In Search of Lost Time Aggregation

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. In Search of Lost Time Aggregation Edmund Crawley 2019-075 Please cite this paper as: Crawley, Edmund (2019). “In Search of Lost Time Aggregation,” Finance and Economics DiscussionSeries2019-075. Washington: BoardofGovernorsoftheFederalReserveSystem, https://doi.org/10.17016/FEDS.2019.075. NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

In Search of Lost Time Aggregation By Edmund Crawley∗ October 1, 2019 In 1960, Working noted that time aggregation of a random walk induces serial correlation in the first difference that is not present in the original series. This important contribution has been overlooked in a recent literature analyzing income and consumption in panel data. I examine Blundell, Pistaferri and Preston (2008) as an important example for which time aggregation has quantitatively large effects. Using new techniques to correct for the problem, I find the estimate for the partial insurance to transitory shocks, originally estimated to be 0.05, increases to 0.24. This larger estimate resolves the dissonance between the low partial consumption insurance estimates of Blundell, Pistaferri and Preston (2008) and the high marginal propensities to consume found in the natural experiment literature. JEL: C18, D12, D31, D91, E21 Keywords: Income, Consumption, Time Aggregation ∗ Federal Reserve Board, 20th Street and Constitution Avenue N.W., Washington, DC 20551, edmund.s.crawley@frb.gov. Theanalysisandconclusionssetfortharethoseoftheauthoraloneanddonot indicateconcurrencebyothermembersoftheresearchstaffortheBoardofGovernors. Manythanksto ChrisCarrollforsupportandguidance. 1

2 OCTOBER 2019 In a short note in Econometrica, Working (1960) made the simple but important point that time aggregation can induce serial correlation that is not present in the original series. This fact was readily absorbed by the macroeconomic literature, where time aggregated series are common1 and a small literature has grown around how to account for time aggregation in various settings.2 However, the effect of time aggregation has been overlooked in much of the literaturestudyingthecovariancestructureofhouseholdincomeandconsumption dynamics.3 This oversight can result in significant bias. I examine Blundell, Pistaferri andPreston (2008) (henceforth BPP)not only as away todemonstrate newtechniquestoovercomethebias, butalsobecausetheconsumptionresponses to transitory and permanent income shocks are of significant economic interest in themselves. Indeed, Kaplan and Violante (2010) argue that “the BPP insurance coefficients should become central in quantitative macroeconomics”. Using the same Panel Study of Income Dynmics (PSID) data as in BPP, I update their underlying model to account for time aggregation. I find the estimate for partial insurance to transitory shocks, originally estimated in BPP to be 0.05, to be 0.24 when time aggregation is accounted for. This new estimate resolves the dissonance between BPP’s “full insurance of transitory shocks” and a parallel literature that, using natural experiments, finds large consumption responses to transitory income shocks.4 While this paper will focus on the implications of time aggregation for the methodology in BPP, the techniques can be applied to a broad swath of the literature. 1ForanexampleseeCampbellandMankiw(1989) 2AsampleofthisliteratureincludesAmemiyaandWu(1972),Weiss(1984)andDrostandNijman (1993). 3TheliteraturegoesbacktoearlyworksuchasHause(1973),WeissandLillard(1979)andMaCurdy (1982)thatlookatthecovariancestructureoftheincomeprocess. FollowingBPP,anumberofpapers havelookedatincomeandconsumptiontogether,forexampleArellano,BlundellandBonhomme(2017) . 4AsmallsampleofthisliteratureincludesParkeretal.(2013),AgarwalandQian(2014)andSahm, Shapiro and Slemrod (2010). Consumers also answer that they have a high marginal propensity to consume when asked, see Fuster, Kaplan and Zafar (2018) and Jappelli and Pistaferri (2014). For an overviewoftheentireliteratureonconsumptionresponsestoincomeshocks,seeJappelliandPistaferri (2010). Note the dissonance between BPP and the natural experiment literature is also addressed by Commault (2017). In contrast to this paper, her approach makes structural changes to the underlying modelbutdoesnotaddresstimeaggregation.

VOL. NO. IN SEARCH OF LOST TIME AGGREGATION 3 $100,000 $75,000 $50,000 $25,000 Permanent Income Flow Observed Annual Income $0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Time Figure 1. Income Flow and Observed Income I. What is Time Aggregation? Time aggregation occurs when a time series is observed at a lower frequency thantheunderlyingdatathatgeneratesit. Forexample, incomeisoftenobserved at an annual frequency when it may in fact consist of paychecks arriving at a monthly, biweekly or irregular timetable. To transform income into an annual frequency, we sum up all the income that was received by a household during the year. The key insight of Working (1960) is that even if there is no correlation between changes in income at the underlying frequency, changes in the resulting time aggregated series will show positive autocorrelation. The intuition behind thiscanbeseeninfigure1,showingtheincomeprocessofahouseholdthatbegins with an annual salary of $50,000 and receives a permanent pay rise to $100,000 mid-way through the second year. The solid line shows this jump in income

4 OCTOBER 2019 flow occurring just once. The crosses show the income we actually observe in annual data. During the second year the household receives an annual $50,000 salary for six months, followed by $100,000 in the second six months, resulting in a reported income of $75,000 for the entire year. The single shock to income therefore appears in the time aggregated data as two increases. In this way, an income change in one year is positively correlated with an income change in the following year, even if the underlying income process follows a random walk. II. Modelling Time Aggregation in Blundell, Pistaferri and Preston (2008) A. The Model in Discrete Time Without Time Aggregation Here I briefly describe the method used by Blundell, Pistaferri and Preston (2008)toestimatehouseholdconsumptionresponsestopermanentandtransitory income shocks. The model described here is a simplified version of the original in order to highlight the role played by time aggregation.5 Thecoreofthemodelistheassumptionsmadeontheincomeandconsumption processes. Unexplained log income growth for household i follows the process: ∆y = ζ +∆ν i,t i,t i,t where ζ (the change in permanent income) and ν (transitory income) are each i,t i,t mean zero, finite variance, i.i.d. and independent of each other. The unexplained change in log consumption is modeled as a random walk that moves in response to permanent and transitory income shocks: ∆c = φζ +ψν i,t i,t i,t whereφandψ arethepartial insurance parameters. Avalueofzeroforeitherφor 5InthissimplifiedmodelIassumeinsuranceparametersareconstantacrossbothtimeandhouseholds, thatthetransitorycomponentofincomehasnopersistence, andthattherearenotasteshocks. These elementsarereintroducedinsectionIIIinwhichIshowthequantitativeeffectoftimeaggregation.

