Consumption Heterogeneity: Micro Drivers and Macro Implications

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Consumption Heterogeneity: Micro Drivers and Macro Implications Edmund Crawley and Andreas Kuchler 2020-005 Please cite this paper as: Crawley, Edmund, and Andreas Kuchler (2020). “Consumption Heterogeneity: Micro Drivers and Macro Implications,” Finance and Economics Discussion Series 2020-005. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2020.005. 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.

CONSUMPTION HETEROGENEITY: MICRO DRIVERS AND MACRO IMPLICATIONS December 23, 2019 Edmund Crawley1 Andreas Kuchler2 Federal Reserve Board Danmarks Nationalbank Abstract This paper explores the microfoundations of consumption models and quanti(cid:28)es the macro implications of consumption heterogeneity. We propose a new empirical method to estimate the response of consumption to permanent and transitory income shocks for di(cid:27)erent groups of households. We then apply this method to administrative data from Denmark. The large sample size, along with detailed household balance sheet information, allows us to (cid:28)nely divide the population along relevant dimensions. We (cid:28)nd that households that stand to lose from an interest rate hike are signi(cid:28)cantly more responsive to income shocks than those that stand to gain. Following a 1-percentage-point interest rate increase, we estimate that consumption growth decreases by a 1/4 percentage point through this interest rate exposure channel alone, making this channel substantially larger than the intertemporal substitution channel that is at the core of representative agent New Keynesian models. Keywords Uncertainty, Consumption Dynamics, MPC JEL codes D12, D31, D91, E21 1Crawley: Federal Reserve Board, Constitution Avenue & 20th Street NW, Washington, DC 20551, USA, edmund.s.crawley@frb.gov. 2Kuchler: Danmarks Nationalbank, Havnegade 5, 1093 Copenhagen K, Denmark, aku@nationalbanken.dk. Viewpoints and conclusions stated in this paper are the responsibility of the authors alone and do not necessarily re(cid:29)ect the viewpoints of Danmarks Nationalbank or the Federal Reserve Board. The authors wish to thank colleagues andparticipantsatconferencesandseminarsatBanquedeFrance,BLS,BocconiUniversity,DanishMinistryofFinance, Danmarks Nationalbank, Deutsche Bundesbank, European Central Bank, Federal Reserve Board, FDIC, Georgetown University,JohnsHopkinsUniversity,UniversityofBergamo,UniversityofCopenhagen,andtheBoston,SanFrancisco, andNewYorkFedsforusefulcommentsandsuggestions.

1 Introduction How do di(cid:27)erences in household consumption behavior a(cid:27)ect the business cycle? Recent heterogeneous agent models suggest that wealth redistribution between households with high and low marginal propensity to consume (MPC) may play a dominant role in propagating macroeconomic shocks, particularly for monetary policy.1 Testing the microfoundations of these models empirically, and quantifying the macroeconomic importance of redistribution, often boils down to measuring how MPCs vary systematically over dimensions such as wealth and exposure to interest rate movements. However, shortcomings both in the empirical methods used to measure MPCs and in the available data have limited the literature’s ability to do this reliably. In this paper we overcome some of these shortcomings. We present a new method to measureMPCsfromincomeandconsumptionpaneldata,buildinguponthatofBlundell, Pistaferri, and Preston (2008) (henceforth BPP). We then apply our method to di(cid:27)erent groups of households in administrative data from Denmark. We estimate the average MPC in the economy to be 0.5, close to estimates obtained from natural experiments. By contrast, using the BPP method on our data yields a much lower estimate of 0.12. We follow BPP by imposing identifying restrictions on household income and consumption dynamics.2 We make two improvements to the method. First, we explicitly model the time aggregated nature of income and consumption in our data. Time aggregation, overlooked in much of the household (cid:28)nance literature, can result in signi(cid:28)cant estimation bias if it is not explicitly modeled.3 Second, we allow complete freedom in how consumption responds to a transitory income shock for up to two years, after which we assume no further response. This short-term response contrasts with the assumption in BPP that consumption follows a random walk. The random walk assumption, incompatible with high MPCs, was previously thought to be benign for estimates of consumption responses to transitory shocks. However, this is not true for time aggregated data.4 A consequence of allowing freedom for two years is that our 1See,forexample,Kaplan,Moll,andViolante(2018);Garriga,Kydland,andSustek(2017);andGreenwald(2018). 2The BPP method, and those closely related to it, has become a standard tool in the consumption literature. See, forexample,Violante,Kaplan,andWeidner(2014);Auclert(2019);andManovskiiandHryshko(2017) 3Crawley (2019) shows that estimates of the elasticity of consumption to transitory income shocks using the BPP method depend strongly on the nature of time aggregation in the data. He shows the estimate of the elasticity of consumptiontotransitoryincomeincreasesfromto0.05(cid:22)theestimateinBPP(cid:22)to0.25whenthetimeaggregatednature ofthePSIDdataismodeled. 4KaplanandViolante(2010)showthattheestimateoftheperiodoneconsumptionresponsetotransitoryshocksin BPP is una(cid:27)ected by assumptions on the path of consumption in following periods. This result does not hold for time aggregateddata(cid:22)inourdatawewouldgreatlyoverestimateMPCsifweassumedarandomwalk,seeonlineappendixE. 1

method requires a slightly longer panel than BPP, but we believe the robustness of our estimates to many forms of misspeci(cid:28)cation is worth this cost. OurdataconsistofapanelofincomeandexpenditurefortheentireDanishpopulation, along with details of the interest rate sensitivity of households’ (cid:28)nancial assets and liabilities that we require to estimate the redistribution e(cid:27)ects of monetary policy. Income and wealth data are largely third-party reported to tax authorities and correspondingly accurate. Accordingly, we use the household budget constraint to back out expenditure from income and wealth. Speaking to the microfoundations of consumption behavior, we uncover a clear negative monotonic relation between MPC and liquid wealth. We show that the sign of this relationship is in line with standard bu(cid:27)er-stock models, although the magnitude of MPCs, especially for households with the most liquid assets, is di(cid:30)cult to reconcile with theory. When we include illiquid wealth, such as housing, this monotonic relationship between wealth and MPC no longer holds: Those with close to zero net worth have higher MPCs than both those with negative and positive net worth. That liquid wealth is more predictive of MPCs than net worth is consistent with the wealthy hand-to-mouth model of Violante, Kaplan, and Weidner (2014). Broadly speaking, in our data, we see three groups with distinct MPCs and exposure to interest rate movements: the (cid:16)poor hand-to-mouth,(cid:17) with MPCs around 0.8, who own few assets, liquid or otherwise and are not directly exposed to interest rates; the (cid:16)wealthy hand-to-mouth,(cid:17) with MPCs around 0.5, who typically own houses and have mortgages and other debts whose payments rise with interest rates; and the (cid:16)wealthy,(cid:17) a smaller group with MPCs around 0.25, who typically own houses and also have large liquid bank balances, and whose income rises with interest rates.5 The strength of our method and data over previous studies can be seen when we quantify the size of monetary policy redistribution channels. We follow the decomposition of Auclert (2019) who reports su(cid:30)cient statistics that determine the e(cid:27)ect of monetary policy redistribution on aggregate consumption. However, being limited by the econometric methods he has at hand as well as by publicly available data sources, he (cid:28)nds it challenging to get a clear picture of how MPCs vary over the dimensions he identi(cid:28)es as relevant, such as unhedged interest rate exposure. Weestimatethata1-percentage-pointriseintherealinterestrate, whichredistributes wealthfromthewealthyhand-to-mouthdebtorswhopayinteresttothewealthycreditors 5These groups loosely line up with those of the same name in Violante, Kaplan, and Weidner (2014), who de(cid:28)ne wealthyhand-to-mouthashouseholdswithsigni(cid:28)cantilliquidassetsbutlittleornoliquidassets. Weobservethatthese threegroupsarenaturallyseparatedalongthedimensionofunhedgedinterestrateexposure. See(cid:28)gureVIIIinsection6. 2

who receive it, reduces aggregate consumption growth by 26 basis points through this redistribution channel alone. This channel is absent in representative agent models in which the intertemporal substitution channel instead dominates. We believe the rich detail we are able to provide on the relationships between MPC, home ownership, liquidity, and interest rate exposure could be used to discipline microfounded macroeconomic models going forward. Furthermore, a growing number of large, high-quality panel datasets on income and consumption are becoming available to economists, which increases the value of robust econometric methods that can uncover household behavior.6 Beyond the applications in this paper, our method has a wide variety of potential applications in the consumption, household (cid:28)nance, and labor literatures. 2 Background Theneedforbettermethodsanddatatomeasureconsumptionbehavioratthehousehold level has grown with the increasing recognition that household heterogeneity may play a key role in macroeconomic dynamics. Models of household heterogeneity, with incomplete markets and idiosyncratic risk, began with Bewley (1983), followed by Imrohoro§lu (1989), Huggett (1993), and Aiyagari (1994) studying the steady-state properties of these models. However, computational and methodological limitations, along with early work by Krusell and Smith (1998) showing that the aggregate dynamics of a TFP shock were not much changed in a heterogeneous agent model, have resulted in a slow start for the literature combining heterogeneity with business cycle dynamics.7 Recent methodological advances, particularly Reiter (2009), have overcome some of these computational limitations and a new generation of models have demonstrated that heterogeneity can play a large role in the business cycle. Heterogeneous Agent New Keynesian (HANK) models, combining elements from both the heterogeneous agent and New Keynesian literature, not only match the growing evidence on micro level consumption behavior, but also imply very di(cid:27)erent aggregate dynamics and/or propagation mechanisms following macroeconomic shocks, compared to their representative 6Paneldatafrom(cid:28)nancialaggregationplatformshavebeenhighlyinformativeaboutconsumptionbehavior,although accesstothesedatabyresearchersisveryrestricted. SeeGelman,Kariv,Shapiro,Silverman,andTadelis(2014),Ganong andNoel(2019),andBaker(2018)forexamplesusingU.S.data. OlafssonandPagel(2018)usedatafromIceland. 7KaplanandViolante(2018)provideanoverviewofthetheoreticalliteratureincorporatinghouseholdheterogeneity intomodelsofeconomic(cid:29)uctuations. 3

agent equivalents. In particular, the transmission mechanism of monetary policy can look very di(cid:27)erent in a HANK model.8 While these heterogeneous-agent models make clear the potential importance of heterogeneity in economic (cid:29)uctuations, particularly for monetary and (cid:28)scal policy, their quantitative results hinge on assumptions, such as the tenure of debt instruments and the timing of taxes, that were unimportant in representative agent models. Thus far the ability of these models to help us distinguish transmission channels empirically has been limited. Auclert (2019), in contrast to the fully structural HANK models, takes a simpli(cid:28)ed approach to aggregate dynamics, and one that we will follow in this paper.9 He derives a set of su(cid:30)cient statistics, directly measurable from a suitable dataset, that is highly informative about the relative size of di(cid:27)erent monetary policy transmission channels. His methodology bene(cid:28)ts from being transparent and closely tied to the data, reducing the problem to that of measuring the distribution of MPCs across relevant dimensions of redistribution. However, as we will see in the following section, evidence on how MPCs vary across the population has been hard to come by. 2.1 Empirical Evidence on Heterogeneity in Consumption Behavior Mostmicroempiricalevidenceonconsumptionbehaviorcomesintheformofanestimate of the marginal propensity to consume out of a one-time source of income over the following three months to one year. Table I shows a selection of the population average estimates from the literature. Most of these studies do not have the power to say much, if anything, about heterogeneity within the population. Three methods are used to empirically determine the marginal propensity to consume. The (cid:28)rst is to identify a natural experiment and measure the consumption response to it. For example, Johnson, Parker, and Souleles (2006) use randomly assigned timing of 2001 tax rebates and questions in the U.S. Consumer Expenditure Survey to identify a threemonth aggregate marginal propensity to consume of 0.2 to 0.4. Of the three methods, natural experiments likely have the strongest identi(cid:28)cation, but estimates vary, and there is no strong consensus. Identi(cid:28)cation issues arise as to when exactly households learn about the payment versus when it is received, and the extent to which external validity extends from these natural experiments to the kinds of transitory shocks found 8Forexample,inthemodelofKaplan,Moll,andViolante(2018),theintertemporalsubstitutionchannelisdwarfed byindirectgeneralequilibriume(cid:27)ects,instarkcontrasttoarepresentativeagentmodel. 9Wong (2016) also takes an empirical approach by identifying how the consumption response to monetary policy shocksvarieswithage. 4

in heterogeneous agent models is unclear.10 As most of these studies rely on consumer survey data, they tend to lack power due to high measurement error and low sample sizes. As a result, they have produced very little evidence of how the MPC varies among di(cid:27)erent groups in the economy. A recent paper by Fagereng, Holm, and Natvik (2016) overcomes some of these problems. By using lottery data, the shock to income is truly random.11 They use administrative data from Norway similar to the data we use from Denmark and have a sample of over 30,000 lottery winners over 10 years. As a result, they can identify the MPC for households with di(cid:27)ering liquid wealth, as well as by the size of the lottery win. They (cid:28)nd that households in the lowest quartile of liquid wealth have an MPC of approximately 0.61 over a six-month period, as opposed to 0.45 for households in the highest quartile of liquid wealth. In another study using data from a (cid:28)nancial aggregator, Gelman (2016) identi(cid:28)es large di(cid:27)erences in the impulse response to a tax rebate at a monthly frequency across household quintiles of cash-on-hand. The second method is simply to ask individuals how much of a transitory income change they would consume. Jappelli and Pistaferri (2014) (cid:28)nd an aggregate MPC of 0.48 using the Italian Survey of Household Income and Wealth and are able to identify clear di(cid:27)erences across levels of liquid wealth. Fuster, Kaplan, and Zafar (2018) (cid:28)nd a lower aggregate MPC in the NY Fed’s Survey of Consumer Expectations, but they (cid:28)nd heterogeneity by both size and sign of the shock. Of course the reliability of these studies is limited by the accuracy of households’ own response to the question. The third method, which we will follow, is to impose covariance restrictions on panel data of income and consumption and use these to identify the consumption response to income shocks of di(cid:27)ering persistence. This method has the advantage that it can be used in a panel dataset with no natural experiment, such as the Danish administrative data we use or the PSID. The most well-known paper to use this method is by Blundell, Pistaferri, andPreston(2008), whouseimputednon-durableconsumptiondatabasedon foodexpenditurereportedinPSIDdata. Theyestimateaconsumptionelasticity(closely related to an MPC if households’ consumption level is close to their income) and (cid:28)nd almost no consumption response to transitory shocks; however, as we will discuss in section 3.4, their estimates are strongly downward biased. Our paper also adds to the limited literature on consumption responses to permanent 10Manystudies(cid:28)ndasmallerMPCforpositiveshocksthannegativeshocks(cid:22)forexample,Bunn,LeRoux,Reinold, andSurico(2018). Inthispaperweimplicitlyassumethattheresponseissymmetric. Inreality,ourestimatesrepresent anaverageofpositiveandnegativeshockreactions. 11We should note that even lottery winnings have some problems. First, the results hold for winners of the lottery whomaynotberepresentativeofthewiderpopulation. Secondtheconsumptionresponsetoalotterywinmaybevery di(cid:27)erentthanotherincomeshocks. 5

