IIDENTIFICATION IN ECONOMIES WITH FRICTIONS
ANDREAS TRYPHONIDESUNIVERSITY OF CYPRUS
Abstract.
Identification in heterogeneous agent economies can be obtained by utilizingrestrictions which are consistent with alternative mechanisms. Survey data that are infor-mative about the proportion of agents whose behavior is distorted due to financial frictionsprovides additional information. The latter can shrink the set of admissible models andpreference parameter values. The paper provides identification analysis, simulation evi-dence and an application using Spanish data where the borrowing constrained are identifiedby combining information from different surveys. Results suggest that incomplete marketscan potentially rationalize the gap between micro and macro elasticities for aggregate con-sumption and labor supply.
Keywords: Frictions, Heterogeneity, Set Identification, Aggregated Micro-SurveysJEL Classification: C32, C51,E10, E2
Date : This version: June 2020, Previous versions: Feb 2020, Dec 2017, Feb 2019, Nov2019.
Parts of this paper draw from chapter 2 of my PhD thesis (EUI, Sep 2016), and the paper circulated under the title "Setidentified Economies and Robustness to Misspecification". I thank Fabio Canova, Peter Reinhard Hansen, Frank Schorfheideand Giuseppe Ragusa for comments and suggestions on the thesis chapter. Previous versions of the paper benefited fromdiscussions with George Marios Angeletos, Eric Leeper, Ellen McGrattan, Juan J. Dolado, Thierry Magnac, Manuel Arellano,Raffaella Giacomini, and comments from participants at the SED (St Louis), MMCN (Stanford), the IAAE (Thessaloniki andNicosia), the 30th EEA Congress (Mannheim), T2M (Banque de France), the ES Winter Meetings (Barcelona), the seminarsat the University of Cyprus, Royal Holloway, Humboldt University, Universidad Carlos III, University of Glasgow, Universityof Southampton, the Econometrics and Applied Macro working groups at the EUI. Any errors are my own.
Email : [email protected]
Address : Panepistimiou 1, 2109, Nicosia, Cyprus. a r X i v : . [ ec on . E M ] J u l . Introduction
Frictions in dynamic macroeconomic models are undoubtedly a source of heterogeneity, withappealing theoretical and empirical implications. For example, borrowing constraints mayinhibit the ability of individuals to insure fluctuations in income and this generates hetero-geneous consumption outcomes. At the same time, this can have important implications forthe persistence of macroeconomic time series.Nevertheless, the specification of the underlying mechanism in heterogeneous agent (HA)economies can be consequential for parameter identification and policy conclusions in general.A natural question to ask is whether we can still obtain meaningful identification withoutrelying on a particular mechanism, as this can lead to more credible inference and theresulting estimates being portable across models. In general, by knowing the extent to whichthis class of models is identified can aid our understanding in how specific assumptions maylead to particular empirical results.Towards this goal, the paper employs identifying restrictions that are consistent with a va-riety of mechanisms. The restrictions generate moment inequalities and therefore set iden-tification. Moment inequality restrictions have been used to characterize frictions in specificmarkets, see for example Luttmer (1996) and Chetty (2012). To my knowledge, this is thefirst paper that characterizes such restrictions in dynamic stochastic macroeconomic modelswith heterogeneity and thus contributes to the literature that deals with partial identificationin structural macroeconomic models (e.g. Lubik and Schorfheide (2004); Coroneo, Corradi,and Santos Monteiro (2011)) and the literature on applications of moment inequality mod-els . See Pakes, Porter, Ho, and Ishii (2015) and references therein. Partial identification in structural micro mod-els has been applied in a variety of settings, from discrete choice in single agent models, games with multipleequilibria to auctions and networks. For a more comprehensive list of applications of partial identificationsee Molinari (2019) and Ho and Rosen (2017). In working paper (Tryphonides, 2017) I show that this approach can be extended to study other forms offrictions including departures from rational expectations and how this relates to perturbations of the measureimplied by the frictionless model in an information theoretic framework. sing a second order approximation, the paper exemplifies which data - other than macroe-conomic time series - are necessary for identification while it also shows that additionalmicro moments can be potentially useful in order to further constrain the set of admissi-ble models without compromising the original set of assumptions. In particular, employingaggregated survey data that are informative about the extensive margin of adjustment e.g.the proportion of agents whose behavior is distorted over time due to financial frictions canprovide additional