VOL. NO. IN SEARCH OF LOST TIME AGGREGATION 5 ψ impliesfullinsurancetopermanentortransitoryshocksrespectively(consumption does not respond at all to the income shock), while a value of one implies no insurance. The transitory insurance parameter will be of particular interest in thisanalysis. Thecoreoftheempiricalmethodologyistoidentifytheseinsurance parameters in the data from the following identities: Cov(∆c ,∆y +∆y +∆y ) t t−1 t t+1 (1) φ = Cov(∆y ,∆y +∆y +∆y ) t t−1 t t+1 Cov(∆c ,∆y ) t t+1 (2) ψ = Cov(∆y ,∆y ) t t+1 B. The Model in Continuous Time with Time Aggregation In this section I show how time aggregation can significantly bias the partial insurance parameter estimates obtained by equations 1 and 2. The model in this section will be the exact analog of the discrete time model just described, but embedded in continuous time where shocks are spread uniformly throughout the year.6 The main result does not hinge on the use of continuous time, and similar estimates would be obtained by dividing the year into quarters or months.7 Time is continuous and one time unit represents one year. For the income process we will assume two underlying martingale processes, P and Q such that t t for all s > s > s > s > 0: 1 2 3 4 Var(P −P ) = (s −s )σ2 s1 s2 1 2 P Cov(P −P ,P −P ) = 0 s1 s2 s3 s4 P = 0 if s < 0 s and similarly for Q . Brownian motion fits these assumptions, but the slightly t 6There is little formal evidence on the distribution of shocks throughout the year. While this assumptionisunlikelytobestrictlytrue,itismorereasonablethantheimplicitassumptionofBPPthat shocksalloccur1stJanuaryeachyear. 7The autocorrelation of a time aggregated random walk is 0.25 in continuous time, compared to 0.23 for a discrete quarterly model and almost indistinguishable from a discrete monthly model. The theoreticalmomentsarehoweversignificantlymoreelegantincontinuoustime. SeeappendixA.A2

6 OCTOBER 2019 more general definition allows for jumps in the income process, such as getting promoted. Instantaneous income in a period dt is given by:8 (3) dy = P dt+dQ t t t (cid:82)t that is they receive their permanent income flow (P = dP ) multiplied by time t 0 s dt in addition to a one-off transitory income dQ . t Keepingwiththeassumptionthatconsumptionisarandomwalkwithinsurance parameters φ and ψ, instantaneous consumption is given by: (4) dc = φP dt+ψQ dt t t t that is, they consume a proportion φ of their permanent income and a proportion ψofthecumulationofallthetransitoryincometheyhavereceivedintheirlifetime (cid:82)t (Q = dQ ). t 0 s In the Panel Study of Income Dynamics (PSID) data, we observe the total income received over the previous calendar year at time T: (cid:90) T yobs = dy T t T−1 Consumption is measured by a survey at the beginning of the following calendar year, whichImaptoasnapshotofconsumptionexactlyattheendofthecalendar year:9 (5) cobs = φP +ψQ T T T 8AmoreformaltreatmentofhowtorelatethistothelogincomeprocessisgiveninappendixA.A2. 9BPP use data on food consumption to impute total annual consumption. The questionnaire asks about food consumption in a typical week, but unfortunately the timing of this ‘typical week’ is not clear. The questionnaire is usually given at the end of March in the following year. See Altonji and Siow(1987)andHallandMishkin(1982)fordifferingviews. InappendixA.A4Ishowthatthetiming ofthe‘typical’weekcanhavealargeeffectontheresults. Thisisanimportantdrawbacktousingthis method with the PSID data. In Crawley and Kuchler (2018) we use expenditure data imputed from Danishadministrativerecordsinwhichthetimingofexpenditureisveryclearlydefined.

VOL. NO. IN SEARCH OF LOST TIME AGGREGATION 7 The BPP method makes use of the changes in observable income and consumption, which in the time aggregated model relate to: (cid:16)(cid:90) T−1 (cid:90) T (cid:17) ∆yobs = (s−(T −2))dP + (T −s)dP T s s T−2 T−1 (cid:16)(cid:90) T (cid:90) T−1 (cid:17) (6) + dQ − dQ t t T−1 T−2 (cid:90) T (cid:90) T (7) ∆cobs = φ dP +ψ dQ T s s T−1 T−1 We see that these observable income and consumption changes in equations 1 and 2 recover the permanent, but not the transitory insurance parameter: Cov(∆cobs,∆yobs +∆yobs+∆yobs ) (8) T T−1 T T+1 = φ Cov(∆yobs,∆yobs +∆yobs+∆yobs ) T T−1 T T+1 Cov(∆cobs,∆yobs ) (3φ−ψ)σ2 (9) T T+1 = ψ− P Cov(∆yobs,∆yobs ) 6σ2 −σ2 T T+1 Q P Indeed the transitory insurance coefficient bears little relation to the true value of ψ. For example, if permanent and transitory variances are equal, and households follow the permanent income hypothesis (φ = 1, ψ = 0), the estimate for ψ using this method will be negative 0.6. III. Revised BPP Estimates In this section I repeat the BPP estimation proceedure, but with the model moments coming from the continuous time model with time aggregated income. While the core identification in BPP is illustrated in equations 1 and 2, the full estimation proceedure minimizes the distance between all the observable covariances(Cov(∆yobs,∆yobs),Cov(∆cobs,∆cobs)andCov(∆cobs,∆yobs))andtheir T S T S T S model implied equivalents.10 The full set of these model implied moments for the continuous time model, extended to include time varying coefficients, transitory 10I follow the exact same diagonally weighted minimum distance proceedure in BPP as described in appendixDofBlundell,PistaferriandPreston(2008)

