Online Robustness Appendix to Are Household Surveys Like Tax Forms: Evidence from the Self Employed

Transcription

1 Online Robustness Aendix to Are Household Surveys Like Tax Forms: Evidence from the Self Emloyed October 01 Erik Hurst University of Chicago Geng Li Board of Governors of the Federal Reserve System Benjamin Pugsley Federal Reserve Bank of New York

2 Online Robustness Aendix This document serves as the online robustness aendix to our aer Are Household Surveys Like Tax Forms: Evidence from Income Underreorting of the Self-Emloyed. In this document we resent details on two additional toics that are referenced within the main text. First, we outline the rocedure we used to test for the effects of income underreorting by the self emloyed on estimates of recautionary savings. Second, we discuss how our estimating methodology relates to that of Pissarides and Weber (1989). 1. Estimating the Effect of Income Mismeasurement By Self Emloyed on Precautionary Savings Estimates To estimate the effects of income underreorting of the self emloyed on estimates of recautionary savings, we draw on the secification from Hurst et al. (010). The goal of the Hurst et al. aer was to show how the estimates of recautionary savings fall to close to zero when the self emloyed are excluded from the analysis. The rocedure used in the Hurst et al. aer was nearly identical to the rocedure used by Carroll and Samwick (1997, 1998) to rovide micro data estimates of the imortance of recautionary savings for younger households. The emirical strategy of estimating the size of recautionary balances using micro data is based on the following secification: ln( W ) = α + ασ + ασ + α ln( y ) + Z β+ u (R1) ermy transy it 0 1 it it 3 it it it where ln(w it ) is the log of a measure of household i s wealth in eriod t, ln(y it ) is the log of i s ermanent income in t, and σ ermy it and transy σ it are, resectively, measures of the variance of ermanent shocks and transitory shocks to i s income. The Z vector includes additional controls designed to cature otential household differences in references and the hum-shaed rofile of wealth over the life cycle.

3 According to the recautionary saving model, wealth is a function not only of ermanent income, but also of uninsurable income risk faced by the household. Almost all emirical studies designed to estimate the size of recautionary balances using micro data roxy uninsurable risk with either the variance of income, the variance of consumtion, or they exloit actual job loss or exectations of future job loss. For our aer, we follow Carroll and Samwick (1997, 1998) by using anel data to distinguish between the variance of ermanent and transitory shocks to income. To estimate (R1), we use data from the PSID. We examine accumulated household wealth in either 1984 or This broadens the analysis erformed in Carroll and Samwick (1997, 1998), which only analyzed household wealth accumulation within the PSID using 1984 wealth data. The measure of wealth used is total net worth, which is defined as the sum of checking and savings accounts, bonds, stocks and mutual funds (including IRAs), home equity, other real estate, business equity, cars and other vehicles, and other assets, minus the value of all debts. Since we use logs, we exclude households who have negative or zero net worth in our samle, which amount to a little more than five ercent of our samle. Following equation (R1), we regress the log of household wealth in year t (either 1984 or 1994) on both ermanent income and measures of the variance of income. We construct ermanent income for each household by taking the seven-year average of non-caital income around the eriod for which we are measuring their wealth. Secifically, when exlaining 1984 (1994) wealth holdings, we define ermanent income as the average of non-caital income between the years of 1981 and 1987 (1991 and 1997). We use anel data from the PSID to

4 comute the variances of ermanent and transitory shocks to income. We follow the same rocedure ut forth by Carroll and Samwick (1997, 1998). 1 Since both ermanent income and the variances of ermanent and transitory income are measured with error, we instrument for these variables using a large instrument set. As suggested by Carroll and Samwick (1997, 1998), we use occuation dummies and these dummies interacted with age and age squared, as well as industry dummies. In addition, we use the unemloyment rate in the county of residence during the rior year, the variance in the county unemloyment rate over the samle eriod, and a dummy for whether the head belongs to a union. When estimating (R1), we also include additional controls (Z) to cature additional reasons why household wealth may differ across households. The Z vector includes the following demograhics: age, age squared, race, gender, marital status, and educational attainment. In addition, we exloit the anel dimension of the PSID to control for ast income and wealth shocks exerienced by households. Secifically, we include year dummies, along with two dummies for whether the household head was unemloyed during the year when the wealth data were collected and whether they were unemloyed any time during the rior four years ( or ). Households that are more likely to face high income risk are also more likely to have been hit by ast negative income shocks, and this may weaken the estimated relationshi between wealth and risk. We also include dummies for ast ositive shocks, such as having received inheritances or other lum-sum ayments. These were the same included when Hurst et al. (010) estimated their version of (R1). 1 See the data aendix to Hurst et al. (010) for a detailed summary of how the income variances were comuted.