information that can tighten the corresponding bounds. Furthermore, themoment inequality approach is robust to under-predicting this margin in the data.I provide identification analysis both in an extended partial equilibrium analytical exampleand in a general equilibrium setting, as well as a simulation experiment where the datagenerating process is the Krusell and Smith (1998) economy. The approach is applied to abenchmark heterogeneous agent model with aggregate risk for the Spanish economy, wherethe proportion of constrained consumers is identified by combining information from differentsurveys. Both simulation and empirical results suggest that controlling for variations in theextensive margin is indeed informative for preferences.These results also contribute to the understanding that since these models are highly non-linear, certain moments can provide information about preferences without necessarily havinga first order predictive power for the level of macroeconomic time series. The classic case isthat of the borrowing constrained - these individuals might represent a small proportion of thepopulation, so their consumption choices may not substantially affect the level of aggregateconsumption, yet fluctuations in this proportion are informative for preferences.The paper also contributes to the strand of the literature that is related to the intricaciesof calibration in representative agents models when the data is generated by heterogeneousagent economies. Browning, Hansen, and Heckman (1999) cast doubt on using microeco-nomic studies as the mapping to aggregate models is affected by aggregation and sampleselection issues. Guvenen (2006) finds that estimates of the elasticity of intertemporal sub-stitution using aggregate data reflect the behavior of poorer agents while Constantinides andDuffie (1996) find that cross sectional heterogeneity is another source of bias. Chang, Kim, nd Schorfheide (2013) show that structural parameters are not invariant to policy whenheterogeneity is ignored.This paper’s empirical strategy delivers heterogeneity-consistent estimates that account forthe lack of policy invariance. Survey data control both for the extensive margin and forconsumption variance, whose variation would otherwise be absorbed by the structural pa-rameters, bridging the gap between micro and macro elasticities. This is relevant both forthe equity premium puzzle (Mehra and Prescott, 1985), in the sense of obviating the needfor a high degree of risk aversion to rationalize the comovement of consumption growth andreturns, and the "Frisch elasticity puzzle" (Chetty, Guren, Manoli, and Weber, 2011), whichconcerns the gap between labor supply elasticities estimated using individual and aggregatedata.Regarding the first, several mechanisms have been considered as potential resolutions (e.g.uninsurable income risk and borrowing constraints, limited asset market participation amongothers (Constantinides, Donaldson, and Mehra (2002); Brav, Constantinides, and Geczy(2002))) and yet results are sensitive to the particular model setups, parameterizations andwider implications (see e.g. Mehra and Prescott (2008)). This paper does not take a stanceon the mechanism, and yet it provides evidence that incomplete markets and borrowingconstraints can account for variation that would be wrongly attributed to risk aversion.This holds despite the presence of external habits in consumption.Regarding the "Frisch elasticity puzzle", empirical results also suggest that incomplete mar-kets and borrowing constraints can bring the macro (aggregate) Frisch elasticity closer tomicro estimates. The paper also exemplifies how the identification of this elasticity basedon aggregate data depends on heterogeneity and the borrowing constrained. In the absenceof heterogeneity, higher estimates of the Frisch elasticity are admissible, in line with theresults of Domeij and Flodén (2006) who show that in the presence of borrowing constraints,housholds may increase labor supply after a negative wage shock. Nevertheless, due to aggre-gation, identification will also depend on the response of the mass of borrowing constrainedand the presence of heterogeneity in consumption. he rest of the paper is organized as follows. Section 2 presents the general model environ-ment. Section 3 demonstrates analytically how identification can be obtained in a partialequilibrium context and why the extensive margin is informative. Section 4 presents thesecond order approximation of the HA model and the corresponding aggregate identifyingrestrictions. Section 5 provides identification analysis, Section 6 presents the simulationexperiment, Section 7 contains the empirical study and Section 8 concludes. The Appen-dix contains the general identification results and a diagrammatic exposition, proofs, themixed frequency model used to measure the borrowing constrained, data explanations andtransformations and an additional simulation result.Finally, a word on notation. T signifies the length of aggregate data, and N the number ofagents. θ P Θ signifies the parameters of interest with Θ I the identified set and Θ CSα the 1 ´ α level confidence set. Bold capital letters Y τ denote a matrix with observations of vector Y t up to time τ , t Y j u j ď τ . K is the orthogonal complement.2. Model Environment