8 OCTOBER 2019 persistence and taste shocks, can be found in appendix A.A1 and appendix A.A3. Table 1—Minimum-Distance Partial Insurance and Variance Estimates BPP Time Agg. Persistence Type: None MA(1) None Uniform Linear Decay ψ 0.0503 0.0501 0.2421 0.2510 0.2403 (Partial insurance tran. shock) (0.0505) (0.0430) (0.0431) (0.0428) (0.0417) φ 0.4692 0.6456 0.3384 0.3287 0.3516 (Partial insurance perm. shock) (0.0598) (0.0941) (0.0471) (0.0580) (0.0627) Table1showstheestimatesforthetransitoryandpermanentinsuranceparameters, firstusingBPP’soriginalmethodandthenwithtimeaggregation. Asthere isnoequivalenttoanMA(1)processincontinuoustime,Iconsidertwoalternative ways to introduce persistence in the transitory shock, as well as reporting results assuming no persistence. First I assume a transitory shock provides a stream of income uniformly distributed over a short period of time (to be estimated). Second I assume the stream of income decays linearly over a short period.11 The time aggregated results are not very sensitive to these assumptions. The top row of table 1 gives the main result showing the transitory insurance parameter increases from 0.05 in BPP to 0.24 with time aggregation. This new estimate is much more in line with the literature that estimates MPCs using natural experiments. Note that this large consumption response to transitory shocks, along with a lifetime budget constraint, is incompatible with the assumption common in both modelsthatconsumptionmovesasarandomwalk. Thismisspecificationperhaps explains why the permanent insurance parameter appears to be relatively small in the time aggregated model. In Crawley and Kuchler (2018) we suggest ways to approach this. 11SeeappendixA.A3fordetails.

VOL. NO. IN SEARCH OF LOST TIME AGGREGATION 9 IV. Conclusion This paper highlights the importance of time aggregation when working with panel data, especially when analyzing the covariance matrix of income and consumption growth. It also resolves the dissonance between BPP’s estimates of transitory income insurance and the natural experiment literature on marginal propensity to consume. Going forward, I hope the methods used here to correct for the time aggregation problem can be useful for researchers, especially as more and more high quality panel datasets on income and consumption become available. REFERENCES Agarwal, Sumit, and Wenlan Qian.2014. “Consumption andDebt Response to Unanticipated Income Shocks: Evidence from a Natural Experiment in Singapore.” American Economic Review, 104(12): 4205–4230. Altonji, Joseph G., and Aloysius Siow. 1987. “Testing the Response of ConsumptiontoIncomeChangeswith(Noisy)PanelData.”The Quarterly Journal of Economics, 102(2): 293–328. Amemiya, Takeshi, and Roland Y. Wu.1972.“TheEffectofAggregationon Prediction in the Autoregressive Model.” Journal of the American Statistical Association, 67(339): 628–632. Arellano, Manuel, Richard Blundell, and St´ephane Bonhomme. 2017. “Earnings and Consumption Dynamics: A Nonlinear Panel Data Framework.” Econometrica, 85(3): 693–734. Blundell, Richard, Luigi Pistaferri, and Ian Preston. 2008. “Consumption Inequality and Partial Insurance.” American Economic Review, 98(5): 1887– 1921.

10 OCTOBER 2019 Campbell, John Y., and N. Gregory Mankiw.1989.“Consumption,Income andInterestRates: ReinterpretingtheTimeSeriesEvidence.”NationalBureau of Economic Research, Inc NBER Chapters. Commault, Jeanne. 2017. “How Does Consumption Respond to a Transitory Income Shock? Reconciling Natural Experiments and Structural Estimations.” Crawley, Edmund, and Andreas Kuchler. 2018. “Consumption Heterogeneity: Micro Drivers and Macro Implications.” Drost, FeikeC., andTheoE.Nijman.1993.“TemporalAggregationofGarch Processes.” Econometrica, 61(4): 909–927. Fuster, Andreas, Greg Kaplan, and Basit Zafar. 2018. “What Would You Do With $500? Spending Responses to Gains, Losses, News and Loans.” National Bureau of Economic Research Working Paper 24386. Hall, Robert, and Frederic Mishkin. 1982. “The Sensitivity of Consumption to Transitory Income: Estimates from Panel Data on Households.” Econometrica, 50(2): 461–81. Hause, John. 1973. “The Covariance Structure of Earnings and the On the Job Training Hypothesis.” National Bureau of Economic Research Working Paper 25. Jappelli, Tullio, and Luigi Pistaferri. 2010. “The Consumption Response to Income Changes.” Annual Review of Economics, 2(1): 479–506. Jappelli, Tullio, and Luigi Pistaferri. 2014. “Fiscal Policy and MPC Heterogeneity.” American Economic Journal: Macroeconomics, 6(4): 107–136. Kaplan, Greg, and Giovanni L. Violante. 2010. “How Much Consumption Insurance beyond Self-Insurance?” American Economic Journal: Macroeconomics, 2(4): 53–87.

VOL. NO. IN SEARCH OF LOST TIME AGGREGATION 11 MaCurdy, ThomasE.1982.“Theuseoftimeseriesprocessestomodeltheerror structure of earnings in a longitudinal data analysis.” Journal of Econometrics, 18(1): 83 – 114. Parker, Jonathan A, Nicholas S Souleles, David S Johnson, and Robert McClelland. 2013. “Consumer spending and the economic stimulus payments of 2008.” The American Economic Review, 103(6): 2530–2553. Sahm, Claudia R., Matthew D. Shapiro, and Joel B. Slemrod. 2010. “Household Response to the 2008 Tax Rebate: Survey Evidence and Aggregate Implications.” Tax Policy and the Economy, 24: 69–110. Weiss, Andrew A. 1984. “Systematic sampling and temporal aggregation in time series models.” Journal of Econometrics, 26(3): 271–281. Weiss, Yoram, and Lee Lillard. 1979. “Components of Variation in Panel Earnings Data: American Scientists 1960-70.” Econometrica, 47: 437–54. Working, Holbrook. 1960. “Note on the Correlation of First Differences of Averages in a Random Chain.” Econometrica, 28(4): 916–918. Mathematical Appendix A1. Identification in the Full Model In this appendix I calculate the full set of identifying equations for the nonstationary model with measurement error in consumption and taste shocks. Appendix A.A3 extends these to add persistence in the transitory shock. I am interested in the full set of observable covariances: Cov(∆yobs,∆yobs) T S Cov(∆cobs,∆cobs) T S Cov(∆cobs,∆yobs) T S