5 Lastly, similar to Carroll and Samwick, we restrict our samle to households whose head is between the ages of 6 and 50 in the year in which the wealth is measured. A detailed descrition of other restrictions we used in constructing our final samle is reorted in the data aendix to Hurst et al. (010). Our final samle includes,144 households. The base results in the aer are identical to the ones reorted in Hurst et al (010). To assess the effects of the underreorting of income by the self emloyed on the recautionary savings estimates, we inflated the income measures of the self emloyed by 5 ercent. Otherwise, the secification was identical to the base secification.. Comarison to Pissarides and Weber (1989) Method Pissarides and Weber (1989), PW hereafter, use a similar Engel curve-based aroach to detect income underreorting of the self-emloyed. The main difference between their identification method and the method in this aer is the treatment of transient income volatility of emloyees and the self-emloyed. Our estimates of underreorting decrease only slightly after accounting for differences of income volatility using the PW method. income as The Hurst, Li, Pugsley (01), HLP hereafter, identification method models reorted log y = logκ + log y +Ω X + ν k k with E[ ν log y, X ] = 0 for k = W, S, and logκ = 0. This embeds two imortant restrictions: (1) conditional on a level of ermanent income log W y, the exected transient log deviations in income are equal to zero, and () that the fraction reorted is constant within

6 grous at 1 and κ for workers and self-emloyed resectively. Under these assumtions, κ is identified as γ κ = ex β where γ and β are the coefficients from the Engel curve regression. PW make the following arametric assumtions in their model of reorted income with log y = logκ + log y + ν, ν µ + σ ζiwt if k = W = µ + σ ζ ist if k = S, with ζ indeendent standard normal and µ σ = µ + σ. The urose of this reorting assumtion is to ensure that E ( ν ) y ex y, X, k does not deend on grou k, i.e., conditional on a level of ermanent income (and other individual characteristics), exected annual income is equal across grous. HLP instead assume that log E y + ν y, X, k does not deend on k. The distinction is relevant if there are large differences in income volatility across grous because of the Jensen s inequality correction. Further, PW allow underreorting to vary within grou by assuming logκ 0 if k = W = µ ist + σω κ ist if k = S logκ can also reresent the average of the log fraction reorted, so long as individual deviations from logκ are zero on average.

7 with ω indeendent standard normals. 3 Again, the uncertainty is relevant when comuting E[ κ ] the exected fraction reorted from logκ. Under these assumtions, E[ κ ] is identified as γ 1 E[ κ] = ex ( µ µ ) σ κ β + + where µ µ is the volatility correction needed to make the additional variance of log selfemloyed transient income fluctuations a mean reserving sread in levels, and the 1 σ κ adjusts for the additional uncertainty in reorted income due to variation in underreorting among the self-emloyed. 1 µ µ + σκ term can be identified off of differences in reorted income The ( ) volatility assuming income underreorting differences are uncorrelated with transient income volatility for the self-emloyed. Let σ yk denote the variance of the error from a regression of reorted income on all the covariates and exogenous instruments. The error includes the unredictable comonent of ermanent income η ik as well as reorting and transient income shocks σ = Var[ η + σ ζ + σ ω ] yk ik k ik κ ik with σ κw = 0. Because ermanent income is uncorrelated with reorting and transient income shocks the errors, then assuming that the variance of ermanent income shocks is equal for both grous then σ σ = σ + σ ρ σ σ σ. ys yw κ κ κ 3 PW are actually interested in k = 1/ κ the adjustment factor to aly to reorted income. Of course, both variables are log normal, only the sign of