skcohS emocnI morf emusnoC ot ytisneporP lanigraM eht fo setamitsE I elbaT erusaeM noitpmusnoC elpmaS/tnevE dohteM noziroH ECP latoT selbarudnoN skcohS tnenamreP 29(cid:21)0891 :elpmaS noitamitsE 3 ∼ 56.0 (cid:63))8002( notserP dna ,irrefatsiP ,llednulB kcohS ecirP enilosaG 1 ∼ 0.1 ,satsuoK ,viraK ,oknehcindoroG ,namleG )6102( siledaT dna ,namrevliS ,oripahS skcohS yrotisnarT 1102 dnediviD htworG eropagniS 1 m01 09.0 )4102( naiQ dna lawragA 29(cid:21)0891 :elpmaS noitamitsE 3 50.0 (cid:63))8002( notserP dna ,irrefatsiP ,llednulB 59(cid:21)5891 ,ataD FPCE hsinapS 1 0 ∼ )1002( odalloC dna gninworB tuC xaT 3002 1 y1 63.0 )5002( reniehS dna ,notpuL ,odanoroC snoitatcepxE .snoC yevruS deF YN 2 m3 13.0(cid:21)80.0 )8102( rafaZ dna ,nalpaK ,retsuF sunoB ’snareteV 6391 1 y1 57.0(cid:21)6.0 )2102( namsuaH 1002(cid:21)0891 ,XEC 1 57.0(cid:21)6.0 0 ∼ (cid:63))3002( heisH 0102 ,ylatI 2 84.0 )4102( irrefatsiP dna illeppaJ tiderC xaT dlihC 3002 1 m3 52.0 ∼ )9002( seleluoS dna ,rekraP ,nosnhoJ 78(cid:21)0891 :elpmaS noitamitsE 3 5.0(cid:21)2.0 (cid:63))6991( idrasuL 39(cid:21)0891 :elpmaS noitamitsE 1 m3 2.0 )9991( rekraP sulumitS cimonocE 8002 1 m3 09.0(cid:21)05.0 03.0(cid:21)21.0 dnallelCcM dna ,nosnhoJ ,seleluoS ,rekraP )3102( sulumitS cimonocE 8002 1 y1 3/1 ∼ )0102( dormelS dna ,oripahS ,mhaS sulumitS cimonocE 8002 1 y1 3/1 ∼ )9002( dormelS dna oripahS 19(cid:21)0891 :elpmaS noitamitsE 1 m3 46.0(cid:21)43.0 90.0(cid:21)540.0 )9991( seleluoS stuC xaT nagaeR ehT 1 y1 9.0(cid:21)6.0 )2002( seleluoS s0891 ylraE eht fo .emocniotnoitpmusnocfoyticitsalE (cid:63) snoitcirtserecnairavoC)3noitseuqyevruS)2tnemirepxelarutaN)1 :sdohteM .)7102(etihWdna,akoukoT,kelacalS,llorraCmorfdetpadasielbatsihT 6

shocks to income. Natural experiments for permanent shocks are rare. Gelman, Gorodnichenko, Kariv, Koustas, Shapiro, Silverman, and Tadelis (2016) use shocks to gasoline prices as a proxy for a permanent shock to income and (cid:28)nd an MPC close to 1 across the population. BPP (cid:28)nd a consumption elasticity to permanent shocks to income around 0.65 (the permanent shock elasticity is less a(cid:27)ected by the time aggregation problem). For a more complete overview of the literature on consumption responses to income changes, see Jappelli and Pistaferri (2010). 3 Empirical Strategy Wetakeareducedformapproachtoestimatefourparameters: thevarianceofpermanent and transitory income shocks and the marginal propensity to consume out of permanent and transitory income shocks. To do so, we will make identifying restrictions on income and consumption dynamics. Speci(cid:28)cally, we will assume that income is made up of a permanent component that moves as a random walk and a transitory component with persistence of less than two years. For consumption, we assume it responds permanently to a permanent income shock but has a short-lived response of no more than two years to a transitory income shock. Our model will be in continuous time in order to correctly account for the time aggregated nature of our data. These restrictions allow us to calculate a set of observable moments with which we can estimate the four parameters of interest using GMM. Before diving into the methodological details, in the next section 3.1 we build some intuition on where identi(cid:28)cation of the transitory and permanent MPCs comes from. 3.1 Methodology Intuition In this section we present some very simple regressions of expenditure growth on income growth and compare them with what we would expect in some very well understood baseline models. We look at the estimate of βN in the model ∆Nc = αN +βN∆Ny +ε it it it where N, the number of years over which growth is measured, varies from 1 to 10. The identi(cid:28)cation of permanent and transitory MPCs in section 3.2 will come from the fact that transitory income shocks make up a relatively large proportion of the variance of income growth over a short period, while permanent income shocks dominate the 7

variance of income growth over a long period. While the coe(cid:30)cients betaN in this section do not represent MPCs, we would expect their values for large N to be close the the permanent MPC, and for small N to be closer to the transitory MPC. Figure I shows what we would expect to see under three baseline models where households are subject to both transitory and permanent income shocks, as well as what we observe in the data. First, the blue horizontal line at zero shows what we would see in a complete markets model. With complete markets, all idiosyncratic shocks to income are insured against, resulting in no relation between idiosyncratic income and consumption growth. Second, the green horizontal line at 0.75 shows what we would see in a Solow model. Households in the Solow model do not optimize, but instead spend a constant proportion of their income each period(cid:22)in this case set at 0.75(cid:22)regardless of shock persistence. Third, the red, upward-sloping line shows the results for a typical bu(cid:27)er-stock saving model.12 In this model, the consumption response to income growth over a one year period is small because households self-insure against the transitory shocks that dominate at this frequency. As the time period over which income growth is measured increases, the observed income growth is proportionally more permanent and self-insurance is not possible. The red line asymptotes toward 1.0 as N gets large. The gray line, along with 95% con(cid:28)dence intervals, shows the results of these regressions using all households in the Danish sample. It is striking that the data appear to be closest to the Solow model, with only a small decrease in the regression coe(cid:30)cient over short periods. However, aggregating all households in this way hides a large degree of heterogeneity, particularly across households with di(cid:27)erent levels of liquid wealth. The two black lines show the regression coe(cid:30)cients where the sample is restricted to households in the lowest and highest quintiles of liquid wealth (averaged over the observed period), respectively. For households in the lowest quintile there is no evidence of consumption smoothing: the consumption response to income growth over one year is both high and almost identical to that over 10 years, strongly suggesting the MPCs for this group out of transitory and permanent shocks are similar and high. Indeed, this is what we (cid:28)nd in section 5, and the result is robust to misspeci(cid:28)cation. Households in the top quintile of liquid wealth show a clear upward slope in (cid:28)gure I, indicating a substantial degree of consumption smoothing. The fact that the regression coe(cid:30)cient for this group appears to asymptote well below 1 also suggests, in contrast to standard bu(cid:27)er-stock models, that the MPC out of permanent shocks for liquid households is signi(cid:28)cantly lower than 1. 12TheseresultscomefromtheheterogeneousbetamodelinCarroll,Slacalek,Tokuoka,andWhite(2017),calibrated tomatchthedistributionofliquidwealthinDenmark. 8

l l l l l l l l l l 2 4 6 8 10 0.1 8.0 6.0 4.0 2.0 0.0 Regressing Consumption Growth on Income Growth N, Years of Growth tneiciffeoC noissergeR , N b Least Liquid l l l l l l l l l ll l l l l l l l l l ll lll ll ll ll ll ll l l All Householdsl l l l l l l l l l Complete Markets l Most Liquid Solow l Buffer−Stock l Data Relatively more Relatively more transitory variance permanent variance Figure I Regression Coe(cid:30)cients of Consumption Growth on Income Growth 9

3.2 Identifying Restrictions Next, we describe the restrictions we impose on income and consumption dynamics that allow us to identify the variance of permanent and transitory income shocks, as well as how households respond to them. 3.2.1 Income Dynamics Our identi(cid:28)cation of permanent and transitory income variance follows the methodology of Carroll and Samwick (1997) closely. As in their approach, we assume idiosyncratic income is composed of permanent and transitory components, where the permanent component follows a random walk and the transitory component persists for no more than two years. Our main innovation is to account for time aggregation by setting their discrete time model in continuous time and aggregating income over each year appropriately. We choose to model the level income process, rather than the log income process as in Carroll and Samwick (1997), because this allows more direct estimates of marginal propensities to consume.13 Our model is set in continuous time where each time period represents one year. We de(cid:28)ne two independent martingale processes (possibly with jumps), P and Q , where t t P will represent the (cid:29)ow of permanent income at time t and the change in Q provides t t the transitory impulses that generate the transitory income. We assume that 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 . That is, these martingales have independent increments. As a t useful benchmark, two Brownian motions satisfy these criteria. The natural generalization of the MA(2) transitory income process from Carroll and Samwick (1997) is to allow for a generically shaped transitory income shock that decays to zero in under two years.14 Figure II shows an example of such a transitory income shape f(t), but the model also allows for completely transitory shocks in which case f(t) would be a delta function with all the income from the transitory shocks arriving as a mass at the time of the shock. In this model the (cid:29)ow of income arriving at time 13InonlineappendixJwegetqualitativelysimilarresultsusingamodeloflogincomeandexpenditure. 14Previousstudieshavefoundlittleevidenceoftransitorydynamicslastingmorethanoneyear,buttobeconservative andinlinewithBPPweallowtransitoryincometopersistforuptotwoyears. 10

0 1 2 3 4 5 2.1 8.0 4.0 0.0 Generic Transitory Impulse Response, f(t) Time emocnI Figure II Generic Transitory Shock Impulse Response t is given by the (cid:29)ow of permanent income and the sum of income arising from any transitory shocks to income that have occurred in the previous two years: (cid:90) t y = P + f(t−s)dQ t t s t−2 We do not observe y directly but instead y¯ , the time aggregated income over each one t T year period: (cid:90) T y¯ = y dt for T ∈ {1,2,3...} (1) T t T−1 Taking the Nth di(cid:27)erence for N ≥ 3 we get: (cid:90) T (cid:90) T−N ∆Ny¯ = y dt− y dt T t t T−1 T−N−1 (cid:90) T (cid:90) T−N = (T −s)dP +(P −P )+ (s−(T −2))dP s T−1 T−N s T−1 T−N−1 (cid:16) (cid:90) T (cid:90) t (cid:90) T−N (cid:90) t (cid:17) + f(t−s)dQ dt− f(t−s)dQ dt (2) t t T−1 t−2 T−N−1 t−2 The variance of time aggregated income of an N year period is therefore:15 1 Var(∆Ny¯ ) = (N − )σ2 +2Var(y˜) for N ≥ 3 (3) T 3 P This is similar to the discrete time model in Carroll and Samwick (1997) except that the coe(cid:30)cient on permanent variance is N − 1 in place of N. The transitory variance 3 identi(cid:28)ed is the variance of (cid:16)total(cid:17) transitory income received in the year, y˜, where this 15SeeonlineappendixAforfulldetailsofthisderivation. 11

is de(cid:28)ned as16 (cid:90) T (cid:90) t y˜ = f(t−s)dQ dt (4) T s T−1 t−2 Equation 3 shows that the variance of income growth grows linearly with the number of years of growth beyond three years. This result comes from the fact that the transitory component adds variance at the beginning and end of the growth period, but any transitory shock to income that occurs in the middle of the period does not a(cid:27)ect income growth, as it will have decayed by the end of the measured period. 3.2.2 Consumption Dynamics Our approach will be to extend the identi(cid:28)cation of income variance by using growth over three, four and (cid:28)ve years to also identify the covariance of income and consumption. In contrast to Blundell, Pistaferri, and Preston (2008), who assume that consumption follows a random walk, we will instead assume that the impulse response to a transitory shock follows a generic path, g(t), that, like the transitory income shock, has fallen to zero two years after the news of the shock. Figure III shows possible paths for both income and consumption, along with the alternative random walk impulse response of BPP. The best evidence for the speed at which the consumption response decays comes from Gelman (2016) and Fagereng, Holm, and Natvik (2016), both of which show that the response has entirely (or almost entirely) decayed two years after the shock. 17 We will maintain the assumption from BPP that the consumption response to a permanent shock to income follows a random walk proportional to the permanent shock. Under these assumptions the instantaneous (cid:29)ow of consumption is given by: (cid:90) t c = φP + g(t−s)dQ t s s t−2 and the covariance of time aggregated income and consumption growth over N ≥ 3 years is given by 1 Cov(∆Nc¯,∆Ny¯ ) = φ(N − )σ2 +2Cov(c˜,y˜) for N ≥ 3 (5) T T 3 p 16In the discrete time MA(2) model, yt =pt+εt+θ1εt−1+θ2εt−2, di(cid:27)erent de(cid:28)nitions of transitory variance are used. Carroll and Samwick (1997) estimate (1+θ 1 2+θ 2 2)σε while Blundell, Pistaferri, and Preston (2008) estimate σε. Ourde(cid:28)nitionofVar(y˜)isthecontinuoustimeanalogofCarrollandSamwick(1997). Itisnotclearwhattheanalogof σε wouldbeincontinuoustime. 17InsectionF,wewillshowhowthisassumptionmaypotentiallybiasthetransitoryconsumptionresponsedown,but thatthisbiasissmall,especiallyforallbutthemostliquidhouseholds. 12

0 1 2 3 4 5 2.1 8.0 4.0 0.0 Generic Transitory Impulse Responses, f(t) and g(t) Time noitpmusnoC/emocnI Income f(t) Consumption g(t) BPP Random Walk Figure III Generic Transitory Shock Impulse Response wheretotaltransitoryincome,y˜,isgivenbyequation4andtotaltransitoryconsumption, c˜, is de(cid:28)ned by (cid:90) T (cid:90) t c˜ = g(t−s)dQ dt (6) T s T−1 t−2 3.3 Minimum Distance Estimation Using the equations for variance (3) and covariance (5) of observed income and consumption growth over N years for at least two di(cid:27)erent values of N, we are able to estimate the following four unknowns in which we are interested:18 1. σ2 Variance of permanent shocks p 2. σ2 = Var(y˜) Variance of transitory income received in a year q˜ 3. φ Marginal Propensity to eXpend (MPX) w.r.t. permanent income 4. ψ = Cov(c˜,y˜) Regression coe(cid:30)cient of transitory consumption w.r.t. transitory Var(y˜) income over a year (MPX out of transitory income). Our panel data cover 13 years and we choose to use growth over three, four and (cid:28)ve years to balance greater identi(cid:28)cation (longer growth periods give more power) with three identi(cid:28)cation problems that grow with N. First, many households drop out of the 18We have a total of 96 moments (we have eight consecutive (cid:28)ve-year periods, each of which has three three-year growthperiods,twofour-yeargrowthperiods,andone(cid:28)ve-yeargrowthperiod. 8×(3+2+1)=48. Eachofthesegrowth periodshasbothavarianceandacovariancemoment,48×2=96). Withonlyfourparameterstoestimate,thesystem isoveridenti(cid:28)ed. WestronglyrejectthenulloftheSargen-HansenJ-testwhenrunonourdata,butthisisnotsurprising giventhesamplesizeofourdata. 13

sample if we demand they have reliable data for too many consecutive years. Second, if the permanent shock in fact decays slowly over time (e.g. is in fact AR(1)), the bias this introduces will be larger for large N. Third, the validity of running the regressions in levels (rather than logs) is reduced over large N when the potential for the variance of income to change signi(cid:28)cantly from the start to the end of the sample is high. In section 7 and online appendix J we test the importance of these issues. We follow Blundell, Pistaferri, and Preston (2008) and use diagonally weighted minimum distance estimation, although our results are not signi(cid:28)cantly changed by using other popular weighting methods.19 As the main part of our analysis will focus on the parameter ψ, it is worth describing exactly what this is and why we have labeled it the marginal propensity to expend out of transitory income. If we were able to exactly observe transitory income and consumption resulting from transitory income, then ψ would be the regression coe(cid:30)cient of this transitory consumption on transitory income. If transitory income shocks have no persistence, this is approximately a six-month MPX (on average, the shock will happen six months into the year so that the regression will pick up the change in consumption in the following six months). If transitory income shocks have a little persistence (online appendix B shows evidence of a small amount of transitory income persistence), ψ can only loosely be interpreted as the MPX to an income shock, and the reader should bear in mind that the true interpretation is, (cid:16)if income is higher by one unit this year due to transitory factors, then consumption this year will be expected to be higher by ψ units.(cid:17) 3.4 A Brief Introduction to the Time Aggregation Problem As previously explained, our identi(cid:28)cation comes from the shape of income and consumption covariance over increasing periods of time. An obvious question is why we have chosen not to use the well-known methodology of Blundell, Pistaferri, and Preston (2008), who achieve identi(cid:28)cation of transitory shocks from the fact that a transitory shock in period t will mean-revert in period t + 1.20 Unfortunately, the method is not robust to the time aggregation problem of Working (1960). While macroeconomists have long been aware of the importance of time aggregation in time series regressions 19Asoursamplesizeislarge,themotivationforusingdiagonallyweightedminimumdistance(DWMD)overoptimal minimumdistance(OMD)issmall;seeAltonjiandSegal(1996). WegetverysimilarresultsusingOMD.Ingeneral,our resultsmaybesubjecttomisspeci(cid:28)cationproblems,butthesamplesizeofourdatameansthatstandarderrorsaresmall. 20Kaplan and Violante (2010) show in discrete time simulations that the methodology works reasonably well for standard calibrations of bu(cid:27)er-stock models and end up concluding, (cid:16)The BPP insurance coe(cid:30)cients should become central in quantitative macroeconomics.(cid:17) However, some recent papers such as Commault (2017) and Hryshko and Manovskii(2018)havepointedtootherpotentialproblemsofthemethodology. 14