12 OCTOBER 2019 forallT andS in{1,2,...}. Ifurthermaketheassumptionthatwhilethevariance ofthepermanentandtransitoryshocksandinsurancecoefficientscanchangefrom year to year, within each year these are constant. The variance the permanent shock in year T is σ2 and the transitory shock σ2 . I use equation 6 for P,T Q,T the change in observable log income, and extend equation 7 for the change in observable log consumption to include taste shocks (ξ ) and measurement error t (u ): T (cid:90) T (cid:90) T (cid:90) T ∆cobs = φ dP +ψ dQ + dξ +u −u T s s s T T−1 T−1 T−1 T−1 These two equations allow for the calculation of all the required identification equations: (cid:16)(cid:90) T−1 (cid:90) T (cid:17) Var(∆yobs) = E (s−(T −2))2dP dP + (T −s)2dP dP T s s s s T−2 T−1 (cid:16)(cid:90) T (cid:90) T−1 (cid:17) +E dQ dQ + dQ dQ t t t t T−1 T−2 1 1 (A1) = σ2 + σ2 +σ2 +σ2 3 P,T 3 P,T−1 Q,T Q,T−1 (cid:16)(cid:90) T (cid:17) (cid:16)(cid:90) T (cid:17) Cov(∆yobs,∆yobs ) = E (T −s)(s−(T −1))dP dP −E dQ dQ T T+1 s s t t T−1 T−1 1 = σ2 −σ2 6 P,T Q,T 1 Cov(∆yobs,∆yobs ) = σ2 −σ2 T T−1 6 P,T−1 Q,T−1 Cov(∆yobs,∆yobs) = 0 ∀S,T such that |S −T| > 1 T S

VOL. NO. IN SEARCH OF LOST TIME AGGREGATION 13 (cid:16)(cid:90) T (cid:17) (cid:16)(cid:90) T (cid:17) (cid:16)(cid:90) T (cid:17) Var∆cobs = φ2E dP dP +ψ2E dQ dQ +E dξ dξ +σ2 +σ2 T s s s s s s u,T u,T−1 T−1 T−1 T−1 = φ2σ2 +ψ2σ2 +σ2 +σ2 +σ2 P,T Q,T ξ,T u,T u,T−1 Cov(∆cobs,∆cobs ) = −σ2 T T+1 u,T Cov(∆cobs,∆cobs ) = −σ2 T T−1 u,T−1 Cov(∆cobs,∆cobs) = 0 ∀S,T such that |S −T| > 1 T S (cid:16) (cid:90) T (cid:90) T (cid:17) Cov(∆cobs,∆yobs) = E φ (T −s)dP dP +ψ dQ dQ T T T s s T s s T−1 T−1 1 = φ σ2 +ψ σ2 2 T P,T T Q,T (cid:16) (cid:90) T (cid:90) T (cid:17) Cov(∆cobs,∆yobs ) = E φ (s−(T −1))dP dP −ψ dQ dQ T T+1 T s s T s s T−1 T−1 1 = φ σ2 −ψ σ2 2 T P,T T Q,T Cov(∆cobs,∆yobs ) = 0 T T−1 Cov(∆cobs,∆yobs) = 0 ∀S,T such that |S −T| > 1 T S A2. Continuous Time Model as Limit of Discrete Model with m Sub-periods The identifying equations in the paper are calculated using a ‘log’ income process that does not directly align with any real-world concept of income. In the datawetakelogsonthesumofincomeovertheentireyear,buttheprocessweuse in the model informally aligns with log income over an instantaneous period dt. This is a problem as transitory income arrive as a point mass, making it difficult to interpret what the ‘log’ income process really represents. Here I show how the identifying equations can be derived as the limit of discrete time model with m sub-periods. I show that in the limit the variance of observed log income growth is the same as derived in the informal model (to a first order approximation). The rest of the identifying equations can be shown in the same way. Let p for t ∈ R+ be a martingale process (possibly with jumps) with int

14 OCTOBER 2019 dependent stationary increments and ν be such that E(ept−pt−1) = eν. Define permanent income as: P = ept−tν t (cid:16) (cid:17) Note that E Pt+s = 1 for all s ≥ 0. Define the variance of log permanent shocks Pt to be: σ2 = Var (cid:16) log (cid:0)P t+1(cid:1) (cid:17) = Var(p −p ) P P t+1 t t We will assume changes in permanent income over a one year period are small enough such that: (cid:16)P (cid:17) (cid:16)P −P (cid:17) t+1 t+1 t Var = Var P P t t (cid:16) (cid:0) P t+1 −P t(cid:1) (cid:17) ≈ Var log 1+ P t = Var (cid:16) log (cid:0)P t+1(cid:1) (cid:17) = σ2 P P t Fortransitoryshocks, wedefineanincreasingstochasticprocess, Θ , whichalso t hasindependentstationaryincrements. Theincrementsinthisprocesswilldefine the transitory shocks. We set the expectation of increments, and the variance of the log of an increment of length 1 as: E(Θ −Θ ) = s t+s t (cid:16) (cid:17) Var log (cid:0) Θ −Θ (cid:1) = σ2 t+1 t Θ Notethatforthistobewelldefined,Θ mustnotonlybeincreasingbutalsoitsint crementsarealmostsurelystrictlypositive(sothatlogoftheincrementisdefined almost everywhere). Examples of such a stochastic process would be a gamma process, or a process that increases linearly with time (non-stochastically) but is