8 1 by assumtion, then With µ µ = ( σ σ ) 1 1 µ µ + σκ = ( σ ys σ yw + ρκσ σκ) after cancelling the σ terms. If we assume ρ = 0, that underreorting and income volatility κ are uncorrelated for the self emloyed, then κ γ 1 E[ κ] = ex + ( σ ys σ yw ) β So under these assumtions, the HLP estimates of underreorting are biased u to the extent that self-emloyed transient income volatility exceeds that for workers. We estimate E[1 κ] using the estimated coefficients from the Engel curve regression 1 and an adjustment ( σ ys σ yw ) estimated from residuals of reorted earnings on the controls and instruments using the 1 year and 3 year averages from the PSID. 4 Table R1 shows the original estimate of 1 κ, and the adjusted value of E[1 κ] after correcting for differences in transient income volatility using the PW method. Since income volatility is higher among the self-emloyed, the adjustment attenuates the estimate of underreorting: the unreorted fraction falls from 3 ercent to 1 ercent instrumenting for total family income in the one year samle. The effect is more modest in the 3 year samle we use a 3 year average of total family income in lace of the instrumented annual total family income. Although the correction under the PW assumtions is small, the effect may be even smaller when differences in ermanent income volatility are considered. If the unredictable 4 This assumes that ermanent log income volatility is constant across grous, to the extent that ermanent log income volatility of the self emloyed is greater than that of workers, estimates of 1 ( σ σ ) ys yw are biased u.

9 comonent of ermanent income is more volatile for the self emloyed, i.e., σ ositive, this further weakens the PW correction. σ is ηs ηw We try to estimate this difference using residuals of consumtion on the controls and instruments. The error includes unredictable comonent of ermanent income scaled by the income elasticity, as well as other unobserved indeendent determinants of consumtion. If we assume that the second comonent of consumtion volatility is constant across grous then we can estimate the σ then σ from residuals normalized by the ηs ηw β. When this quantity is non zero ( σ σ ( σ σ )) 1 [ κ] = γ ex + ys yw S W β E η η The last row of Table R1 shows the corrected estimates of E[1 κ] after incororating the estimate of σ σ. Underreorting increases from 1 to 3 ercent using the one year ηs ηw samle. With this adjustment, while estimated underreorting is still smaller than the HLP estimates, the differences are small in magnitude. Overall, the effect of adjusting for the differences in volatility between the grous and exlicitly incororating heterogeneity in underreorting has small quantitative effects documented in Table R1. We have also assumed that the fraction reorted is uncorrelated with the transient income shock ρ κ = 0. If underreorting is higher (fraction reorted is lower) during good years, this would actually increase estimates of underreorting.

10 References Carroll, Christoher., and Andrew Samwick (1997), The Nature of Precautionary Wealth, Journal of Monetary Economics, 40(1), Carroll, Christoher and Andrew Samwick (1998). How Imortant is Precautionary Saving? Review of Economics and Statistics, 80(3), Hurst, Erik, Annamaria Lusardi, Arthur Kennickell, and Francisco Torralba (01). The Imortance of Business Owners in Assessing the Size of Precautionary Savings, Review of Economics and Statistics, 9(1), Pissarides, Christoher and Guglielmo Weber (1989). An Exenditure Based Estimate of Britain's Black Economy, Journal of Public Economics, 39(1),

11 Table R1: Alternative Estimates of 1-κ with PW Income Volatility Adjustment Labor + Business Income Total Family Income ˆ σ ˆ σ Estimate 1 Year 3 Year Averages 1 Year 3 Year Averages ys yw ˆβ ˆ γ ˆ σ ˆ σ cs cw HLP 1 κ 37.7 % 36.0% 31.5 % 9.4 % PW E[1 κ] 8.0 % 9.1 % 0.9 % 1.8 % PW E[1 κ] with ermanent income adjustment 30.4 % 3.7 % 3.4 % 5.6 %