(see Campbell and Mankiw (1989) for a well-known example), the problem has been overlooked by the household (cid:28)nance and labor economics literature.21 We will therefore brie(cid:29)y describe the problem here. For a more detailed account with particular attention to BPP, see Crawley (2019). Time aggregation occurs when a time series is observed at a lower frequency than the underlying data that generates it. For example, income is often observed at an annual frequency when it may in fact consist of paychecks arriving at a monthly, biweekly or irregular frequency. 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 underlying intuition can be seen in (cid:28)gure IV, which shows the income process of a household that begins 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 (cid:29)ow occurring just once. The crosses show the income we actuallyobserveinannualdata. Duringthesecondyearthehouseholdreceivesanannual $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.22 While it would be possible to stick very closely to the original BPP model and adjust the covariance restrictions to take account of the time aggregation problem,23 we have found that in practice the underlying assumptions made by BPP (in particular that consumption follows a random walk) do not (cid:28)t with the data. The random walk assumption was previously thought to be benign. Not only were the estimates of the consumption response to transitory shocks in BPP small and consistent with such an assumption, Kaplan and Violante (2010) show that without time aggregation, the BPP method correctly identi(cid:28)es the transitory consumption response in the period of the income shock regardless of the consumption dynamics going forward. This fact is again not robust to the time aggregation problem. With time aggregation taken into 21Forexamples,seeMo(cid:30)ttandGottschalk(2012);MeghirandPistaferri(2004);NielsenandVissing-jorgensen(2004); Heathcote, Perri, and Violante (2010); and more recent quantile regression approaches such as Arellano, Blundell, and Bonhomme(2017). 22If all permanent shocks to income occurred on January 1 each year, then this would not hold. Low, Meghir, and Pistaferri (2010) show that a signi(cid:28)cant portion of permanent income variance is explained by job mobility, which can occuratanypointintheyear. 23Crawley(2019)takesthismorestraightforwardapproachusingthesamePSIDdataasusedinBPP. 15

$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 IV The Time Aggregation Problem account, theestimatesarehighlysensitivetoassumptionsaboutshort-termconsumption dynamics.24 Therefore we have chosen to attain identi(cid:28)cation in a manner similar to Carroll and Samwick (1997), which allows us to be agnostic about the exact short-term dynamics of income and consumption. 4 Data Our panel data on income and expenditure comes from Danish registry data from 2003- 2015. These data have a number of advantages over survey-based measures. First, the sample contains millions of households rather than thousands. Second, households are required by law to report their data, so there is much less risk of selection bias through drop outs. Third, measurement error in income data is largely eradicated, as employees’ income data is third party reported by their employer, compared to survey data where self-reported income has been shown to be particularly unreliable for irregular income.25 24SeeonlineappendixEtoseehowthesedi(cid:27)erentassumptionchangetheestimates. 25SeeDavid,Marquis,Moore,Stinson,andWelniak(1997)forasurveyofincomemeasurementerrorissuesinsurvey data. 16

4.1 Income Weareinterestedinincomeandconsumptiondecisionsatthehouseholdlevel. Wede(cid:28)ne a household as having either one or two adult members. Two adults are considered to be in the same household if they are living together and (i) are married to each other or have entered into a registered partnership, (ii) have at least one common child registered in the Civil Registration System or, (iii) are of opposite sex and have an age di(cid:27)erence of 15 years or less, are not closely related and live in a household with no other adults.26 In the panel data, an individual’s household will change if he or she gets married or divorced, which leads to some selection bias given that we require households to survive for at least (cid:28)ve years. Following the literature, our baseline results will be reported using thelaborincomeoftheheadofhousehold.27 Wewilluseaftertaxandtransferincome, as we are interested in the consumption response to these changes in income, although the method could be used to measure the extent of consumption insurance provided by the tax and transfer system. Our data come from the administrative records from the tax authority. The tax reporting system in Denmark is highly automated and individuals bear little of the reporting burden. For employees, income is reported by their employer and is thought to be highly accurate. The underground economy in Denmark is small. We remove business owners from the sample, as their income may be less accurately reported but, more importantly, the expenditure imputation method does not work well for them (see section 4.2). We work with the residual of income after controlling for observable characteristics of households that may a(cid:27)ect their income and consumption. To start, we remove households in the top and bottom 1% of the income distribution. We then normalize by average household income over the observed period and regress income on dummies for age, year, highest level of education, marital status, homeowner status, and number of children along with interaction of age with education, marital status, and homeowner status. We take the change in the residuals of this regression to be the unexpected income change for a household from one year to the next and remove households in the top and bottom 1% of the unexpected income change distribution. 26Adultslivingatthesameaddressbutnotmeetingoneofthethreecriteriaareregardedasseparatefamilies. Children livingwiththeirparentsareregardedasmembersoftheirparents’familyiftheyareunder25yearsold,haveneverbeen married or entered into a registered partnership, and do not themselves have children. A family meeting these criteria can consist of only two generations. If three or more generations live at the same address, the two younger generations areconsideredonefamily,whilethemembersoftheeldestgenerationconstituteaseparatefamily. 27SeeMo(cid:30)ttandZhang(2018)foranoverviewoftheliteratureonincomevolatilityinthePSID.Incontrasttothe PSIDliterature,wede(cid:28)netheheadofhouseholdasthehighestearneroverthe13-yearperiodinoursample. Webelieve thisde(cid:28)nitionbetter(cid:28)tsthesocialstructureinDenmark. 17

4.2 Imputed Expenditure We use data on expenditure, and will therefore refer to the Marginal Propensity to eXpend (MPX), for the rest of the paper to re(cid:29)ect the fact that our measure includes durables.28 Our expenditure data come from imputing expenditure from income and wealth. Along with other Scandinavian countries, Denmark is unusual in that tax reportingincludesinformationaboutwealthalongwithincome, alegacyfromthewealth tax that was phased out between 1989 and 1997. Following the methodology from Browning and Leth-Petersen (2003) and Fagereng and Halvorsen (2017), we impute expenditure using the identity ¯ ¯ ¯ ¯ C ≡ Y −S = Y −P −∆NW t t t t t t ¯ ¯ ¯ where C , Y , and S are the sum of expenditure, income, and savings over the year t, t t t respectively. P is contributions to privately administered pension schemes, for which we t have very accurate data due to tax deductibility, ∆NW is the change in (non-pension, t non-housing) net worth measured at the end of years t and t − 1. Banks and brokers are required to report the value of their clients’ accounts on December 31 each year, and the tax reporting year runs from January 1 to December 31, so the data for income and wealth reported in the tax returns match that required to use this identity to impute consumption. The method works well for households with simple (cid:28)nancial lives. One of the biggest problems with the method is its inability to handle capital gains well. The income used in the imputation includes all labor income and capital income; however, it excludes capital gains. The value of assets will vary due to savings from reported income but, also due to capital gains and losses, which we handle in a number of ways. First, we completely exclude housing wealth, treating housing as an o(cid:27)-balance-sheet asset for this calculation.29 The problem with treating housing in this way is that we must exclude households in years in which they are involved in a housing transaction. For the self-employed, it is also di(cid:30)cult to distinguish between expenditure and investment in their business, so we exclude all households that receive more than a trivial amount of their income from business ventures. Finally, households that hold signi(cid:28)cant equity investmentsarelikelytoseesizablecapitalgainsandlosses. Wemakeanaiveadjustment byassumingthattheyholdadiversi(cid:28)edindexofstocks. Whilethisassumptionwilllikely 28Seesection7.2forhowdurablesa(cid:27)ectourestimates. 29Weexcludehousingwealthfromnetworthonlyfortheimputationprocedure,whichcorrectlyidenti(cid:28)esexpenditure for years in which there is no housing transaction(cid:22)knowing house prices with precision in these years would not help imputeexpenditure. Ourmeasureofnetwealthwherewealthitselfisofinterest,forexampletableII,includeshousing. 18

lead to signi(cid:28)cant measurement error for these individuals, the concern is mitigated (cid:28)rst by the fact that stock holding is much more unusual in Denmark than in the United States, for example. Only around 10% of households hold any stocks, and for many of those households stocks make up only a small proportion of their total wealth. Furthermore, as we will explain in section 7.1, measurement error in consumption is not a concern unless it is correlated with changes in income. However, Baker, Kueng, Pagel, and Meyer (2018) show, in a German dataset, that the relation between income and imputation error is economically small. Another concern with the imputation method is transfers of wealth(cid:22)say, between family members or friends. Indeed imputed expenditure is negative for approximately 3% of households, which may explain a proportion of that. We discard both income and expenditure data for households in years in which their expenditure is negative. In online appendix J we test the robustness of our results to sample selection bias problems that these issues may give rise to. As with income, we work with the residual of expenditure after normalizing by mean household income and controlling for the same observable features as income. We follow exactly the same steps as described in section 4.1. Abildgren, Kuchler, Rasmussen, and Sorensen (2018) show that the mean levels of expenditure from this imputation method are close to those from the national accounts (see (cid:28)gure V). They (cid:28)nd relatively large di(cid:27)erences at the household level between the consumer survey and imputed expenditure, although it is not clear that this is a problem with the imputation method as opposed to the survey measure. Indeed, for car purchases, for which highly accurate register data are available, the consumer survey shows signi(cid:28)cant underreporting, consistent with Koijen, Nieuwerburgh, and Vestman (2014), who (cid:28)nd 30% underreporting of car purchases in the Swedish consumer survey. We believe that, with the exception of transaction-level data reported by (cid:28)nancial aggregation applications, the imputation method we use results in some of the highest quality expenditure data available to researchers for the types of questions we are addressing. 4.3 Sample Selection As our methodology requires income uncertainty to be relatively constant through the observed period and the young and old are likely to have predictable income trends unobservable to the econometrician, we limit the sample to households headed by an 19

Figure V Imputed Register Measure and National Account Measure of Expenditure (from Abildgren, Kuchler, Rasmussen, and Sorensen (2018)) individual between the ages of 30 and 55 in 2008.30 Our (cid:28)nal sample contains 7.7 million observations from 2004 to 2015 from an age group population totaling 18.1 million. The selection criterion that reduces the sample size the most is the requirement that a household does not make a housing transaction for a period of (cid:28)ve years. Table II shows summary statistics for all Danish households whose head (cid:28)ts into this age group as a whole as well as the sample we use in estimation. It is reassuring that both the mean and medianvaluesforafter-taxincomeandconsumptionaresimilarintheestimationsample and the population. Our estimation sample has much lower standard deviations as a mechanical result of excluding the top and bottom 1% of the income and consumption distributions that contain extreme values. Sample selection shows up in homeownership and car ownership, as we exclude those households that buy a house at the end of a (cid:28)ve year period but who otherwise would be counted as renters. As a result, our sample is, on average one year older than the population. Unhedged Interest Rate Exposure (URE) and Net Nominal Position (NNP) will be discussed in section 6, but again the signi(cid:28)cant di(cid:27)erences here are due to the housing transaction criteria. 5 Income and Consumption Characteristics by Household Wealth Liquidity constraints are the key microfoundation for the lack of consumption smoothing in heterogeneous agent models. In this section we look at the empirical relation between 30OnlineappendixBshowstheassumptionholdsforthisagegroup. 20

Table II Summary Statistics Estimation Sample Population (Age 30-55) Mean Median Std Dev Mean Median Std Dev After Tax Income 59,261 57,804 28,819 58,312 53,304 68,799 Consumption 52,680 48,344 28,581 54,022 46,373 38,126 Liquid Assets 18,438 6,856 33,016 23,331 6,578 81,473 Net Worth 74,937 19,115 157,295 85,799 12,952 564,404 Homeowner 0.57 1.00 0.50 0.50 1.00 0.50 Car Owner 0.66 1.00 0.47 0.55 1.00 0.50 Higher Education 0.31 0.00 0.46 0.33 0.00 0.47 Age 43.5 44.0 7.1 42.5 42.0 7.3 URE -28,052 -12,627 108,382 -47,589 -19,374 243,604 NNP -109,685 -65,810 156,523 -158,321 -85,207 542,498 No. Household-year obs 7,664,360 18,050,340 Notes: Valuesare2015USD.Agereferstotheagein2008ofthemainincomeearnerinthehousehold. Forthepurposesof calculationofconsumptioninthepopulation, topandbottom1%intermsofconsumptionhavebeenexcluded. UREand NNPcanonlybecalculatedintheperiod2009-2015duetomortgageinformationbeinginsu(cid:30)cientlydetailedintheprevious years. liquid wealth and the marginal propensity to expend (MPX) out of both permanent and transitory shocks to income. We (cid:28)nd a strong monotonic negative relation. We also look at net wealth and (cid:28)nd such a monotonic relation no longer holds. Using our entire estimation sample we (cid:28)nd a mean MPX out of transitory shocks of 0.50 and a mean MPX out of permanent shocks of 0.72. However, these averaged results hide a signi(cid:28)cant amount of heterogeneity. From the standpoint of consumption theory it is the ability of households to self-insure with their own wealth that mostly determines how much they smooth their consumption over shocks. We divide our estimation sample into quintiles according to both liquid wealth (which we de(cid:28)ne as bank deposits31) and net wealth. In each case, wealth is measured as the mean household wealth holdings over the entire sample period. Figure VI shows the estimated income variances and MPXs for households in each quintile of liquid wealth.32 Looking at the left-hand variance panel (cid:28)rst, it is noticeable that income uncertainty, and particularly permanent income uncertainty, is highest for households in the lowest quintile of liquid wealth. This low bu(cid:27)er-stock, despite high incomevolatility,providessomeevidencetowardstheideathatheterogeneoustastes(e.g. discountfactorsofriskaversion)maybemoreimportantthanincomeriskindetermining 31Theresultsarelittlechangedusinganyotherde(cid:28)nitionofliquidwealthaslongashousinganddebtsareexcluded. SeeonlineappendixJ. 32Forthesegraphs, andallsimilaronesinthispaper, the95%con(cid:28)denceintervalsareshownaboveandbeloweach quantileestimate. 21

Permanent and Transitory Variance by Liquid Wealth Quantile ecnairaV kcohS MPX by Liquid Wealth Quantile 0.008 s s p 2 q 2 P Tr e a r n m s a it n o e ry n t V V a a r r 0.006 0.004 0.002 0.000 $0−2,000 $2,000−6,000 $6,000−12,000 $12,000−30,000 > $30,000 XPM 1.0 f Permanent MPX y Transitory MPX 0.8 0.6 0.4 0.2 0.0 $0−2,000 $2,000−6,000 $6,000−12,000 $12,000−30,000 > $30,000 Figure VI Variance and MPX by Liquid Wealth Quintile wealth held for precautionary saving. For households in the top three quintiles of liquid wealth, the similarity of their level of income risk is remarkable. Note that in contrast to standard estimates of the U.S. income process, permanent income variance in Denmark is slightly higher than transitory variance, likely due to the high levels of social insurance available in Denmark. The variance level, at just over 0.002 for these top three quintiles, represents a standard deviation of just below 5% of permanent income per year. Note that the estimates of income variance we obtain are highly sensitive to our treatment of outliers, but our MPX estimates do not change.33 The right-hand panel of (cid:28)gure VI shows our estimates for the MPX out of permanent and transitory shocks by liquid wealth quintile. The lowest wealth quintile, who hold less than $2,000 in bank deposits on average over the sample period, look somewhat like hand-to-mouth consumers. They respond almost equally to permanent and transitory shocks, spending over 80% of income shocks in the year that it arrives. However, the fact that both permanent and transitory MPXs are very similar and signi(cid:28)cantly less than 1 suggests that these households may be more accurately modeled as saving in an illiquid asset such as housing or a pension following a rule of thumb (say, 20% of income) and then living hand to mouth on the remainder. As the quintile of liquid wealth increases, the MPX out of both transitory and permanent income decreases. In the top quintile, formed of households that maintained a mean bank balance above $30,000, the MPX out of permanent shocks is 0.57 and out of transitory shocks 0.23. From the point of 33SeeonlineappendixJforevidenceofthis. 22