VOL. NO. IN SEARCH OF LOST TIME AGGREGATION 15 also subject to positive shocks that arrive as a Poisson process. The stochastic part of this process has no Brownian motion component as this would necessarily lead to non-zero probability of a decreasing increment. We will use these two processes to define an income process in discrete time with m intervals per period, and then look at the limit as m → ∞. Define θ t,m for t ∈ {1, 2, 3...} to be the increment of Θ from t− 1 to t: m m m t m θ = Θ −Θ t,m t 1−1 m Income is defined for each period t ∈ {1, 2, 3...} as: m m m Y = P θ t,m t t,m Therefore the underlying income process has a pure division into permanent and transitory shocks. Income is observed for T ∈ {1,2,3...} as the sum of income in each of the subperiods: m−1 (cid:88) Y¯ = P θ T,m T− i T− i ,m m m i=0 Note that for m = 1 this the same as the underlying income process, with permanent and transitory variance as defined above. We are interested in the log of observable income growth: ∆y¯ = logY¯ −logY¯ T,m T,m T−1,m (cid:32)m−1 (cid:33) (cid:32)m−1 (cid:33) (cid:88) (cid:88) = log P θ −log P θ T− i T− i ,m T−1− i T−1− i ,m m m m m i=0 i=0 (cid:32)m−1 P (cid:33) (cid:32)m−1 P (cid:33) (cid:88) T− i (cid:88) T−1− i = log mθ −log mθ P T− i ,m P T−1− i ,m T−1 m T−1 m i=0 i=0 As P and Θ have independent increments, the covariance between each of the t t

16 OCTOBER 2019 two parts of the sum above is 0. Therefore: (cid:32) (cid:32)m−1 P (cid:33)(cid:33) (cid:32) (cid:32)m−1 P (cid:33)(cid:33) Var (cid:16) ∆1y¯ (cid:17) = Var log (cid:88) T− m i θ +Var log (cid:88) T−1− m i θ T,m P T− i ,m P T−1− i ,m T−1 m T−1 m i=0 i=0 We will treat each of these two variances individually. We begin by looking at the variable: (cid:32)m−1 P (cid:33) (cid:32)m−1 m−1 P (cid:33) (cid:88) T− i (cid:88) (cid:88) (cid:16) T− i (cid:17) log mθ = log θ + m −1 θ P T− i ,m T− i ,m P T− i ,m T−1 m m T−1 m i=0 i=0 i=0 (cid:32) m−1 P θ (cid:33) (cid:16) (cid:17) (cid:88) (cid:16) T− i (cid:17) T− i ,m = log Θ −Θ +log 1+ m −1 m T T−1 i=0 P T−1 (cid:80)m l= − 0 1θ T− l ,m m m−1 P θ (cid:16) (cid:17) (cid:88) (cid:16) T− i (cid:17) T− i ,m ≈ log Θ −Θ + m −1 m T T−1 i=0 P T−1 (cid:80)m l= − 0 1θ T− l ,m m Where the approximation comes from the fact that the shocks to permanent income in a one year period are small. Defining P t ζ = t,m P t−1 m

VOL. NO. IN SEARCH OF LOST TIME AGGREGATION 17 we have that (cid:32) (cid:32)m−1 P (cid:33)(cid:33) (cid:32)m−1 m−1 θ (cid:33) Var log (cid:88) T− m i θ ≈ σ2 +Var (cid:88) (cid:16) (cid:89) ζ −1 (cid:17) T− m i ,m i=0 P T−1 T− m i ,m Θ i=0 j=i T− m j (cid:80)m l= − 0 1θ T− l ,m m (cid:34)m−1 m−1 θ (cid:35)2 = σ2 +E (cid:88) (cid:16) (cid:89) ζ −1 (cid:17) T− m i ,m Θ i=0 j=i T− m j (cid:80)m l= − 0 1θ T− l ,m m (cid:34)m−1(cid:32) m−1 (cid:32) θ (cid:33)2 = σ2 +E (cid:88) (cid:16) (cid:89) ζ −1 (cid:17)2 T− m i ,m Θ i=0 j=i T− m j (cid:80)m l= − 0 1θ T− l ,m m m−1 m−1 θ θ (cid:33)(cid:35) (cid:88)(cid:16) (cid:89) (cid:17)(cid:16) (cid:89) (cid:17) T−k,m T− i ,m +2 ζ −1 ζ −1 m m T−j T−j (cid:16) (cid:17)2 k<i j=k m j=i m (cid:80)m l= − 0 1θ T− l ,m m = σ2 + σ P 2 m (cid:88) −1(cid:32) iE (cid:32) θ T− m i ,m (cid:33)2 +2 (cid:88) (m−1−i)E (cid:32) θ T− m k,m θ T− m i ,m (cid:33)(cid:33) Θ m i=0 (cid:80)m l= − 0 1θ T− m l ,m k<i (cid:16) (cid:80)m l= − 0 1θ T− l ,m (cid:17)2 m = σ2 + σ P 2 m(m−1) E (cid:32) θ T− m i ,m (cid:33)2 Θ m 2 (cid:80)m−1θ l=0 T− l ,m m +2 σ P 2 m (cid:88) −1 i(m−1−i)E (cid:32) θ T− m k,m θ T− m i ,m (cid:33) m i=1 (cid:16) (cid:80)m l= − 0 1θ T− l ,m (cid:17)2 m (cid:32) θ (cid:33)2 = σ2 +σ2 m−1 E T− m i ,m Θ P 2 (cid:80)m−1θ l=0 T− l ,m m (cid:34) (cid:35) (cid:32) (cid:33) θ θ +σ2 (m−1)2− (m−1)(2m−1) E T− m k,m T− m i ,m P 3 (cid:16) (cid:80)m−1θ (cid:17)2 l=0 T− l ,m m Note that: (cid:32)m−1 θ (cid:33)2 1 = E (cid:88) T− m i ,m (cid:80)m−1θ i=0 l=0 T− l ,m m m−1 (cid:32) θ (cid:33)2 (cid:32) θ θ (cid:33) = (cid:88) E T− m i ,m +2 (cid:88) E T− m k,m T− m i ,m i=0 (cid:80)m l= − 0 1θ T− m l ,m k<i (cid:16) (cid:80)m l= − 0 1θ T− l ,m (cid:17)2 m

