
  • High school education recovers faster than college education
  • Married household heads recover faster than single
  • Metro area household heads recover faster than non-metro

Future Work

  • Connect income resiliency to intergenerational measures
  • Mechanisms of heterogeneity results
  • Non-linear model for income

Conclusion

  • Model income shock recovery speed
  • Income dynamics with measurement error
  • Policy relevant characteristics for heterogeneity in recovery speed
  • Mechanism: shocks are the driving factor
  • Accounts for measurement error in unbiased recovery speed distribution

Appendix

Mean Reverting Measurement Error

(a) Mean-reverting ME

(b) Main Results

Figure 12: Main Sample Recovery Speed Distribution of 10% Shock with Mean-Reverting and non-Mean Reverting Measurement Error (ME) by ME Type for 2008 data. A recovery period is four months.

Empirical Analysis

Table 7: 2008 Empirical Recovery times.
2008
Proportion small shocks 0.57
Proportion medium shocks 0.03
Proportion large shocks 0.03
Proportion any shocks 0.64
Proportion recover first small shock 0.18
Proportion recover first medium shock 0.11
Proportion recover first large shock 0.16
Proportion recover first shock 0.18
Recovery time (months) first small shock 1.23
Recovery time (months) first medium shock 3.13
Recovery time (months) first large shock 2.52
Recovery time (months) first shock 1.18
Number of small shocks 8.48
Number medium shocks 0.49
Number large shocks 0.48
Number any shocks 9.45

Robustness across Sample Types

Table 8: Robustness across Sample Types for the Recovery Proportion on a 10% Shock with the Upper Bound of the Projection Error Variance.
Recovered
Employed 0.56
No change in HH 0.54
Allow self employed 0.66
No change in marital 0.58
Main Sample 0.56

Figure 13: Robustness across Sample Types for a 10% Shock on the Upper Bound of Projection Error Variance. A recovery period is four months.