Permanent and Transitory Variance by Net Wealth Quantile ecnairaV kcohS MPX by Net Wealth Quantile 0.006 s s p 2 q 2 P Tr e a r n m s a it n o e ry n t V V a a r r 0.005 0.004 0.003 0.002 0.001 0.000 < −20,000 $−20,000−3,000 $3,000−62,000 $62,000−182,000 > $182,000 XPM 1.0 f Permanent MPX y Transitory MPX 0.8 0.6 0.4 0.2 0.0 < −20,000 $−20,000−3,000 $3,000−62,000 $62,000−182,000 > $182,000 Figure VII Variance and MPX by Net Wealth Quintile view of theory, the responsiveness of spending out of permanent shocks in this quintile is low, while that of transitory shocks is high. Figure VII shows the estimates for households grouped by quintiles of net wealth. Here the pattern is slightly di(cid:27)erent. The quintile with the highest MPX out of both transitory and permanent income is the second lowest, the quintile that contains zero net worth. Households in the lowest quintile(cid:22)those with over $20,000 in net debt(cid:22) do not seem to distinguish between permanent and transitory income shocks in their consumption responses, but their MPX for both is about 10 percentage points lower than the quintile with close to zero net wealth. The pattern for quintiles 3 to 5 looks similar to that for liquid wealth: the MPX out of transitory shocks falls sharply to around 0.28, while that out of permanent shocks also falls but more slowly to 0.62. These results are broadly in line with the literature. The population mean of 0.5 for transitory MPX is a little higher than most estimates from table I, but, bearing in mind that our estimate includes durables and is best compared to a six-month MPC, it is certainly not an outlier. The MPX out of permanent shocks of 0.72 is also between the BPP estimate of 0.6534 and the estimate of 1.0 from Gelman, Gorodnichenko, Kariv, Koustas, Shapiro, Silverman, and Tadelis (2016). The strength of the relationship between liquid wealth and MPC is similar to that found in Gelman (2016) and stronger than in Fagereng, Holm, and Natvik (2016). 34Thepermanent(cid:16)insurance(cid:17) coe(cid:30)cientestimatedbyBPPdoesnotsu(cid:27)erasmuchfromthetimeaggregationproblem asthetransitorycoe(cid:30)cient. 23

6 Monetary Policy and the Redistribution Channel Auclert (2019) lays out a clear and intuitive theory as to how heterogeneity in the MPC out of transitory shocks a(cid:27)ects the transmission mechanism of monetary policy. He identi(cid:28)es channels through which monetary policy can act, including those involving redistribution. He then uses this theory to identify a small set of su(cid:30)cient statistics that help distinguish which of these channels are of quantitative importance. While these statistics in theory are highly informative about the transmission mechanism of monetary policy, good data and MPC estimation methods are required to estimate them convincingly. Auclert states, (cid:16)As administrative quality household surveys become available and more sophisticated identi(cid:28)cation methods for MPCs arise, a priority for future work is to re(cid:28)ne the estimates I provide here.(cid:17) We are able to bring our new MPC estimation method, along with administrative data from Denmark, in order to estimate Auclert’s su(cid:30)cient statistics. Our data have two signi(cid:28)cant advantages over previous e(cid:27)orts.35 First, our sample is large, containing most households in Denmark. Second, we have detailed balance sheet information for the households. Furthermore, we are able to identify interest rate risk and nominal positions held by (cid:28)rms, foreigners, and the government so that the aggregate position is zero, as required in equilibrium. This allows us to avoid some problematic assumptions otherwise needed for aggregation of household data. 6.1 Distribution of MPX across NNP, URE, and Income In this section we de(cid:28)ne a household’s Net Nominal Position (NNP) and Unhedged Interest Rate Exposure (URE), and examine the distribution of households’ transitory MPX along these dimensions, as well as income. In section 6.2 we show how these distributions can be used to quantify the importance of heterogeneity for monetary policy transmission. The redistribution e(cid:27)ects of monetary policy depend crucially on two household characteristics: • Net Nominal Position (NNP) is the net value of a household’s nominal assets and liabilities. Its relevance for analyzing the redistributive e(cid:27)ects of monetary policy comes from the fact that an unexpected rise in the price level will decrease 35Fagereng,Holm,andNatvik(2016)alsoestimateAuclert’ssu(cid:30)cientstatistics,imputingMPCsfromlotterywinnings inNorway,buttheyarelimitedbysamplesize. Ampudia,Georgarakos,Slacalek,Tristani,Vermeulen,andViolante(2018) lookatdi(cid:27)erencesinAuclert’sstatisticsbetweenEuropeancountriesbutdonotattempttoestimateMPCs. 24

the real wealth of households with positive nominal assets, redistributing it to those with negative NNP (who now have less real debt). In our administrative data, we observe directly held nominal positions and their composition at the household level, including bank deposits and loans, bond holdings, and mortgages. In aggregate, the directly held NNP position of the household sector is negative, which as we will see is balanced in the national accounts by the (cid:28)nancial sector as well as foreigners. • Unhedged Interest Rate Exposure (URE) is the di(cid:27)erence between all maturing assets (including income) and liabilities (including planned consumption).36 For example, households with a large variable rate mortgage will likely have very negative URE. For these households, the entire value of their mortgage will be adjusted to the new rate. When the interest rate rises for one period they will see their disposable income (after mortgage payments) go down, and hence if they have a high MPX their spending will also decrease. To calculate URE, we assume all household bank deposits and loans have a variable rate that changes instantaneously. For mortgage debt we directly observe the amount resetting over the following year and assume that the new rate will only apply for half of the year.37 For all other assets and liabilities we assume a maturity of (cid:28)ve years. As with NNP we (cid:28)nd households on aggregate have a negative URE position in our data, and this is counterbalanced by the interest rate position of the (cid:28)nancial sector.38 Figure VIII shows how the MPX varies across households, according to their URE, NNP, and income. In each case, the value on the x-axis has been divided by the mean level of expenditure in the sample. The top panel shows the estimated MPX for each decile of unhedged interest rate exposure. The deciles on the left contain households most negatively exposed to a rise in interest rates, those in the middle deciles have little exposure, while the two top deciles on the right have the most to gain from an interest rate rise. We have included in this (cid:28)gure data on both rates of homeownership 36We de(cid:28)ne (cid:16)maturing(cid:17) assets and liabilities as those that are due to have their interest rates reset, even if they contractually exist for a longer period. For example, a 30 year variable rate mortgage with annual interest rate (cid:28)xation periodsis(cid:16)maturing(cid:17) eachyearinourde(cid:28)nition. 37See online appendix C for more details on the Danish mortgage market. Note that the prevalence of (cid:28)xed-rate mortgages will strongly in(cid:29)uence the distribution of URE. To the extent that the United States has more (cid:28)xed-rate mortgages than Denmark, the interest rate exposure channel is likely to be smaller in the United States. Furthermore, householddebttoincomeisabouttwiceaslargeinDenmarkastheUnitedStates. 38SeeonlineappendixDformoredetailsonhowwecalculateNNPandUREpositions. 25

and median liquid assets for each decile. A pattern emerges in which we can roughly categorize the deciles into three groups following Violante, Kaplan, and Weidner (2014): • Wealthy Hand-to-Mouth: The (cid:28)rst (cid:28)ve deciles contain households with high homeownership rates but few liquid assets. These households have relatively high MPXs, and it is likely that their wealth is locked up in illiquid assets (mostly housing) and that they have large mortgages. • Poor Hand-to-Mouth: The next three deciles tend to be renters with very little in the way of liquid assets either. These households have very high MPXs and are close to being truly hand-to-mouth. As they have few assets, they have very little exposure to interest rates and cannot easily substitute consumption between periods; therefore, their consumption behavior is likely not a(cid:27)ected by changes in interest rates directly.39 • Wealthy: The top two deciles contain households that are both likely to be homeowners and hold very large liquid asset balances. These are likely to be households that own their house outright without a mortgage and have been able to build up a large stock of liquid assets. Relative to the other deciles, they have low MPX and are likely able to use their assets to e(cid:27)ectively smooth consumption. The distribution of MPX with NNP follows a similar pattern. As mortgages in Denmark are a mixture of (cid:28)xed and variable rates (see online appendix C for details on the Danish mortgage market), we can think of a typical household with negative URE or NNP as having a large mortgage, while those with positive URE or NNP are wealthy households with lots of liquid wealth. This pattern has not been evident in previous attempts to measure the distribution of MPX across these dimensions. Most importantly for the theory, the average MPX for those with negative URE and NNP positions is signi(cid:28)cantly greater than for those with positive URE or NNP. Note that the mean levels of both URE and NNP are negative for the households in our estimation sample, so even a constant (positive) MPX would result in interest rate hikes reducing their expenditure if not balanced by indirectly held exposures.40 The (cid:28)nal chart in (cid:28)gure VIII shows the distribution of MPX with total household income. There is a clear downward trend. If the income of lower-income households 39Neither the interest rate exposure channel nor the intertemporal substitution channel will have much impact on theirconsumption. Monetarypolicycouldimpacttheirexpenditurestronglythroughincomee(cid:27)ects. 40Incontrast,Auclert(2019)(cid:28)ndsameanpositiveUREacrosshouseholds. Webelievethedi(cid:27)erenceispartlydueto theprevalenceof(cid:28)xed-ratemortgagesintheUnitedStates,butalsoduetounderreportingofexpenditures,especiallyin thePSIDdata. 26

decreases more than that of high-income households during a monetary policy contraction, then expenditure will go down by more than the mean income-weighted MPX that would be the result of a representative agent model.41 6.2 Theoretical Setup and Su(cid:30)cient Statistics Auclert’s method is to consider individual households’ consumption response to a monetary policy shock in which (i) the real rate of interest changes for one period by dR, (ii) the price level makes a one-time change of dP and then remains at the new level, and (iii) aggregate income makes a transitory change of dY. While the dynamics here are clearly stylized, and in particular lack any lag in the economy’s response, we believe such a simple experiment to be highly informative as to the relative sizes of each transmission channel. Auclert (2019) divides the e(cid:27)ect of monetary policy on aggregate consumption into (cid:28)ve distinct channels: AggregateIncomeChannel EarningsHeterogeneityChannel FisherChannel (cid:122) (cid:125)(cid:124) (cid:123) (cid:122) (cid:125)(cid:124) (cid:123) (cid:122) (cid:125)(cid:124) (cid:123) dC dY dY dP = M +γE −E Y P C Y Y P dR dR +E −σS (7) R R R (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) InterestRateExposureChannel IntertemporalSubstitutionChannel where σ is the elasticity of intertemporal substitution, γ is the elasticity of relative income to aggregate income, and the (cid:28)ve su(cid:30)cient statistics, M, E , E , E , and S, are Y P R measurable in the data and de(cid:28)ned in table III.42 In (cid:28)gure VIII, we estimated MPX’s for households with heads between the ages of 30 and 55, excluding the young and the old. Furthermore, some of the URE and NNP exposures are held indirectly on households’ balance sheets through pension funds and corporations, or by the government and foreigners, so that the URE and NNP exposure in our sample does not aggregate to zero. We allocate the aggregate URE and NNP exposure from our sample into seven bins so that the total exposure across the economy iszero. Thesebinsincludehouseholdswith(i)young(<30)and(ii)old(>55)heads, and exposures held by households indirectly through (iii) pension funds, (iv) government, (v) non(cid:28)nancial corporates, (vi) (cid:28)nancials, and (vii) exposures held by the rest of the world. Within each of these bins we assume no heterogeneity so that the MPX with respect to 41For comparison, the distribution of MPX out of permanent income shocks across these three dimensions can be foundinonlineappendixK. 42Here we are making the simplifying assumptions that σ and γ are common for all households; see Auclert (2019) foradiscussion. 27

MPX by URE Decile URE/Mean Expenditure etar pihsrenwoemoH ro XPM 1.0 Transitory MPX Homeownership Liquid Assets (Right Axis) 0.8 $ 60,000 0.6 Poor Hand−to−Mouth $ 40,000 0.4 Wealthy Hand−to−Mouth $ 20,000 0.2 Wealthy 0.0 $ 0 2 8 7 9 7 7 6 1 8 3 6. 8 2. 4 1. 5 0. 9 0. 5 0. 2 0. 0 0. 0. 4 2. 2 − − − − − − − MPX by NNP Quantile NNP/Mean Expenditure XPM MPX by Income Quantile 1.0 Transitory MPX 0.8 0.6 0.4 0.2 0.0 −16.37 −6.38 −4.66 −3.41 −2.31 −1.31 −0.56 −0.14 0.09 1.92 Income/Mean Expenditure XPM 1.0 Transitory MPX 0.8 0.6 0.4 0.2 0.0 0.33 0.58 0.73 0.88 1.07 1.3 1.5 1.69 1.93 2.92 Figure VIII MPX Distribution by URE, NNP, and Income 28

these exposures is constant. This assumption is conservative, and likely underestimates the size of the heterogeneous agent channels. Our assumptions on the level of these MPXs can be seen in table IV. Table III Su(cid:30)cient Statistics De(cid:28)nitions Statistic De(cid:28)nition Description (cid:34) (cid:35) M 1 (cid:80) MPXY + (cid:80) MPX Y Income-weighted MPX C i i j j i∈Incomedeciles j∈{young,old} E M−MPXY Redistribution elasticity for Y Y (cid:34) C (cid:35) E 1 (cid:80) MPXNNP + (cid:80) MPX NNP Redistribution elasticity for P P C i i j j i∈NNPdeciles j∈bins (cid:34) (cid:35) E 1 (cid:80) MPXURE + (cid:80) MPX URE Redistribution elasticity for R R C i i j j i∈UREdeciles j∈bins (cid:34) (cid:35) S 1− 1 (cid:80) MPXC + (cid:80) MPX C Hicksian scaling factor C i i j j i∈Consumptiondeciles j∈{young,old} Note: MPXisthemeanMPXoverallhouseholdsintheeconomy. Y andCareaggregatehouseholdincomeandconsumption respectively. BinsreferstothesevencategoriesforwhichwehaveallocatedUREandNNPexposuresoutsideourestimationsample. {young,old}arethetwobinsthatcontainyoungandoldhouseholds(theother(cid:28)vebinsareonlyrelevantforUREandNNPexposuresas Y andCmeasurehousehold incomeandconsumption). We de(cid:28)ne E as R (cid:34) (cid:35) 1 (cid:88) (cid:88) E = MPX URE + MPX URE (8) R i i j j C i∈UREdeciles j∈bins where i sums over the 10 deciles of URE, j over the seven bins de(cid:28)ned above, and C is aggregate household expenditure in the economy. This method of dealing with the fact that aggregate exposure does not equal zero in the estimation sample is di(cid:27)erent than the approach taken by Auclert. He assumes the residual exposure is distributed equally across households in the sample. By making use of the national accounts, we believe we are able to get a better handle on the likely MPXs to attach to this residual exposure. 6.3 Out of Sample MPX The assumptions we make about the MPX of the young and the old, as well as out of indirectly held URE and NNP exposures, are shown in table IV. In each case we believe we have made conservative choices that will underestimate the size of the interest rate exposure channel of monetary policy. For the young we choose an MPX of 0.5, in line with the rest of the population. As the young have aggregate negative exposures, choosing an MPX on the low side is conservative. Similarly, for the old we choose an MPX of 0.5, which is on the high side for this age group. The assumption that there is no heterogeneity in MPX within these groups is also a very conservative assumption. 29