18 OCTOBER 2019 So that (cid:32) θ θ (cid:33) (cid:32) θ (cid:33)2 E T− m k,m T− m i ,m = 1 − 1 E T− m i ,m (cid:16) (cid:80) l m = − 0 1θ T− l ,m (cid:17)2 m(m−1) m−1 (cid:80)m l= − 0 1θ T− m l ,m m This gives: (cid:32) (cid:32)m−1 P (cid:33)(cid:33) (cid:32)m−1 m−1 θ (cid:33) Var log (cid:88) T− m i θ ≈ σ2 +Var (cid:88) (cid:16) (cid:89) ζ −1 (cid:17) T− m i ,m i=0 P T−1 T− m i ,m Θ i=0 j=i T− m j (cid:80)m l= − 0 1θ T− l ,m m (cid:32) θ (cid:33)2 ≈ σ2 + m−2 σ2 + m+1 E T− m i ,m σ2 Θ 3m P 6 (cid:80)m−1θ P l=0 T− l ,m m 1 → σ2 + σ2 as m → ∞ Θ 3 P A very similar calculation shows that: (cid:32) (cid:32)m−1 P (cid:33)(cid:33) Var log (cid:88) T−1− m i θ → σ2 + 1 σ2 as m → ∞ P T−1− i ,m Θ 3 P T−1 m i=0 Putting these together gives: (cid:16) (cid:17) 2 Var ∆y¯ → σ2 +2σ2 as m → ∞ T,m 3 P Θ (cid:16) (cid:17) This is the same as the identifying equation for Var ∆yobs (equation A1 from T appendix A.A1, assuming shock variances are constant over time), and the rest of the identifying equations can be shown as the limit of the discrete time model in a similar way. A3. Persistence in Transitory Shock This appendix shows how to extend the time aggregated model to include persistence in the transitory shock.

VOL. NO. IN SEARCH OF LOST TIME AGGREGATION 19 Linear Decay Model I will walk though the derivation of the moments for the linear decay model in detail and then just list the moments for the uniform model. In the linear decay model, a shock of size 1 will arrive with a flow intensity of 2 and over the τ subsequenttimeτ thetotalflowoftransitoryincomewillsumto1. Instantaneous income can be written as: (cid:16)(cid:90) t (cid:17) (cid:16)(cid:90) t 2 (cid:17) dy = dP dt+ (s−(t−τ))dQ dt t s s τ 0 t−τ So that the observable change in income is given by: (cid:90) T (cid:90) T−1 ∆yobs = y dt− y dt T t t T−1 T−2 (cid:90) T (cid:90) t (cid:90) T−1(cid:90) t = dP dt− dP dt s s T−1 0 T−2 0 (cid:90) T (cid:90) t 2 (cid:90) T−1(cid:90) t 2 + (s−(t−τ))dQ dt− (s−(t−τ))dQ dt s s τ τ T−1 t−τ T−2 t−τ (cid:16)(cid:90) T−1 (cid:90) T (cid:17) = (s−(T −2))dP + (T −s)dP s s T−2 T−1 2(cid:16)(cid:90) T 1(cid:16) (s−(T −τ))2(cid:17) (cid:90) T−τ 1 (cid:90) T−1 1(s−(T −1−τ))2 (cid:17) + τ − dQ + τdQ + dQ s s s τ 2 τ 2 2 τ T−τ T−1 T−1−τ 2(cid:16)(cid:90) T−1 1(cid:16) (s−(T −1−τ))2(cid:17) (cid:90) T−1−τ 1 − τ − dQ + τdQ s s τ 2 τ 2 T−1−τ T−2 (cid:90) T−2 1(s−(T −2−τ))2 (cid:17) + dQ s 2 τ T−2−τ (cid:90) T−1 (cid:90) T = (s−(T −2))dP + (T −s)dP s s T−2 T−1 (cid:90) T (cid:16)s−(T −τ)(cid:17)2 (cid:90) T−τ + 1− dQ + dQ s s τ T−τ T−1 (cid:90) T−1 (cid:16)s−(T −1−τ)(cid:17)2 − 1−2 dQ s τ T−1−τ (cid:90) T−1−τ (cid:90) T−2 (cid:16)s−(T −2−τ)(cid:17)2 − dQ − dQ s s τ T−2 T−2−τ

20 OCTOBER 2019 The full set of identification equations used in this model are: (cid:16)(cid:90) T−1 (cid:90) T (cid:17) Var(∆yobs) = E (s−(T −2))2dP dP + (T −s)2dP dP T s s s s T−2 T−1 (cid:16)(cid:90) T (cid:16) (cid:16)s−(T −τ)(cid:17)2(cid:17)2 (cid:90) T−τ (cid:17) +E 1− dQ dQ + dQ Q s s s s τ T−τ T−1 (cid:16)(cid:90) T−1 (cid:16) (cid:16)s−(T −1−τ)(cid:17)2(cid:17)2 (cid:17) +E 1−2 dQ dQ s s τ T−1−τ (cid:16)(cid:90) T−1−τ (cid:90) T−2 (cid:16)s−(T −2−τ)(cid:17)4 (cid:17) +E dQ dQ + dQ dQ s s s s τ T−2 T−2−τ 1 1 = σ2 + σ2 3 P,T 3 P,T−1 8 + τσ2 +(1−τ)σ2 15 Q,T Q,T 7 + τσ2 15 Q,T−1 1 +(1−τ)σ2 + τσ2 Q,T−1 5 Q,T−2 = 1 σ2 + 1 σ2 + (cid:0) 1− 7 τ (cid:1) σ2 +(1− 8 τ)σ2 + 1 τσ2 3 P,T 3 P,T−1 15 Q,T 15 Q,T−1 5 Q,T−2 (cid:16)(cid:90) T (cid:17) Cov(∆yobs,∆yobs ) = E (T −s)(s−(T −1))dP dP T T+1 s s T−1 (cid:16)(cid:90) T (cid:16) (cid:16)s−(T −τ)(cid:17)2(cid:17)(cid:16) (cid:16)s−(T −τ)(cid:17)2(cid:17) (cid:17) −E 1− 1−2 dQ dQ s s τ τ T−τ (cid:16)(cid:90) T−τ (cid:17) −E dQ Q s s T−1 (cid:16)(cid:90) T−1 (cid:16) (cid:16)s−(T −1−τ)(cid:17)2(cid:17)(cid:16)s−(T −1−τ)(cid:17)2 (cid:17) +E 1−2 dQ dQ s s τ τ T−1−τ 1 2 1 = σ2 − τσ2 −(1−τ)σ2 − σ2 6 P,T 5 Q,T Q,T 15 Q,T−1 (cid:16)(cid:90) T (cid:16) (cid:16)s−(T −τ)(cid:17)2(cid:17)(cid:16)s−(T −τ)(cid:17)2 (cid:17) Cov(∆yobs,∆yobs ) = −E 1− dQ dQ T T+2 τ τ s s T−τ 2 = − τσ2 15 Q,T