The theoretical model features households maximizing expected discounted lifetime utilitysubject to a general intertemporal budget constraint. Each household receives total income y i,t p l i,t q which might depend on labor hours supplied ( l i,t ) and makes consumption p c i,t q ,labor supply and investment decisions. In particular, the household chooses to store wealthin J assets (cid:32) a ji,t ( j “ ..J which earn gross returns equal to R ji,t “ ` r j p a jt , X t q .Furthermore, there is a general constraint technology g p . q that restricts trades between theseassets which is increasing in wealth , while ν ji,t is the cost of accessing the j th asset.Letting β P p , q be the discount factor and assuming an instantaneous utility function U p c i,t , l i,t ; ω, X t q , where ω signifies preference related parameters and X t an aggregate state This includes popular restrictions e.g. on short selling and non-collateralized borrowing ( a jt ` ě ´ a ) andcollateralized borrowing p a jt ` ě ´ ř Jk ‰ j w k a kt { t ` q where w k summarize restrictions on quantities acrossassets. ector , the household problem becomes as follows:max t c i,t ,l i,t ,a i,t ` u t “ E ÿ t “ β t U p c i,t , l i,t ; ω, X t q s.t. c i,t ´ y i,t p l i,t q “ J ÿ j “ p ` r j p a ji,t , X t qq a ji,t ´ J ÿ j “ a ji,t ` ν ji,t ` c i,t ą g i ´(cid:32) a ji,t , a ji,t ` ( Jj “ ¯ ě U p c i,t ; l i,t , ω, X t q , the Euler equation forany asset j is distorted by the non-negative Lagrange multipliers on the occasionally bindingliquidity constraint, denoted by µ i,t , which are positive when the borrowing constraint isbinding: U p c i,t ; l i,t , ω, X t q“ β E t ν ji,t ` ˜ ` r j p a ji,t ` , X t ` q ` B r j p a ji,t ` , X t ` qB a ji,t ` a ji,t ` ¸ U p c i,t ` ; l i,t ` , ω, X t ` q` µ i,t B g i ˆ! a ji,t , a ji,t ` ) Jj “ ˙ B a ji,t ` ` E t µ i,t ` B g i ˆ! a ji,t ` , a ji,t ` ) Jj “ ˙ B a ji,t ` which can be simplified to U p c i,t ; l i,t , ω, X t q“ β E t ν ji,t ` ˜ ` r j p a ji,t ` , X t ` q ` B r j p a ji,t ` , X t ` qB a ji,t ` a ji,t ` ¸ U p c i,t ` ; l i,t ` , ω, X t q ` λ i,t where λ i,t summarizes the distortions . Given this setup, I next focus on what can be learnedfrom the data in terms of the preference parameters p β, ω q and the unobserved endogenousvariable λ i,t . For example, with external habits past aggregate consumption, C t ´ , belongs to X t Note that λ i,t is positive as µ i,t is positive and g i is increasing in wealth. . Analytical Example
The more formal identification analysis later on will focus on aggregate time series andsurvey data. However, for exposition purposes it is instructive first to show analytically how identification can be obtained in a partial equilibrium framework, how much additionalinformation survey data can provide and why informativeness depends on the true model,as in any identification question.Since (1) holds for all assets, for identification purposes it suffices to look at the condition fora subset of assets where we do not need to take a stance on how returns vary with asset levelsand how much it costs to access them e.g. the subset of conditions in (1) that correspondto a savings account (or a riskless government bond): U p c i,t ; l i,t , ω, X t q “ β p ` r q E t U p c i,t ` ; l i,t ` , ω, X t ` q ` λ i,t (1)There are alternative ways of achieving identification. For example, if cross section or paneldata is available, one can split the sample into unconstrained and constrained householdsusing some criterion. In his seminal paper, Zeldes (1989) splits the sample using a cut-offrule on the end of period nonhuman wealth and discusses different rules that could have beenused, as well as liquidity constraints arising from credit market imperfections and illiquidassets as e.g. in Pissarides (1978). The importance of illiquid assets has been recently revivedby the heterogeneous agent literature, see for example Kaplan, Violante, and Weidner (2014)and references therein, where they also identify groups of wealthy and poor hand to mouthconsumers using different rules.However, this approach cannot be readily applied to time series data, as is the case foraggregate data which is the focus in the rest of the paper. More importantly, alternativechoices can lead to sample selection and thus whimsical conclusions on whether liquidityconstraints are present. An alternative way is a structural treatment which amounts toemploying an estimated income process and computing optimal consumption. One would herefore place much more faith on the technology of the borrowing constraint and the natureof idiosyncratic income risk.A more credible way to identify p β, ω q is to use the more general implication of