Table IV Aggregating Redistribution Elasticities MPX NNP URE E component E component P R Sample See Distribution -204 -61 -0.78 -0.29 Young 0.5 -32 -15 -0.12 -0.06 Old 0.5 -23 6 -0.09 0.02 Pension Funds 0.1 137 37 0.10 0.03 Government 0.0 -85 -23 0.00 0.00 Non-(cid:28)nancial Corp. 0.1 -49 -13 -0.04 -0.01 Financial Sector 0.1 223 61 0.17 0.05 Rest of World 0.0 33 9 0.00 0.00 Total 0 -0 -0.75 -0.26 Notes: NNPandUREnumbersareinbillionsof2015USD.PensionFundsincludesspecialsavingsuchaschildren’ssavings accounts. SeeappendixDfordetail. Much of the URE and NNP exposure is not held directly on the balance sheet of households but instead indirectly through pension funds, corporates, and the government. There is signi(cid:28)cant evidence that the MPX out of shocks to the value of pension wealth, stocks, or the government balance sheet is substantially lower than the MPX from income. We choose to use the estimate from Maggio, Kermani, and Majlesi (2019) that households’ MPX from changes in stock market wealth is about 10%. This choice is the most quantitatively important for the su(cid:30)cient statistics, as the bin containing the most exposure is the (cid:28)nancial sector, which is positively exposed to interest rate increases. This positive interest rate exposure may seem surprising because banks are typically thought to have long-term assets and short-term debt that would result in negative URE exposure. However, our (cid:28)ndings are in line with those of Landier, Sraer, and Thesmar (2013), who (cid:28)nd that the (cid:28)nancial sector bene(cid:28)ts from interest rate hikes overall. An important caveat is due here: we focus on the MPX out of changes in the assets indirectly held by households through the (cid:28)nancial sector and do not assume any spending or lending response at the bank level. This assumption may be reasonable in good times when banks are not credit constrained, but will hold during a banking crisis. Financial frictions could possibly result in monetary policy being much less e(cid:27)ective during a banking crisis as the interest rate exposure channel to household spending is counterbalancedbyachannelfrombankbalancesheetinterestrateexposuretolending.43 We choose an MPX of zero for government and the rest of the world. There is no 43Itshouldbenotedthatouranalysisisallonthehouseholdside. Evidencesuggeststhat(cid:28)rmsarealsosensitiveto changesincash(cid:29)ow;forexample,seeBlanchard,Lopez-deSilanes,andShleifer(1994). 30

evidence that households respond in any signi(cid:28)cant way to changes in the government’s balance sheet, and furthermore a low MPX is a conservative assumption for the size of the heterogeneous agent channels. As Denmark is a very small part of the world economy, we assume that foreigners spend a negligible proportion of their wealth there. 6.4 Results Our estimates of the (cid:28)ve su(cid:30)cient statistics, along with their standard errors, are shown in table V. The aggregate income channel is summarized by M that we estimate to be 0.52. This means that if income for all households in the economy increased by 1%, aggregate consumption growth would increase by 52 basis points, broadly in line with calibrationsofsaver-spendermodelsdesignedto(cid:28)tevidencefromCampbellandMankiw (1989). We (cid:28)nd little role for the redistribution e(cid:27)ect of income, E , despite the clear Y negative correlation between income and MPX seen in (cid:28)gure VIII.44 S, the Hicksian scaling factor, is 0.49, which reduces the size of the intertemporal substitution channel by close to half. The two most interesting statistics are E and E , both of which act through redis- P R tribution from households with low MPX to those with high MPX. E is estimated to P be negative 0.75 suggesting that a one-time increase in the price level of 1% increases aggregate consumption growth by 75 basis points due to redistribution from those with large nominal assets to those with large nominal debts. This Fisher channel of monetary policyisemphasizedinDoepkeandSchneider(2006). Theinterestrateexposurechannel is also large. We estimate E to be negative 0.26, suggesting that a 1% increase in the R interest rate decreases aggregate household expenditure growth by 26 basis points. For both of these channels, but particularly the interest rate exposure channel, it is informative to compare them to the size of the intertemporal substitution channel. An increase in the real interest rate reduces aggregate consumption today by σS multiplied by the percent point change in the rate. Reliable estimates of σ have been elusive to the economics profession, but there is very little evidence of a large positive number. Havranek (2015) provides a meta-study of the elasticity of intertemporal substitution and (cid:28)nds a mean of zero from studies using macrodata, and 0.3 to 0.4 for those using microdata. Many of these microlevel studies su(cid:27)er from identi(cid:28)cation problems.45 Best, Cloyne, Ilzetzki, and Kleven (2019) make use of mortgage notches in the United 44Patterson(2019)(cid:28)ndsalargerroleforincomeheterogeneitybydividinghouseholdsintogroupsthathavedi(cid:27)ering incomesensitivitytoaggregateincome. 45SeeCarroll(2001)foracritiqueofstudiesoftheelasticityofintertemporalsubstitution. 31

Table V Su(cid:30)cient Statistics M E E E S Y P R 0.52 -0.03 -0.75 -0.26 0.49 (0.01) (0.01) (0.03) (0.03) (0.01) Kingdom to overcome some of these problems. They estimate the average elasticity of intertemporal substitution to be 0.1, which would result in a size of the intertemporal substitution channel of monetary policy being 0.05, over (cid:28)ve times smaller than our estimate of the interest rate exposure channel.46 These estimates are signi(cid:28)cantly di(cid:27)erent to the results found in Auclert (2019). Our estimate of the Fisher channel is an order of magnitude larger, while our estimate of the interest rate exposure channel is over twice as large.47 7 Robustness In the online appendix we address a number of identi(cid:28)cation concerns.48 Here we focus on two that we think deserve highlighting: measurement error and durable expenditure. 7.1 Measurement Error Our identi(cid:28)cation comes from estimating Var(∆Ny¯) and Cov(∆Nc¯,∆Ny¯) using our observeddata. ForunbiasedestimatesofVar(∆Ny¯)werequirenomeasurementerrorinour observed changes in labor income. For unbiased estimation of Cov(∆Nc¯,∆Ny¯) we only require (further to no measurement error in income growth) that the measurement error in expenditure growth is uncorrelated with labor income growth. As our expenditure is imputed from income and changes in assets, this is potentially more of a concern than would be the case in survey data in which questions about consumption are not directly linked to those on income. We will examine potential sources of error in labor income and imputed consumption. 46Ourdecompositiondoesnotalloweasycomparisonoftheinterestrateexposurechannelwiththeaggregateincome channel,aswedonotmakeassumptionsabouthowmuchaggregateincomechanges. Cloyne,Ferreira,andSurico(2019) comparemortgagorswithoutrighthomeownersand(cid:28)ndtheaggregateincomechannelislargerthanthedirecte(cid:27)ectof highermortgagepayments. 47The di(cid:27)erences are mainly driven by two factors: (i) Our methodology uncovers more heterogeneity in MPX; (ii) we allocate URE and NNP exposures to sectors according to the national accounts, as opposed to assuming lump-sum rebatestohouseholds. 48Online appendix F looks a consumption persistence beyond two years; G goes into more detail on durables; H considersalternativeincomeprocesses;Ilooksattime-varyingrisk;andJlooksatalternativedatade(cid:28)nitions. 32

7.1.1 Labor Income For most workers, labor income is well measured. Third party reporting, along with a high level of trust in government institutions, means that underreporting is likely very low. The black economy in Denmark is small, and to the extent that any growth in unreported income is uncorrelated with growth in reported income this will not bias our estimates.49 In contrast to survey data, in which measurement error in income is likely to downwardly bias transitory MPX estimates, this is of very little concern in our data. 7.1.2 Imputed Expenditure Expenditure is calculated as the residual of total household income (including interest and dividends) after pension contributions and the change in net wealth have been deducted. For households with simple (cid:28)nancial lives (which we believe (cid:28)ts most of the Danish population), this should work well. There are a few scenarios that merit further investigation. • Stock Market Capital Gains: Only 10% of Danish households directly own stocks or mutual funds.50 In online appendix J we show that the qualitative patterns we observe are unchanged even when we completely remove these households from the sample. For households that do own stocks, we assume the return they receive is equal to a diversi(cid:28)ed portfolio of Danish stocks. Given that di(cid:27)erent householdswillhavetheirownidiosyncraticportfolios, thismethodologywillresult in signi(cid:28)cant measurement error. Baker, Kueng, Pagel, and Meyer (2018) show that the size of this measurement error is not only correlated with income and wealth, butalsowiththebusinesscycle. Furthermore, Fagereng, Guiso, Malacrino, and Pistaferri (2019) show that some groups of investors consistently outperform the market, which would lead us to consistently underestimate their expenditure. Our concern, however, is that the change in measurement error of expenditure be correlated with the change in labor income. Consistently underestimating expenditure by the same amount is therefore not a problem for us. Furthermore, as we have removed all aggregate e(cid:27)ects from the labor income residuals that we use in estimation, any measurement error correlated with the business cycle will be uncorrelated with our measure of changes in labor income. We see two 49Suchincomemayshowupasachangeinnetwealthandhenceexpenditure,butmeasurementerrorinthechange inexpenditureuncorrelatedwiththechangeinlaborincomewillnotbiasourMPXestimates. 50In our calculation we directly observe (cid:29)ows in and out of pension accounts, so these can be treated as o(cid:27) balance sheetinwhichcapitalgainsdonota(cid:27)ectourexpenditurecalculation. 33

potential ways in which mis-measuring stock returns may bias our results. First, if households have signi(cid:28)cantly invested in the stock of the (cid:28)rm they work for, which is likely only to be the case for high-level management. Second, to the extent that households invest their labor income gains halfway through the year, we will underestimate expenditure for those whose income increases, and overestimate it for those whose income decreases, leading us to underestimate the MPX. The size of this bias is limited by the size of excess expected returns, so our MPX estimate will be biased by no more than a few percentage points. • Family and Friends Transfers: If a household receives a transfer of money from their parents, for example, imputed expenditure will be lower than true expenditure by this amount. Large transfers typically occur upon death of a parent, which is likely to be uncorrelated with the household head’s labor income, orwhenpurchasingahouse(cid:22)yearsthatwehavealreadyexcludedfromoursample. However, to the extent that friends and family actively insure each other’s labor income, our MPX estimates will be upward biased. • O(cid:27)-Balance-Sheet Assets: A larger concern is that some forms of saving may be hidden o(cid:27) balance sheet. Our imputation method would interpret o(cid:27)-balancesheet saving as expenditure, so our estimate of the MPX would increase one-to-one for each percentage point of saving out of income shocks performed o(cid:27) balance sheet. All Danish banks and brokers are required to report their clients’ holdings, so o(cid:27)-balance-sheet assets are likely to be either o(cid:27)shore or non(cid:28)nancial assets. Such o(cid:27)-balance-sheet saving would be a large concern if we were focused on the expenditure of the super wealthy 0.1%, but is less so when dividing the population into quintiles or deciles as we have done. 7.2 Durables A critique of our empirical methodology is that it does not take account of durable goods, while our data include all spending (except on real estate) and therefore include large and durable goods such as cars and home improvements. The empirical model assumes that in response to a transitive income shock, expenditure increases temporarily for up to two years, which is entirely consistent with a model that includes durable goods. However, the model assumes that in response to a permanent shock to income, expenditure increases once to a new permanent level. A model that included durable goods would instead imply a large one-o(cid:27) expenditure on durable goods to get the 34

MPX by Liquid Wealth Quantile XPM MPX by Liquid Wealth Quantile 1.0 0.8 0.6 0.4 0.2 0.0 XPM MPX by Liquid Wealth Quantile 1.0 0.8 0.6 0.4 0.2 0.0 $0−2,000 $2,000−6,000 $6,000−12,000 $12,000−30,000 $30,000+ XPM 1.0 All Expenditure f Permanent MPX Excluding Cars y Transitory MPX Nondurable Proxy 0.8 0.6 0.4 0.2 0.0 $0−2,000 $2,000−6,000 $6,000−12,000 $12,000−30,000 $30,000+ Figure IX MPX Removing Cars and Using the Nondurable Proxy Panel households up to their desired (cid:29)ow of durable good services, followed by a decrease back to a permanent level of spending that accounts for replenishing the higher level of depreciating durable goods. In online appendix G we address this problem in two ways. First we show that our MPX estimates are unbiased in a simple model that includes durables, as long as we interpret the MPX out of transitory shocks to include durable expenditure (the correct de(cid:28)nition for understanding aggregate demand) and the MPX out of permanent shocks to include only the consumption (cid:29)ow from durables. Second, we are able to construct a nondurable consumption proxy for each household using registry data on car purchases. This proxy has very large measurement error, but will result in unbiased estimates of the MPC (excluding durables) to both permanent and transitory shocks. The estimated MPCs by liquid wealth quintile are shown in (cid:28)gure IX. The (cid:28)gure shows the estimates usingthenondurableproxyare, asexpected, lowerthanthoseincludingallexpenditures, although the change in magnitude is similar in size to the overall fraction of durable expenditure, suggesting durables do not play a special role in expenditures following transitory shocks. For the top quintile, durables do appear to play an outsized role, accounting for about a third of the expenditure response to transitory shocks. 35

8 Conclusion In this paper we have presented a new method to measure the sensitivity of consumption to permanent and transitory income shocks for di(cid:27)erent groups of households. Our focus has been to use this method to test the microfoundations of heterogeneous agent models and quantify the importance of consumption heterogeneity for monetary policy. With administrative data from Denmark, we have been able to dig into the distribution of MPC across wealth, URE and NNP in far more detail than has previously been possible. We (cid:28)nd that MPCs vary systematically along these dimensions and in ways that are important for monetary policy transmission, although the current generation of heterogeneous agent models struggle to (cid:28)t the high sensitivity to income that we observe. Our hope is that the method we present in this paper, or variants of it, can also be of use to economists in a variety of (cid:28)elds. More and more high-quality microdata on consumption are becoming available, such as the administrative data used here, or the even more detailed transaction-level data available from (cid:28)nancial aggregators. If this trend continues, as we hope it will, methods such as ours will become even more valuable in bridging the gap between models and data. 36

References Abildgren, Kim, Andreas Kuchler, America Solange Lohmann Rasmussen, and Henrik Sejerbo Sorensen (2018): (cid:16)Consistency between Household-Level Consumption Data from Registers and Surveys,(cid:17) Danmarks Nationalbank Working Paper, no. 131. Agarwal, Sumit, and Wenlan Qian (2014): (cid:16)Consumption and Debt Response to UnanticipatedIncomeShocks: EvidencefromaNaturalExperimentinSingapore,(cid:17) American Economic Review, 104(12), 4205(cid:21)30. Aiyagari, S. Rao (1994): (cid:16)Uninsured Idiosyncratic Risk and Aggregate Saving,(cid:17) Quarterly Journal of Economics, 109, 659(cid:21)684. Altonji, Joseph, and Lewis M. Segal (1996): (cid:16)Small-Sample Bias in GMM Estimation of Covariance Structures,(cid:17) Journal of Business & Economic Statistics, 14(3), 353(cid:21)66. Ampudia, Miguel, Dimitris Georgarakos, Jiri Slacalek, Oreste Tristani, Philip Vermeulen, and Giovanni Violante (2018): (cid:16)Monetary Policy and Household Inequality,(cid:17) SSRN Scholarly Paper ID 3223542, Social Science Research Network, Rochester, NY. Arellano, Manuel, Richard Blundell, and StØphane Bonhomme (2017): (cid:16)Earnings and Consumption Dynamics: A Nonlinear Panel Data Framework,(cid:17) Econometrica, 85(3), 693(cid:21)734. Auclert, Adrien (2019): (cid:16)Monetary Policy and the Redistribution Channel,(cid:17) American Economic Review, 109(6), 2333(cid:21)67. Baker, Scott R. (2018): (cid:16)Debt and the Response to Household Income Shocks: Validation and Application of Linked Financial Account Data,(cid:17) Journal of Political Economy, 126(4), 1504(cid:21)57. Baker, Scott R., Lorenz Kueng, Michaela Pagel, and Steffen Meyer (2018): (cid:16)Measurement Error in Imputed Consumption,(cid:17) SSRN Scholarly Paper ID 3179117, Social Science Research Network, Rochester, NY. Best, Michael Carlos, James S Cloyne, Ethan Ilzetzki, and Henrik J Kleven (2019): (cid:16)Estimating the Elasticity of Intertemporal Substitution Using Mortgage Notches,(cid:17) Review of Economic Studies, Forthcoming. Bewley, Truman (1983): (cid:16)A Di(cid:30)culty with the Optimum Quantity of Money,(cid:17) Econometrica, 51(5), 1485(cid:21)504. 37