VOL. NO. IN SEARCH OF LOST TIME AGGREGATION 21 The above equations also work for Cov(∆yobs,∆yobs ) and Cov(∆yobs,∆yobs ) T T−1 T T−2 due to symmetry. Cov(∆yobs,∆yobs) = 0 ∀S,T such that |S −T| > 2 T S The covariance matrix Cov(∆cobs,∆cobs) is the same as in appendix A.A1. T S (cid:16)(cid:90) T (cid:17) Cov(∆cobs,∆yobs) = φ E (T −s)dP dP T T T s s T−1 (cid:16)(cid:90) T (cid:16) (cid:16)s−(T −τ)(cid:17)2(cid:17) (cid:90) T−τ (cid:17) +ψ E 1− dQ dQ + dQ dQ T s s s s τ T−τ T−1 1 1 = φ σ2 +ψ (1− τ)σ2 2 T P,T T 3 Q,T (cid:16)(cid:90) T (cid:17) Cov(∆cobs,∆yobs ) = φ E (s−(T −1))dP dP T T+1 T s s T−1 (cid:16)(cid:90) T (cid:16) (cid:16)s−(T −τ)(cid:17)2(cid:17) (cid:90) T−τ (cid:17) −ψ E 1−2 dQ dQ + dQ dQ T s s s s τ T−τ T−1 1 2 = φ σ2 −(1− τ)ψ σ2 2 T P,T 3 T Q,T (cid:16)(cid:90) T (cid:16)s−(T −τ)(cid:17)2 (cid:17) Cov(∆cobs,∆yobs ) = −ψ E dQ dQ T T+2 T τ s s T−τ 1 = − ψ τσ2 5 T Q,T The Uniform Model In the uniform model, transitory shocks consist of a constant flow of income that lasts for a time period τ. The full set of moments for this model are: Var(∆yobs) = 1 σ2 + 1 σ2 + (cid:0) 1− 2 τ (cid:1) σ2 +(1− 2 τ)σ2 + 1 τσ2 T 3 P,T 3 P,T−1 3 Q,T 3 Q,T−1 3 Q,T−2

22 OCTOBER 2019 1 1 1 Cov(∆yobs,∆yobs ) = σ2 − τσ2 −(1−τ)σ2 − σ2 T T+1 6 P,T 6 Q,T Q,T 15 Q,T−1 1 Cov(∆yobs,∆yobs ) = − τσ2 T T+2 6 Q,T The above equations also work for Cov(∆yobs,∆yobs ) and Cov(∆yobs,∆yobs ) T T−1 T T−2 due to symmetry. Cov(∆yobs,∆yobs) = 0 ∀S,T such that |S −T| > 2 T S The covariance matrix Cov(∆cobs,∆cobs) is the same as in appendix A.A1. T S 1 1 Cov(∆cobs,∆yobs) = φ σ2 +ψ (1− τ)σ2 T T 2 T P,T T 2 Q,T 1 Cov(∆cobs,∆yobs ) = φ σ2 −(1−τ)ψ σ2 T T+1 2 T P,T T Q,T 1 Cov(∆cobs,∆yobs ) = − ψ τσ2 T T+2 2 T Q,T A4. Effect of Timing of Consumption in the PSID BPP impute annual consumption from the question in the PSID asking about food consumption in a ‘typical’ week. Unfortunately it is not clear if this relates to an average of the previous calendar year, or some more recent week closer to when the interview was conducted (normally in March of the following year). In the paper I have assumed the answer gives a snapshot of consumption at the end ofthecalendaryear. HereIshowthatassumingthe‘typical’weekisanaverageof consumption over the previous calendar year, the identifying equation from BPP for transitory insurance coefficient is different again, and still significantly biased. Under this new assumption the equation for the permanent insurance coefficient

VOL. NO. IN SEARCH OF LOST TIME AGGREGATION 23 is unbiased as before: Cov(∆cobs,∆yobs +∆yobs+∆yobs ) T T−1 T T+1 = φ Cov(∆yobs,∆yobs +∆yobs+∆yobs ) T T−1 T T+1 While the identifying equation for the transitory insurance coefficient is: Cov(∆cobs,∆yobs ) −φ1σ2 + 1ψσ2 T T+1 = 6 P 2 Q (cid:54)= ψ Cov(∆yobs,∆yobs ) −1σ2 +σ2 T T+1 6 P Q Under the permanent income hypothesis with φ = 1, ψ = 0 and permanent and transitory variances approximately equal, the BPP estimate of ψ would be -0.2. A5. Other Tables from the BPP paper Table A1 replicates Table 6 from the original BPP paper. Table A2 replicates Table 7 from the original BPP paper. Table A3 replicates Table 8 from the original BPP paper.