Blanchard, Olivier Jean, Florencio Lopez-de Silanes, and Andrei Shleifer (1994): (cid:16)What Do Firms Do with Cash Windfalls?,(cid:17) Journal of Financial Economics, 36(3), 337(cid:21)60. Blundell, Richard, Luigi Pistaferri, and Ian Preston (2008): (cid:16)Consumption Inequality and Partial Insurance,(cid:17) American Economic Review, 98(5), 1887(cid:21)1921. Browning, Martin, and M. Dolores Collado (2001): (cid:16)The Response of Expenditures to Anticipated Income Changes: Panel Data Estimates,(cid:17) American Economic Review, 91(3), 681(cid:21)92. Browning, Martin, and Słren Leth-Petersen (2003): (cid:16)Imputing Consumption from Income and Wealth Information,(cid:17) Economic Journal, 113(488), F282(cid:21)F301. Bunn, Philip, Jeanne Le Roux, Kate Reinold, and Paolo Surico (2018): (cid:16)The Consumption Response to Positive and Negative Income Shocks,(cid:17) Journal of Monetary Economics, 96, 1(cid:21)15. Campbell, JohnY., andN.GregoryMankiw(1989): (cid:16)Consumption,IncomeandInterest Rates: Reinterpreting the Time Series Evidence,(cid:17) NBER Chapters, National Bureau of Economic Research. Carroll, Christopher, and Andrew Samwick (1997): (cid:16)The Nature of Precautionary Wealth,(cid:17) Journal of Monetary Economics, 40(1), 41(cid:21)71. Carroll, Christopher, Jiri Slacalek, Kiichi Tokuoka, and Matthew N. White (2017): (cid:16)TheDistributionofWealthandtheMarginalPropensitytoConsume,(cid:17) Quantitative Economics, 8(3), 977(cid:21)1020. Carroll, Christopher Dixon (2001): (cid:16)Death to the Log-Linearized Consumption Euler Equation! (And Very Poor Health to the Second-Order Approximation),(cid:17) Advances in Macroeconomics, 1(1). Cloyne, James, Clodomiro Ferreira, and Paolo Surico (2019): (cid:16)Monetary Policy when Households Have Debt: New Evidence on the Transmission Mechanism,(cid:17) Review of Economic Studies, Forthcoming. Commault, Jeanne(2017): (cid:16)HowDoesConsumptionRespondtoaTransitoryIncomeShock? Reconciling Natural Experiments and Structural Estimations,(cid:17) Working Paper. Coronado, Julia Lynn, Joseph P. Lupton, and Louise M. Sheiner (2005): (cid:16)The Household Spending Response to the 2003 Tax Cut: Evidence from Survey Data,(cid:17) FEDS Discussion Paper 32, Federal Reserve Board. 38

Crawley, Edmund (2019): (cid:16)In Search of Lost Time Aggregation,(cid:17) Finance and Economics Discussion Series 2019(cid:21)075, Washington: Board of Governors of the Federal Reserve System. David, Martin, Kent H. Marquis, Jeffrey C Moore, Linda L. Stinson, and Edward J. Welniak(1997): (cid:16)IncomeMeasurementErrorinSurveys: AReview-Semantic Scholar,(cid:17) Discussion paper, U.S. Bureau of the Census, Washington. Doepke, Matthias, and Martin Schneider (2006): (cid:16)In(cid:29)ation and the Redistribution of Nominal Wealth,(cid:17) Journal of Political Economy, 114(6), 1069(cid:21)97. Fagereng, Andreas, Luigi Guiso, Davide Malacrino, and Luigi Pistaferri (2019): (cid:16)Heterogeneity and Persistence in Returns to Wealth,(cid:17) Econometrica, Forthcoming. Fagereng, Andreas, and Elin Halvorsen (2017): (cid:16)Imputing Consumption from Norwegian Income and Wealth Registry Data,(cid:17) Journal of Economic and Social Measurement, 42(1), 67(cid:21)100. Fagereng, Andreas, Martin B. Holm, and Gisle J. Natvik (2016): (cid:16)MPC HeterogeneityandHouseholdBalanceSheets,(cid:17) DiscussionPaper,StatisticsNorway,Research Department. Fuster, Andreas, Greg Kaplan, and Basit Zafar (2018): (cid:16)What Would You Do With $500? SpendingResponsestoGains,Losses,NewsandLoans,(cid:17) NBERWorkingPaper24386, National Bureau of Economic Research. Ganong, Peter, and Pascal Noel (2019): (cid:16)Consumer Spending during Unemployment: Positive and Normative Implications,(cid:17) American Economic Review, 109(7), 2383(cid:21)2424. Garriga, Carlos, Finn E. Kydland, and Roman Sustek (2017): (cid:16)Mortgages and Monetary Policy,(cid:17) The Review of Financial Studies, 30(10), 3337(cid:21)3375. Gelman, Michael (2016): (cid:16)What Drives Heterogeneity in the Marginal Propensity to Consume? Temporary Shocks vs Persistent Characteristics,(cid:17) Working Paper. Gelman, Michael, Yuriy Gorodnichenko, Shachar Kariv, Dmitri Koustas, Matthew D. Shapiro, Dan Silverman, and Steven Tadelis(2016): (cid:16)TheResponseof Consumer Spending to Changes in Gasoline Prices,(cid:17) NBER Working Paper 22969, National Bureau of Economic Research. Gelman, Michael, Shachar Kariv, Matthew D. Shapiro, Dan Silverman, and Steven Tadelis (2014): (cid:16)Harnessing Naturally Occurring Data to Measure the Response of Spending to Income,(cid:17) Science, 345(6193), 212(cid:21)15. 39

Greenwald, Daniel (2018): (cid:16)The Mortgage Credit Channel of Macroeconomic Transmission,(cid:17) SSRN Scholarly Paper ID 2735491, Social Science Research Network, Rochester, NY. Guvenen, Fatih (2007): (cid:16)Learning Your Earning: Are Labor Income Shocks Really Very Persistent?,(cid:17) American Economic Review, 97(3), 687(cid:21)712. (2009): (cid:16)An Empirical Investigation of labor income Processes,(cid:17) Review of Economic Dynamics, 12(1), 58(cid:21)79. Hausman, Joshua K. (2012): (cid:16)Fiscal Policy and Economic Recovery: The Case of the 1936 Veterans’ Bonus,(cid:17) mimeo, University of California, Berkeley. Havranek, Tomas (2015): (cid:16)Measuring Intertemporal Substitution: The Importance of Method Choices and Selective Reporting,(cid:17) Journal of the European Economic Association, 13(6), 1180(cid:21)1204. Heathcote, Jonathan, Fabrizio Perri, and Giovanni L. Violante (2010): (cid:16)Unequal We Stand: An Empirical Analysis of Economic Inequality in the United States, 1967-2006,(cid:17) Review of Economic Dynamics, 13(1), 15(cid:21)51. Hryshko, Dmytro, and Iourii Manovskii (2018): (cid:16)Income Dynamics and Consumption Insurance,(cid:17) Working Paper. Hsieh, Chang-Tai (2003): (cid:16)Do Consumers React to Anticipated Income Changes? Evidence from the Alaska Permanent Fund,(cid:17) American Economic Review, 99(1), 397(cid:21)405. Huggett, Mark (1993): (cid:16)The risk-free rate in heterogeneous-agent incomplete-insurance economies,(cid:17) Journal of Economic Dynamics and Control, 17(5), 953(cid:21)69. Imrohoro§lu, Ay‡e (1989): (cid:16)Cost of Business Cycles with Indivisibilities and Liquidity Constraints,(cid:17) Journal of Political Economy, 97(6), 1364(cid:21)83. Jappelli, Tullio, and Luigi Pistaferri (2010): (cid:16)The Consumption Response to Income Changes,(cid:17) Annual Review of Economics, 2(1), 479(cid:21)506. (2014): (cid:16)Fiscal Policy and MPC Heterogeneity,(cid:17) American Economic Journal: Macroeconomics, 6(4), 107(cid:21)36. Johnson, David S., Jonathan A. Parker, and Nicholas S. Souleles (2006): (cid:16)Household Expenditure and the Income Tax Rebates of 2001,(cid:17) American Economic Review, 96(5), 1589(cid:21)610. 40

(2009): (cid:16)The Response of Consumer Spending to Rebates During an Expansion: Evidence from the 2003 Child Tax Credit,(cid:17) Working Paper, The Wharton School. Kaplan, Greg, Benjamin Moll, and Giovanni L. Violante (2018): (cid:16)Monetary Policy According to HANK,(cid:17) American Economic Review, 108(3), 697(cid:21)743. Kaplan, Greg, and Giovanni L. Violante (2010): (cid:16)How Much Consumption Insurance beyond Self-Insurance?,(cid:17) American Economic Journal: Macroeconomics, 2(4), 53(cid:21)87. (2018): (cid:16)Microeconomic Heterogeneity and Macroeconomic Shocks,(cid:17) Journal of Economic Perspectives, 32(3), 167(cid:21)94. Koijen, Ralph, Stijn Van Nieuwerburgh, and Roine Vestman (2014): (cid:16)Judging the QualityofSurveyDatabyComparisonwith"Truth"asMeasuredbyAdministrativeRecords: Evidence From Sweden,(cid:17) in NBER Chapters: Improving the Measurement of Consumer Expenditures, pp. 308(cid:21)46. National Bureau of Economic Research. Krusell, Per, and Anthony Smith (1998): (cid:16)Income and Wealth Heterogeneity in the Macroeconomy,(cid:17) Journal of Political Economy, 106(5), 867(cid:21)96. Landier, Augustin, David Sraer, and David Thesmar (2013): (cid:16)Banks’ Exposure to Interest Rate Risk and the Transmission of Monetary Policy,(cid:17) NBER Working Paper 18857, National Bureau of Economic Research. Low, Hamish, Costas Meghir, and Luigi Pistaferri (2010): (cid:16)Wage Risk and Employment Risk over the Life Cycle,(cid:17) American Economic Review, 100(4), 1432(cid:21)67. Lusardi, Annamaria (1996): (cid:16)Permanent Income, Current Income, and Consumption: Evidence from Two Panel Data Sets,(cid:17) Journal of Business and Economic Statistics, 14(1), 81(cid:21)90. Maggio, Marco Di, Amir Kermani, and Kaveh Majlesi (2019): (cid:16)Stock Market Returns and Consumption,(cid:17) Journal of Finance, Forthcoming. Manovskii, Iourii, and Dmytro Hryshko (2017): (cid:16)How Much Consumption Insurance in the U.S.?,(cid:17) Working Paper 1584, Society for Economic Dynamics. Meghir, Costas, and Luigi Pistaferri (2004): (cid:16)Income Variance Dynamics and Heterogeneity,(cid:17) Econometrica, 72(1), 1(cid:21)32. Moffitt, Robert, and Sisi Zhang (2018): (cid:16)Income Volatility and the PSID: Past Research and New Results,(cid:17) AEA Papers and Proceedings, 108, 277(cid:21)80. 41

Moffitt, Robert A., and Peter Gottschalk (2012): (cid:16)Trends in the Transitory Variance of Male Earnings Methods and Evidence,(cid:17) Journal of Human Resources, 47(1), 204(cid:21)36. Nielsen, Helena Skyt, and Annette Vissing-jorgensen (2004): (cid:16)The Impact of Labor IncomeRiskonEducationalChoices: EstimatesandImpliedRiskAversion,(cid:17) WorkingPaper, University of Aarhus. Olafsson, Arna, and Michaela Pagel (2018): (cid:16)The Liquid Hand-to-Mouth: Evidence from Personal Finance Management Software,(cid:17) The Review of Financial Studies, 31(11), 4398(cid:21)446. Parker, Jonathan A. (1999): (cid:16)The Reaction of Household Consumption to Predictable Changes in Social Security Taxes,(cid:17) American Economic Review, 89(4), 959(cid:21)73. Parker, Jonathan A, Nicholas S Souleles, David S Johnson, and Robert McClelland (2013): (cid:16)Consumer Spending and the Economic Stimulus Payments of 2008,(cid:17) American Economic Review, 103(6), 2530(cid:21)53. Patterson, Christina (2019): (cid:16)The Matching Multiplier and the Ampli(cid:28)cation of Recessions,(cid:17) Mimeo, Department of Economics, MIT. Reiter, Michael (2009): (cid:16)Solving heterogeneous-agent models by projection and perturbation,(cid:17) Journal of Economic Dynamics and Control, 33(3), 649(cid:21)665. Sahm, Claudia R., Matthew D. Shapiro, and Joel B. Slemrod (2010): (cid:16)Household Responsetothe2008TaxRebate: SurveyEvidenceandAggregateImplications,(cid:17) Tax Policy and the Economy, 24, 69(cid:21)110. Shapiro, Matthew D., and Joel Slemrod (2009): (cid:16)Did the 2008 Tax Rebates Stimulate Spending?,(cid:17) American Economic Review, 99(2), 374(cid:21)79. Souleles, Nicholas S. (1999): (cid:16)The Response of Household Consumption to Income Tax Refunds,(cid:17) American Economic Review, 89(4), 947(cid:21)58. (2002): (cid:16)Consumer Response to the Reagan Tax Cuts,(cid:17) Journal of Public Economics, 85, 99(cid:21)120. Violante, Gianluca, Greg Kaplan, and Justin Weidner (2014): (cid:16)The Wealthy Handto-Mouth,(cid:17) Brookings Papers on Economic Activity, 2014(1), 77(cid:21)138. Wong, Arlene (2016): (cid:16)Population Aging and the Transmission of Monetary Policy to Consumption,(cid:17) 2016 Meeting Paper 716, Society for Economic Dynamics. 42

Working, Holbrook (1960): (cid:16)Note on the Correlation of First Di(cid:27)erences of Averages in a Random Chain,(cid:17) Econometrica, 28(4), 916(cid:21)18. 43

Online Appendix A Identi(cid:28)cation with Time Aggregation In this section we show how to derive equations 3 and 5 for the variance of income growth and covariance of income and consumption growth. From equation 2 we have (cid:90) T (cid:90) T−N ∆Ny¯ = (T −s)dP +(P −P )+ (s−(T −2))dP T s T−1 T−N s T−1 T−N−1 (cid:16) (cid:90) T (cid:90) t (cid:90) T−N (cid:90) t (cid:17) + f(t−s)dQ dt− f(t−s)dQ dt (9) t t T−1 t−2 T−N−1 t−2 Making use of the independent increment property of P and Q , we get t t (cid:90) T (cid:90) T−N Var(∆Ny¯ ) = (T −s)2σ2ds+(N −1)σ2 + (s−(T −2))2σ2ds T P P P T−1 T−N−1 (cid:16) (cid:90) T (cid:90) t (cid:17) (cid:16) (cid:90) T−N (cid:90) t (cid:17) +Var f(t−s)dQ dt +Var f(t−s)dQ dt t t T−1 t−2 T−N−1 t−2 1 = (N − )σ2 +2Var(y˜) for n ≥ 3 (10) 3 P The equivalent of equation 2 for consumption is (cid:90) T (cid:90) T−N ∆Nc¯ = (T −s)φdP +φ(P −P )+ (s−(T −2))φdP T s T−1 T−N s T−1 T−N−1 (cid:16) (cid:90) T (cid:90) t (cid:90) T−N (cid:90) t (cid:17) + g(t−s)dQ dt− g(t−s)dQ dt (11) t t T−1 t−2 T−N−1 t−2 Again making use of the independent increment property, we can calculate the covariance of income and consumption growth: (cid:90) T (cid:90) T−N Cov(∆Nc¯,∆Ny¯ ) = (T −s)2φσ2ds+φ(N −1)σ2 + (s−(T −2))2φσ2ds T T P P P T−1 T−N−1 (cid:16) (cid:90) T (cid:90) t (cid:90) T (cid:90) t (cid:17) +Cov f(t−s)dQ dt, g(t−s)dQ dt t t T−1 t−2 T−1 t−2 (cid:16) (cid:90) T−N (cid:90) t (cid:90) T−N (cid:90) t (cid:17) +Cov f(t−s)dQ dt, g(t−s)dQ dt t t T−N−1 t−2 T−N−1 t−2 1 = φ(N − )σ2 +2Cov(c˜,y˜) for N ≥ 3 (12) 3 p 44