24 OCTOBER 2019 Table A1—Minimum-Distance Partial Insurance and Variance Estimates WholeSample NoCollege College BPP TimeAgg. BPP TimeAgg. BPP TimeAgg. σ2 1979-1981 0.0103 0.0247 0.0068 0.0234 0.0101 0.0189 P,T (Varianceperm. shock) (0.0034) (0.0043) (0.0037) (0.0063) (0.0053) (0.0050) 1982 0.0208 0.0358 0.0156 0.0290 0.0253 0.0455 (0.0041) (0.0071) (0.0052) (0.0099) (0.0060) (0.0099) 1983 0.0301 0.0333 0.0318 0.0553 0.0234 0.0086 (0.0057) (0.0100) (0.0074) (0.0128) (0.0090) (0.0148) 1984 0.0274 0.0292 0.0334 0.0232 0.0177 0.0361 (0.0049) (0.0114) (0.0073) (0.0131) (0.0060) (0.0161) 1985 0.0295 0.0363 0.0287 0.0504 0.0208 0.0025 (0.0096) (0.0124) (0.0073) (0.0145) (0.0152) (0.0205) 1986 0.0221 0.0327 0.0173 0.0247 0.0311 0.0597 (0.0060) (0.0136) (0.0067) (0.0172) (0.0101) (0.0202) 1987 0.0289 0.0420 0.0202 0.0478 0.0354 0.0229 (0.0063) (0.0143) (0.0073) (0.0182) (0.0098) (0.0211) 1988 0.0158 0.0082 0.0117 -0.0069 0.0183 0.0302 (0.0069) (0.0137) (0.0079) (0.0209) (0.0110) (0.0149) 1989 0.0185 0.0531 0.0107 0.0639 0.0274 0.0414 (0.0059) (0.0129) (0.0101) (0.0214) (0.0061) (0.0149) 1990-92 0.0135 0.0291 0.0093 0.0265 0.0217 0.0291 (0.0042) (0.0042) (0.0045) (0.0063) (0.0065) (0.0057) σ2 1979 0.0379 0.0310 0.0465 0.0364 0.0301 0.0261 Q,T (Variancetrans. shock) (0.0059) (0.0049) (0.0096) (0.0080) (0.0056) (0.0043) 1980 0.0298 0.0240 0.0330 0.0247 0.0283 0.0238 (0.0039) (0.0033) (0.0053) (0.0046) (0.0059) (0.0047) 1981 0.0300 0.0265 0.0363 0.0305 0.0253 0.0222 (0.0035) (0.0032) (0.0053) (0.0048) (0.0046) (0.0040) 1982 0.0287 0.0280 0.0375 0.0332 0.0213 0.0237 (0.0039) (0.0034) (0.0063) (0.0057) (0.0042) (0.0036) 1983 0.0262 0.0276 0.0371 0.0378 0.0185 0.0169 (0.0037) (0.0034) (0.0063) (0.0056) (0.0037) (0.0040) 1984 0.0346 0.0350 0.0404 0.0388 0.0304 0.0315 (0.0039) (0.0038) (0.0059) (0.0058) (0.0051) (0.0046) 1985 0.0450 0.0427 0.0355 0.0338 0.0496 0.0465 (0.0075) (0.0071) (0.0056) (0.0053) (0.0130) (0.0122) 1986 0.0458 0.0404 0.0474 0.0373 0.0452 0.0464 (0.0058) (0.0055) (0.0076) (0.0068) (0.0085) (0.0084) 1987 0.0461 0.0445 0.0520 0.0486 0.0421 0.0385 (0.0054) (0.0053) (0.0082) (0.0078) (0.0071) (0.0069) 1988 0.0399 0.0327 0.0471 0.0360 0.0343 0.0313 (0.0047) (0.0044) (0.0074) (0.0072) (0.0060) (0.0055) 1989 0.0378 0.0343 0.0539 0.0475 0.0219 0.0215 (0.0067) (0.0061) (0.0126) (0.0117) (0.0051) (0.0044) 1990-92 0.0441 0.0359 0.0535 0.0408 0.0345 0.0322 (0.0040) (0.0027) (0.0062) (0.0047) (0.0049) (0.0032) θ 0.1126 N/A 0.1260 N/A 0.1082 N/A (Serialcorrel. trans. shock) (0.0248) (0.0319) (0.0342) σ2 0.0097 0.0122 0.0065 0.0114 0.0132 0.0146 ξ (Varianceunobs. slopeheterog.) (0.0041) (0.0039) (0.0079) (0.0070) (0.0040) (0.0039) φ 0.6456 0.3384 0.9484 0.4365 0.4180 0.2729 (Partialinsuranceperm. shock) (0.0941) (0.0471) (0.1773) (0.0738) (0.0913) (0.0603) ψ 0.0501 0.2421 0.0724 0.2870 0.0260 0.1590 (Partialinsurancetrans. shock) (0.0430) (0.0431) (0.0593) (0.0616) (0.0546) (0.0504)

VOL. NO. IN SEARCH OF LOST TIME AGGREGATION 25 Table A2—Minimum-Distance Partial Insurance and Variance Estimates Consumption: Nondurable Nondurable Nondurable Income: net income earnings only male earnings Sample: baseline baseline baseline BPP Time Agg. BPP Time Agg. BPP Time Agg. φ 0.6456 0.3384 0.3101 0.1761 0.2240 0.1232 (Partial insurance perm. shock) (0.0941) (0.0471) (0.0572) (0.0339) (0.0492) (0.0316) ψ 0.0501 0.2421 0.0630 0.1625 0.0502 0.1181 (Partial insurance trans. shock) (0.0430) (0.0431) (0.0306) (0.0280) (0.0293) (0.0244) Table A3—Minimum-Distance Partial Insurance and Variance Estimates Consumption: Nondurable Nondurable Nondurable Income: net income excluding help net income Sample: baseline baseline low wealth BPP Time Agg. BPP Time Agg. BPP Time Agg. φ 0.6456 0.3384 0.6244 0.3422 0.8339 0.8584 (Partial insurance perm. shock) (0.0941) (0.0471) (0.0891) (0.0466) (0.2762) (0.2498) ψ 0.0501 0.2421 0.0469 0.2404 0.2853 0.4926 (Partial insurance trans. shock) (0.0430) (0.0431) (0.0429) (0.0427) (0.1154) (0.1050) Consumption: Nondurable Total Nondurable Income: net income net income net income Sample: high wealth low wealth baseline+SEO BPP Time Agg. BPP Time Agg. BPP Time Agg. φ 0.6278 0.2691 1.0207 1.0580 0.7663 0.4630 (Partial insurance perm. shock) (0.0998) (0.0420) (0.3426) (0.3099) (0.1028) (0.0499) ψ 0.0088 0.1838 0.3647 0.6185 0.1201 0.3232 (Partial insurance trans. shock) (0.0409) (0.0409) (0.1477) (0.1344) (0.0352) (0.0367)