B Sample Selection We choose to look at households whose head is between the ages of 30 and 55 in 2008, whichisdrivenbythedesiretoremovehouseholdsforwhichtheassumptionthatmostof the income growth is unexpected is not likely to be ful(cid:28)lled. For the old and the young, individual households will likely have a lot of information about their income path that is not available to the econometrician (for example, the year in which they plan to retire, orthefactthattheyareonaspeci(cid:28)ccareertrackwithsetexpectationsofpromotionand pay raises). We also want to remove households whose income volatility is increasing or decreasing sharply. Figures B.1 and B.2 show how our estimates of both income variance and MPX vary with age. The dots represent the point estimate for each age, while the lines are the centered moving averages over the (cid:28)ve nearest age groups. The solid black line shows the total variance of income growth over one year. It should not be surprising that income growth for households with heads in their 20’s is highly volatile. This volatility plateaus around the age of 35 and stays at a constant level until retirement, at which point it temporarily grows before falling to an even lower level. We can see that while both transitory and permanent shocks to income are high early in life, permanent income shocks are particularly high while individuals (cid:28)nd their place in the workforce. From the ages of 30 to 55, both transitory and permanent shocks are approximately the same size and remarkably stable. At retirement, shocks to permanent income rise(cid:22)not surprising, as the model sees retirement itself as a shock(cid:22)even as transitory income variance declines. As the model assumes the variance to permanent and transitory shocks to be constant in the observed period, interpretation of the numbers outside of the 30 to55 age group needs to be treated with care. However, the (cid:28)gure clearly shows that within this age group the assumption of constant variance appears to be a reasonable one. The dotted black line shows the variance of ∆y, assuming no persistence in the transitory component. The fact that this line is slightly above the empirical variance of ∆y is consistent with some persistence in the transitory component of income, justifying our decision to exclude growth over one and two years in our identi(cid:28)cation. The level of both permanent and transitory shock variance for households aged 30 to 55 is approximately 0.0035, re(cid:29)ecting a standard deviation of 6%. Estimates using U.S. data are signi(cid:28)cantly higher, especially for the transitory shock variance (for example, Carroll and Samwick (1997) estimate 0.02 for permanent and 0.04 for transitory). This di(cid:27)erencemaybeduetolowerincomeinequalityinDenmark, moreprogressivetaxation, 45

lll l ll l ll ll lllllllllllllllllllllllll l llllll 30 40 50 60 520.0 020.0 510.0 010.0 500.0 000.0 Permanent and Transitory Variance by Age Age ecnairaV kcohS s 2 Permanent Var p l s 2 Transitory Var q l l Var(D y) l 2 l s 2+2s 2 l 3 p q l l l ll l l l l lllllllllllllllllllllllll lll l l l l l lllllllllllllllllllllllllllllllll l ll ll Figure B.1 Permanent and Transitory Shock Variance by Age and more generous unemployment insurance. The lower transitory variance will also be due to signi(cid:28)cantly reduced measurement error relative to the survey-based U.S. data. C The Danish Mortgage Market MortgageloansinDenmarkareissuedbyspecializedmortgagebanks, whichfully(cid:28)nance loans by issuing bonds. Interest rates are directly determined by sales prices at the bond market. That is, borrowers only pay the bond market interest rate plus a supplementary fee for the mortgage bank. Most loans are issued as 20- or 30-year loans, and households can only obtain loans from mortgage banks for up to 80% of the value at loan origination of properties used as permanent residences. The remaining (more insecure) part of the funding may be provided by commercial banks. The close link between loans and bonds, as well as (cid:28)xed loan-to-value ratios, fast foreclosure procedures, full recourse, etc., mean that mortgage banks do not assume signi(cid:28)cant market risks. The status of Danish covered mortgage bonds as a safe asset class (AAA-rated by, e.g., S&P) implies that borrowers have access to very cheap real estate funding. 46

ll l l ll l l llll l l l l ll lll l ll l l l l lll l l l l l l l l l l l l 30 40 50 60 0.1 8.0 6.0 4.0 2.0 0.0 MPX by Age Age XPM l l ll l l l l l ll l l l l l l l l l ll l ll ll ll l l ll ll l ll l l ll l f Permanent MPX y Transitory MPX Figure B.2 MPX by Age Figure C.1 Mortgage Debt by Type (All Households) Source: Danmarks Nationalbank 47

Figure C.2 Mortgage Debt by Maturity (All Households Excluding Self-employed) Source: Danmarks Nationalbank The Danish mortgage system has been functioning for two centuries, but signi(cid:28)cant liberalization has taken place over the past 20 years. Variable interest loans were (re- )introduced in 1996, while interest only loans were introduced in 2003. These new loan characteristics are by now very popular; see (cid:28)gure C.1. In contrast to the United States, where most mortgage debt is (cid:28)xed rate, 40% of mortgage debt in Denmark is variable rate, with interest (cid:28)xation periods mostly between six months and (cid:28)ve years. Fixedrate loans come with an option for early redemption, which implies that in practice, re(cid:28)nancing of (cid:28)xed-rate mortgages often takes place, both when interest rates decrease and increase. The latter may be attractive because borrowers have the option to repay their loan by purchasing the corresponding amount of bonds. When interest rates increase, the bond value decreases, so the option to repay the loan by purchasing the corresponding amount of bonds in essence acts as an equity insurance. Around one-fourth of the total loan balance is due to have interest rates reset over a 12-month period (see (cid:28)gure C.2). This (cid:28)gure only comprises loans that automatically will have a new interest rate and not active decisions to re(cid:28)nance or extract equity. D Details on the Calculation of NNP and URE The Net Nominal Position (NNP) and Unhedged Interest Rate Exposure (URE) for the various sectors in the Danish economy are calculated from our household-level dataset as well as the (cid:28)nancial accounts from the national accounts statistics. All calculations 48

are based on average values over the years 2009 to 2015, de(cid:29)ated by the consumer price index. D.1 NNP and URE for Households The NNP for households is calculated as (cid:28)nancial assets minus liabilities. As (cid:28)nancial assets, we include bank deposits as well as the market value of securities (excluding shares). Liabilities include all debt to (cid:28)nancial institutions (including credit card debt) as well as publicly administered student debt, tax debt and other debt to government bodies. These data are reported to the tax authorities by (cid:28)nancial institutions on behalf of the households. URE is calculated as annual savings (i.e. after-tax income minus expenditure) plus maturing assets minus maturing liabilities. As maturing assets, we include all bank deposits, thereby assuming that they are (cid:29)oating rate. We assume a maturity of (cid:28)ve yearsforsecuritiesheldbyhouseholdsandthereforeinclude20%ofthevalueofsecurities. Regarding liabilities, we assume that all bank debt is (cid:29)oating rate. According to the interest rate statistics collected by Danmarks Nationalbank since 2013, on average 95% of bank debt from households is (cid:29)oating rate, most of which is tied either to a market reference rate or to the Danmarks Nationalbank rate on certi(cid:28)cates of deposit, with immediate adjustment. For mortgage debt, we have detailed information allowing us to calculate the stock of debt which is due to have interest rates reset over the coming 12 months. Voluntary re(cid:28)nancing of mortgage loans, with or without extraction of additional equity, takes place to a large extent in Denmark. Our measure of maturing liabilities only includes the loans that are contractually due to have their interest rates reset, and we do not attempt to estimate the amount of additional re(cid:28)nancing. For remaining liabilities, which constitute very small amounts, we have no information regarding maturity, so we assume (cid:28)ve years. D.2 Other Sectors NNP for the other sectors in the economy is obtained from the (cid:28)nancial accounts statistics compiled by Danmarks Nationalbank. To most closely resemble the de(cid:28)nition used in the household-level data, we de(cid:28)ne NNP as net assets (i.e., assets minus liabilities) in the following categories: "Currency and deposits", "Securities other than shares", "Loans", and "Trade credits and other accounts receivable/payable". 49

NNP for the whole economy should, in principle, sum to 0. However, the householdlevel microdata on bank deposits that we have access to is exclusive of certain types of savings (specialized children’s savings accounts as well as some forms of pension savings accounts administered by banks), which are included in the (cid:28)nancial accounts statistics. For the age group included in our sample, these types of accounts can be assumed to be largely illiquid. We therefore group those deposits (33 billion USD) together with the assets of pension funds (see table IV).51 URE for non-households is also based on the (cid:28)nancial accounts. In the national accounts, we do not observe the maturity of di(cid:27)erent asset and liability classes. We hold household URE (cid:28)xed at the values from the micro-level data and take advantage of the identity that total URE in the economy must be 0 to calibrate the maturity for the remaining sectors of the economy. This results in a maturity of assets and liabilities for non-households of 3.65 years. E Comparison to Blundell, Pistaferri, and Preston (2008) Are the di(cid:27)erences in our estimates to those in BPP driven by the data or by the method? Here we apply BPP’s original method to moments obtained from the Danish administrative data. Applied to the entire dataset, this method estimates the response to permanent income shocks to be 0.81 and to transitory income shocks to be 0.12. The response to permanent income shocks is in line with our method, while the response to transitory income shocks is signi(cid:28)cantly biased towards zero, although still larger than that obtained using PSID data. Figure E.1 shows the results of using BPP’s method on quintiles of liquid wealth. While the downward sloping pattern is evident, the magnitudes are very di(cid:27)erent to those from our method shown in (cid:28)gure VI in the paper. Our method di(cid:27)ers from BPP in two respects: (cid:28)rst we model time aggregation and second we impose that consumption responses to transitory income shocks are short lived, in contrast to the random walk behavior in BPP. Crawley (2019) shows how just the (cid:28)rst of these, modeling time aggregation, signi(cid:28)cantly changes BPP’s estimates in the PSID data. With the high MPX’s that we estimate in our data, it is clear 51In practice, this amount is calculated as a residual, which may also re(cid:29)ect other minor di(cid:27)erences between the household-leveldataandthenationalaccountsstatistics. Forexample,holdingsofbanknotesandcoinsarenotobserved in the microdata but are allocated based on certain assumptions in the (cid:28)nancial accounts. For our exercise, the impact ofsuchotherdi(cid:27)erencesislikelytobeverysmall. 50

MPX by Liquid Wealth Quantile XPM 1.0 f Permanent MPX y Transitory MPX 0.8 0.6 0.4 0.2 0.0 $0−2,000 $2,000−6,000 $6,000−12,000 $12,000−30,000 > $30,000 Figure E.1 Estimates by Liquid Wealth Quintile Obtained Using BPP’s Methodology that the random walk assumption cannot be valid. If we only make this (cid:28)rst change and estimate again using the Danish administrative data, we estimate the response to permanent income shocks to be 1.18 and to transitory income shocks to be 1.50. These unreasonablyhighestimatescomeasaresultoftheassumptionthatconsumptionfollows a random walk. In particular, the fact that the response to transitory income shocks is signi(cid:28)cantly greater than one comes from the method extending the short term response over the period of one year. This exercise has shown that the original BPP method is both biased and sensitive to assumptions made about the path of consumption. In appendices F, H, and I we show that, in contrast, our method is relatively robust to misspeci(cid:28)cation. F Persistent Consumption Response Our estimation procedure makes the assumption that the consumption response to a transitory income shock decays to zero in a period of two years or less. A slower decay willleadtoadownwardbiasinourestimatesofthetransitoryMPX.FigureF.1showsthe results of our estimation procedure on simulated data under two di(cid:27)erent assumptions about the transitory consumption response. The exponential decay line assumes that the consumption (cid:29)ow following a transitory 51

shock decays exponentially.52 We vary the decay rate to match a range of year 1 MPCs and assume that the entire transitory income is eventually consumed. For high MPCs, and especially those over 0.5, there is very little bias. However, for MPCs signi(cid:28)cantly below 0.5 our method results in downward-biased estimates. This bias arises because low MPCs, combined with exponential consumption decay, result in a relatively stable consumption (cid:29)ow over the (cid:28)rst few years that has not declined close to zero after two years. Empirical evidence suggests that in fact the consumption response to a transitory shock decays quickly in the (cid:28)rst few months and then more slowly after that.53 The (cid:16)Fagereng et al.(cid:17) line in (cid:28)gure F.1 shows the MPC estimate in simulated data in which the consumption response decays according to the estimates made in Fagereng, Holm, and Natvik (2016). In this case, the fast decay in the (cid:28)rst few months results in a smaller bias than the exponential case for low MPCs, while the fact that the decay is slower following these (cid:28)rst months results in a larger bias for high MPCs. Overall it seems likely that our assumption about the persistence of the consumption response leads to a slight downward bias across the range of MPCs. We also show that our MPX estimates are not very sensitive to the choice of N (years of growth in our identi(cid:28)cation equations) between 3 and 6, which lends further support to the fact that assuming a two-year limit does not bias our results too much.54 F.1 Details on Section F Simulations For the simulations in section F we divided each year into 20 sub-intervals. Both permanent and transitory shocks occur each period, and the transitory shocks have no persistence. At an annual frequency the variance of permanent and transitory shocks are equal. Households spend their permanent income each period, along with their consumption response to the history of transitory shocks. For the exponential decay model, this is ∞ (cid:88) c = p +(1−ρ) ρnε t t t−n n=0 In Fagereng, Holm, and Natvik (2016) the T year MPC is estimated as a function: MPC = θ Tθ2 T 1 52Standardbu(cid:27)er-stockmodelsgiverisetoaconsumptionresponsethatdecaysveryclosetoexponentially. 53BothFagereng,Holm,andNatvik(2016)andGelman(2016)provideevidenceforthis. 54UsingN equalto4and5insteadof3,4,and5allowsustoextendtheconsumptionresponseouttothreeyears,at theexpenseoflosingdataandbecomingmoresensitivetomisspeci(cid:28)cationoftheincomeprocess. 52

0.0 0.2 0.4 0.6 0.8 1.0 0.1 8.0 6.0 4.0 2.0 0.0 Bias Due to Persistent Consumption True MPC CPM detamitsE 45 degree line Exponential Decay Fagereng et al. Decay Figure F.1 Bias from Persistent Consumption where θ controls the size of the response and θ the speed of decay. We vary θ and 1 2 1 choose θ = 0.2142 according to their estimate. In this model consumption in period t 2 (measured in sub-intervals) is: ∞ (cid:88)(cid:16) n+1 n (cid:17) c = p +θ ( )θ2 −( )θ2 ε t t 1 t−n 20 20 n=0 We then time aggregate both income and consumption over each 20-sub-interval period, choose a sample of 13 years, and run our estimation procedure with N = 3,4,5. The transitory MPC estimates are shown in (cid:28)gure F.1, and the permanent estimates are shown in (cid:28)gure F.2. The bias in permanent estimates is small across the range of transitory MPCs. F.2 Estimates Using Di(cid:27)erent Values of N Table F.1 ψ Estimates Using Di(cid:27)erent N 53

0.0 0.2 0.4 0.6 0.8 1.0 0.1 8.0 6.0 4.0 2.0 0.0 Bias Due to Persistent Consumption True Transitory MPC CPM tnenamreP detamitsE True Permanent MPC Exponential Decay Fagereng et al. Decay Figure F.2 Bias from Persistent Consumption n 2 1 2 3 4 5 6 1 0.58 0.59 0.59 0.60 0.60 2 0.62 0.62 0.62 0.62 n 3 0.62 0.62 0.63 1 4 0.62 0.64 5 0.68 6 TableF.1showstheestimatesofthetransitoryMPXthatwerecoverfromourestimation sample when we just use N = n ,n in our identi(cid:28)cation equations 3 and 5. Remember 1 2 in our main results we used GMM with N = 3,4,5 and we have circled N = 3,5 to highlight where we get identi(cid:28)cation from in the paper. The purpose of this exercise is to show that the estimation results are not very sensitive to the values of N chosen, providing more evidence that the assumption we made that the transitory consumption response lasts less than two years is not biasing our results signi(cid:28)cantly. In fact, the results are not changed dramatically even when N = 1,2, which suggests the majority of the transitory consumption response is very short-lived. 54

G Durables In this appendix we expand on section 7.2 on durable expenditure. First, it will help to write down a simple model. The model will show that our empirical methodology continuestoestimatetheconsumptionresponsetopermanentandtransitoryshocks, but that these need to be interpreted carefully. The model uses the same income process as section 3.2. Remembering the income process is made up of two martingale processes, P and Q , which may have jumps, instantaneous income is given by t t (cid:16) (cid:90) t (cid:17) dy = dP dt+dQ t s t 0 while instantaneous expenditure now has both a durable and a nondurable component: (cid:16) (cid:90) t (cid:17) dc = φ dP dt+φ dP +ψdQ t nd s d t s 0 Here we have assumed that the expenditure response to transitory shocks is instantaneous, but it would not change things to assume as before that the response decays to zero after two years. However, it is important that the durable component of the expenditure response to permanent shocks occurs instantaneously with the shock (or very soon after). Aggregating income and consumption annually gives (cid:16) (cid:90) T−N (cid:90) T−1 (cid:90) T (cid:17) ∆Ny¯ = (s−(T −N −1))dP + dP + (T −s)dP T s s s T−N−1 T−N T−1 (cid:16) (cid:90) T (cid:90) T−N (cid:17) + dQ − dQ t t T−1 T−N−1 (cid:16) (cid:90) T−N (cid:90) T−1 (cid:90) T (cid:17) ∆Nc¯ = φ (s−(T −N −1))dP + dP + (T −s)dP T nd s s s T−N−1 T−N T−1 (cid:16) (cid:90) T (cid:90) T−N (cid:17) +φ dP − dP d t t T−1 T−N−1 (cid:16) (cid:90) T (cid:90) T−N (cid:17) +ψ dQ − dQ t t T−1 T−N−1 From this we can calculate the covariance: Cov(∆nc¯,∆ny¯ ) = φ Var(∆ny¯ ) T T nd T (cid:32) (cid:33) (cid:90) T (cid:90) T−N +φ (T −s)σ2dt− (s−(T −N −1))σ2dt d P P T−1 T−N−1 55

Bias in y vs Durable Delay Delay, Months saiB s 2 pf 2s 2 d q 0 0 6 12 18 24 Figure G.3 Bias in Transitory MPX with Delay in Durable Goods Purchase (cid:32) (cid:33) (cid:90) T (cid:90) T−N +ψ σ2dt+ σ2dt Q Q T−1 T−N−1 1 = φ (n− )σ2 +0+2ψσ2 nd 3 P Q So the durable component of the covariance cancels out, and our identi(cid:28)cation method correctly identi(cid:28)es φ and ψ but is unable to identify φ . nd d However, ifthereissomedelaybetweenthehouseholdreceivingthepermanentincome shock and purchasing the durable goods, then this introduces an upward bias into the estimate of transitory MPX. The size of the bias grows with the number of months delay between the permanent income shock and the durable goods purchase, plateauing after σ2 12 months at a level of p φ . Figure G.3 shows how this bias increases with the delay. 2σ2 d q In order to quantify how large this bias may be in practice, we make use of the car registry data available in Denmark. Using data on the current value of cars owned by a household, we perform the same residual calculation to (cid:28)nd the change in car value that is unpredictable with the household characteristics we are able to observe. We then constructtwonewexpenditurepanels: oneinwhichweremoveexpendituresoncars, and one in which we make a proxy for non-durable consumption by removing expenditures on cars multiplied by 1 (car purchases make up 42.1% of durable expenditure in 0.421 Denmark): Cnocar = C −∆CarValue T T 1 Cnondurable = C − ∆CarValue T T 0.421 The second, nondurable proxy consumption panel, can be modeled as the true nondurable consumption panel with classical measurement error added. This classical 56

measurement error does not bias our estimates, so we can use this nondurable proxy panel to estimate an unbiased MPC out of transitory shocks, where the MPC does not include durable expenditures. Theresultsofthisexercisecanbeseenin(cid:28)gureIX.Evenwithoutbias,wewouldexpect the nondurable proxy estimates to be lower than those including all expenditures, as the de(cid:28)nition of transitory MPX changes over the three panels to exclude cars and then all durable goods. For the lower quintiles of liquid wealth it therefore looks as though the bias is likely very small, as nondurable goods make up 10% of spending and the MPX estimates are smaller by approximately 10% in this region. For the top quintile of liquid wealth there seems to be some bias, with the estimate of MPX for all expenditures decreasing from 25% to an MPC for nondurable goods of 17%. While there is some evidence that our results may be biased upward for those in the top quintiles of liquid assets, this bias will only have a small e(cid:27)ect on our overall conclusions. As the relevant number for the monetary policy exercise is the MPX rather than the MPC, we have chosen not to adjust our baseline results using this method and accept that a small bias may exist in our data. It should be noted that such a bias will cause the heterogeneous channels of monetary policy to appear smaller than they actually are. H RIP or HIP Income Process? H.1 RIP or HIP Income Process? Our method makes strong assumptions on the income process(cid:22)namely, that there is no persistent idiosyncratic component to income growth and that the process contains a randomwalk. Guvenen(2009)showsthatitisempiricallydi(cid:30)culttodistinguishbetween a ‘Restricted Income Pro(cid:28)le’ (RIP) like this and a ‘Heterogenous Income Pro(cid:28)le’ (HIP) income process, in which (i) shocks to income are much less persistent (e.g., AR(1) with ρ ≈ 0.8), and (ii) households have a persistent idiosyncratic growth component. The reason the RIP and HIP processes are di(cid:30)cult to tell apart is that the two features (i) and (ii) act in opposite directions on the cross-section variance of income growth. The lesspersistentincomeshocksleadthecross-sectionalincomegrowthvariancetonotgrow as fast as the HIP model, while the persistent idiosyncratic growth component leads the same variance to grow at a faster rate. The result is that the increase in variance of income growth over three to four years is approximately the same as the increase from 57

four to (cid:28)ve years. To the extent that the consumption response to these semi-permanent shocks is similar to the response to the idiosyncratic persistent growth component,55 our methodology will continue to provide reasonable estimates of the (cid:16)permanent(cid:17) MPX and the more familiar transitory MPX. Both the Restricted Income Pro(cid:28)le (RIP) and Heterogeneous Income Pro(cid:28)le (HIP) processes can be described by the equations: yi = βih+zi +εh h h i zi = ρzi +ηh h h−1] i where i indexes the worker and h the years of experience. εh represents a transitory i shock to income, while ηh is persistent. βi represents an idiosyncratic persistent growth i factor. In the RIP model, βi = 0 and ρ is usually estimated to be very close to 1 (in this paper we assumed ρ = 1). In the HIP model, βi has a cross-sectional variance σ2 > 0, β and ρ is normally estimated to be signi(cid:28)cantly lower than 1, around 0.8. The reason these are di(cid:30)cult to tell apart is because the theory does not give a strong indication in which model the cross-sectional variance of income growth over N years should grow faster. In the RIP model with ρ = 1, the cross-sectional variance of income growth increases linearly with N. In the RIP model with ρ ≈ 0.8, the growth in the crosssectional variance of income growth will decrease due to the low ρ but increase due to the idiosyncratic βi. Figure H.1 shows the empirical values for income growth variance and the covariance of income and expenditure growth over N years. We have also plotted the (cid:28)tted values for these statistics that are implied by our model when (cid:28)tted to N = 3,4,5 as we do in the paper. We see the empirical variance and covariance decline slightly below the model (cid:28)tted line as N becomes large, which (cid:28)ts with the (cid:28)nding that ρ in the RIP model is usually slightly below 1.0, around 0.98 or 0.99. We also note that around the region where we achieve our identi(cid:28)cation (N = 3,4,5), there is very little curvature in the empirical statistics, and the increase in both variance and covariance is close to linear. While this linearity around N = 3,4,5 cannot help us distinguish between the RIP and HIP process, it does imply that our empirical methodology may be somewhat robust to misspeci(cid:28)cation along this dimension. If we assume that the expenditure response to a change in zi and to the increase from the persistent idiosyncratic growth are equal to h 55SeeGuvenen(2007)foranexampleofwhythismightbethecase: ifhouseholdsdonotknowtheirownidiosyncratic growthex-ante,aBayesianlearningprocesswillbeveryslow,sohouseholds(atleastinitially)willreactinsimilarways tochangesinincomeduetothispersistentgrowthcomponentasatrueincomeshock. 58

l l l l l l l l l l 0 2 4 6 8 10 40.0 30.0 20.0 10.0 00.0 Covariance with Increasing Difference Operator N ecnairavoC/ecnairaV Var(DNy) Empirical Var(DNy) matched to N=3,4,5 Cov(DNy,DNc) Empirical Cov(DNy,DNc) matched to n=3,4,5 l l l l l l l l l l Figure H.1 Variance and Covariance with Years of Growth φ, and the response to a transitory shock is ψ, that is: ∆Nci ≈ φ∆N(βih+zi)+ψ∆Nεh h h i (cid:0) (cid:1) Then, the fact that Var ∆N(βih+zi) grows approximately linearly with N means that h our empirical method will correctly identify φ and ψ. A full investigation of the implications of di(cid:27)erent income processes is beyond the scope of this paper but would be a very useful exercise for future research. I Time-Varying Risk We have assumed that idiosyncratic risk remains constant over time. Given that our sample period covers the great recession, this may not be appropriate. Here we show how the variance of income growth has varied over time, peaking just after the crisis in 2010. In order to test how much this time-varying risk might bias our results, we simulated data with φ = 1 and ψ = 0.5, with permanent variance equal to estimates from the data and transitory variance varying in order to match the time-varying income risk pattern observed in the data. When we run this simulation we (cid:28)nd estimates of φ and ψ within 1% of their true values. Figure I.1 shows how the standard deviation of income growth has changed over the sample period. From trough to peak, the standard deviation increases approximately 10%. In the simulation referred to in section I, we assume that both transitory income and transitory consumption response have no persistence. We divide each year into 59

l l l l l l l l l l l l 2004 2006 2008 2010 2012 2014 01.0 80.0 60.0 40.0 20.0 00.0 Income Growth Standard Deviation by Year Year .dtS htworG emocnI Figure I.1 Standard Deviation of Income Growth 20 sub-periods, choose the variance of permanent shocks to be 0.003, and allow timevarying transitory shocks to match the pattern in (cid:28)gure I.1. We choose values of φ = 1 and ψ = 0.5 and apply our estimation procedure (that assumes constant variance) to the simulated data. We recover estimated values of φ and ψ to be 1.006 and 0.499, respectively. J Robustness As would be clear from the main text, we have made a number of choices regarding both data and variable de(cid:28)nitions as well as more methodological issues. In a series of graphs, this appendix presents a number of robustness checks aimed at assessing the extent to which our results are sensitive to the speci(cid:28)c choices. We begin with a number of robustness checks regarding our imputed expenditure measure, which may su(cid:27)er from measurement error. In (cid:28)gure J.2, we compare our baseline estimates of the MPX to estimates based on di(cid:27)erent sample selection procedures. First, we exclude all households that own stocks corresponding to more than 10,000 USD (10% of households in our sample). Second, we do not remove households that have negative imputed expenditure. We remove those households in our baseline samplebecausenegativeexpenditureisclearlynotagoodestimateofactualexpenditure. However, for example, in the event that negative expenditure arises because of classical measurement error, removal of negative estimates may be asymmetric and introduce 60

Transitory MPX by Liquid Wealth Quintile Quintile XPM Transitory MPX by URE Decile 1.0 Baseline No Stocks Include Neg Cons 0.8 Strict Outliers 0.6 0.4 0.2 0.0 1 2 3 4 5 Decile XPM 1.0 Baseline No Stocks Include Neg Cons 0.8 Strict Outliers 0.6 0.4 0.2 0.0 1 4 7 10 Figure J.2 Robustness of Liquid Wealth and URE Distributions an upward bias in average imputed expenditure. Third, to check that large outliers do not drive our results, we remove observations in the top and bottom 2.5% in terms of level and change of income and expenditure. In the baseline calculations, we use only a 1% cuto(cid:27). Our results are qualitatively unchanged when using these alternative approaches to take account of measurement error. In terms of magnitudes of the estimatedMPXs, thelargestdi(cid:27)erencetothe baselineresultsseemstobefoundwhenwe include negative expenditure estimates. As expected, this makes the largest di(cid:27)erence among the wealthier households. The speci(cid:28)cation of outliers also matters somewhat for the point estimates of MPX in certain groups of households, but di(cid:27)erences are not large. Anotherrobustnesscheckconsistsofspecifyingconsumptionandincomeinlogsrather thaninlevels. Thefundamentaldi(cid:27)erenceisthatthelogspeci(cid:28)cationyieldsanelasticity rather than an MPX. Hence, some di(cid:27)erence between level and log results must be expected for households that only spend a fraction of their annual income (typically wealthier households). Indeed, as expected, (cid:28)gure J.3 demonstrates that results hold qualitatively when specifying income and expenditure in logs rather than in levels, whereas estimated elasticities are higher than the MPXs for the wealthier households and those with high URE. Time-varying income risk may also potentially contribute to di(cid:27)erences between results based on levels and logs. However, as shown in section I, this is not likely to be important in our setting. As discussed in section 4.1, we use labor income of the head of the household as our 61

Transitory MPX by Liquid Wealth Quintile Quintile XPM Transitory MPX by URE Decile 1.0 Baseline Log Total (Elasticity) 0.8 0.6 0.4 0.2 0.0 1 2 3 4 5 Decile XPM 1.0 Baseline Log Total (Elasticity) 0.8 0.6 0.4 0.2 0.0 1 2 3 4 5 6 7 8 9 10 Figure J.3 Results Using Log Income and Expenditure prime measure of income in line with previous literature. Various mechanisms(cid:22)e.g., intra-household income insurance(cid:22)may give rise to di(cid:27)erences between results based on income of the head of household and total household income. However, (cid:28)gure J.4 demonstrates that there is virtually no di(cid:27)erence in our results between using total household income and only the household head’s income. Online appendix L brie(cid:29)y discusses the potential role that intra-household insurance may play, which we leave as an area for future research. Finally, (cid:28)gure VI shows the distribution of MPX by quintile of liquid wealth. It might be argued that the relevant level of liquid wealth is relative to income rather than in absolute terms. Figure J.5 demonstrates that results based on quintiles of liquid wealth divided by permanent income are similar. Also, results (not shown here) where deciles are based on a broader de(cid:28)nition of liquid wealth(cid:22)i.e., including stock and bond holdings(cid:22)are similar to our baseline results. K Distribution of Permanent MPX by NNP, URE, and Income Figure K.1 shows the distribution of both transitory and permanent MPX by NNP, URE and income decile. The transitory numbers are a repeat of (cid:28)gure VIII. 62

Transitory MPX by URE Decile Decile XPM 1.0 Total Head 0.8 0.6 0.4 0.2 0.0 1 2 3 4 5 6 7 8 9 10 Figure J.4 Results Using Total Labor Income and Head Labor Income Transitory MPX by Liquid Wealth Quintile Quintile XPM 1.0 Baseline Liquid Wealth/Permanent Income 0.8 0.6 0.4 0.2 0.0 1 2 3 4 5 Figure J.5 Results Using Quintiles of Liquid Wealth over Permanent Income vs Liquid Wealth 63

MPX by URE Quantile URE/Mean Expenditure XPM MPX by NNP Quantile 1.0 f Permanent MPX y Transitory MPX 0.8 0.6 0.4 0.2 0.0 −6.82 −2.48 −1.57 −0.99 −0.57 −0.27 −0.06 0.1 0.48 2.23 NNP/Mean Expenditure XPM 1.0 f Permanent MPX y Transitory MPX 0.8 0.6 0.4 0.2 0.0 −16.37 −6.38 −4.66 −3.41 −2.31 −1.31 −0.56 −0.14 0.09 1.92 MPX by Income Quantile Income/Mean Expenditure XPM 1.0 f Permanent MPX y Transitory MPX 0.8 0.6 0.4 0.2 0.0 0.33 0.58 0.73 0.88 1.07 1.3 1.5 1.69 1.93 2.92 Figure K.1 MPX Distribution by URE, NNP, and Income 64

Transitory MPX by Liquid Wealth Quintile Quintile XPM 1.0 Total Head Spouse 0.8 0.6 0.4 0.2 0.0 1 2 3 4 5 Figure L.1 Results Using Total, Head, and Spouse Labor Income L Intra-household Income Insurance As discussed in section 4.1, we use labor income of the head of the household as our prime measure of income, in line with previous literature. Figure J.4 demonstrates that results based on total household income and income of the head of household are similar. However, MPXs from transitory shocks to the income of the spouse are lower than MPXs from shocks to total income, in particular for the less wealthy households, as demonstrated in (cid:28)gure L.1. This indicates heterogeneity in the role that intra-household income insurance plays across di(cid:27)erent groups of households. We leave this interesting topic for future research. 65