borrowingconstraints, which would be true across models with different types of heterogeneity andinsurance mechanisms that would give rise to different borrowing limit technologies (speci-fications for g i ) and therefore λ i,t . The implication is that liquidity constraints generate apositive discrepancy between current and next period marginal utility of consumption, as thehousehold cannot smooth consumption as much as it desires . Conditions in (1) thereforegenerate conditional moment inequalities, one of which is the case of the riskless asset : U p c i,t ; l i,t , ω, X t q ě β p ` r q E t U p c i,t ` ; l i,t ` , ω, X t ` q Using income y i,t pě q as an instrument, the following unconditional moment inequalityholds, which does not pin down a unique vector of values for p β, ω q : E rp U p c i,t ; l i,t , ω, X t q ´ β p ` r q U p c i,t ` ; l i,t ` , ω, X t ` qq y i,t s ě X t . I adopt the approximation of Hall(1978) using CRRA utility U p c i,t ; ω q : “ c ´ ωi,t ´ ´ ω , which implies the following law of motion forconsumption: c i,t ` “ ρc i,t ` ˜ λ i,t ` φ i,t ` (cid:15) i,t ` (3)where ˜ λ i,t ” ´p U p c i,t ; ω qq ´ λ i,t , U p c i,t ; ω q is the second derivative of the utility function, ρ “ p β p ` r qq ´ U ci,t ; ω q ci,tU ci,t ; ω q “ p β p ` r qq ω and φ i,t is the residual of this approximation which isset to zero for simplicity . Moreover, (cid:15) i,t ` is the rational forecast error such that E t (cid:15) i,t ` “ Notice also that we did not specify a specific process for labor income risk. It can therefore include anykind of individual specific risk to income as well as any kind of transfers. Similar restrictions also arise in the context of limited commitment and endogenous solvency constraints asin Alvarez and Jermann (2015). In a recent paper Commault (2019) challenged Hall (1978)’s random walk result for consumption in thecase of isoelastic utility. In the Appendix I clarify why the identification analysis is unaffected by thissimplification; in a nutshell, φ i,t has very similar properties to ˜ λ i,t . nd summarizes the uncertainty faced by the otherwise rational individual. Equivalently,consumption growth is equal to∆ c i,t ` “ p ρ ´ q c i,t ` ˜ λ i,t ` (cid:15) i,t ` Since higher income relaxes the budget constraint (
Cov p ˜ λ i,t , y i,t q ă ρ is as follows: ρ ID, : “ ˆ , ` Cov p y i,t , ∆ c i,t ` q Cov p y i,t c i,t q “ p , ` ρ IV s (4)Avoiding to take a stance on the exact nature of λ i, , which can involve a vast amount ofunobserved information, results in set identification. Nevertheless, we are still able to infercertain facts about the household’s preferences and economic behavior and conclusions arevalid across alternative environments.3.1. Bounds on Risk Aversion.
Given the set of admissible values for the reduced formcoefficient ρ , we can recover the implied bounds for the risk aversion parameter, ω ID . Treat-ing p r, β q as known, if β p ` r q “
1, risk aversion is unidentified ( ω ID “ R ` ) as consumptionfollows a random walk. If β p ` r q ‰
1, then risk aversion is set identified and has a veryintuitive interpretation, as it reflects restrictions on preferences implied by the presence ofnon-diversifiable income risk.In particular, for β p ` r q ă ω ID, “ $&% ω P R ` : ω ă } log p β p ` r qq} log ´ Cov p y i,t c i,t q Cov p y i,t c i,t q´} Cov p y i,t ∆ c i,t ` q} ¯ ,.- The stronger the negative correlation, the lower the upper bound on risk aversion, indicatingthat the less risk averse household is not accumulating enough wealth to fully insure againstincome risk.3.2.
The extensive margin as additional information.
Suppose now that we observethe dichotomous response of the household over time to a survey question that asks whether he household is (or expects to be in the near future) financially constrained. An honesthousehold will answer positively whenever λ i,t ą that had the data been generated by the model in (1) e.g. a restriction onborrowing, then consumption growth under incomplete markets becomes as follows:∆ c i,t ` “ p ρ ´ q c i,t ` (cid:15) i,t ` ` p λ i,t ą q ˆ ´ ρc i,t ` ¯ y i ˆ ´ ρ ` r p ´ F y i p x ‹ i qq ˙˙ : “ p ρ ´ q c i,t ` (cid:15) t ` ` p λ i,t ą q p λ c i,t ` λ ,i q (5)where F y i p x ‹ i q is the the probability of hitting the constraint, which depends on the distri-bution of income and cash on hand threshold x ‹ i . The higher this probability is, the higherthe savings rate out of income, increasing thereby consumption growth.The relevant question to ask is whether we can utilize the survey information without know-ing the true data generating process i.e. the form of λ i,t in (5), and whether this leads toeven more precise estimates of risk aversion. First, dichotomous responses to the survey canbe used to estimate P t p λ i,t ą q . Therefore, for v P R and (cid:15) i,t ` with cumulative densityΦ i,t p . q , the distribution function of consumption growth is: P t p ∆ c i,t ` ă v q “ Φ i,t p v ´ p ρ ´ q c i,t q P t p λ i,t “ q ` Φ i,t p v ´ p ρ ´ q c i,t ´ λ i,t q P t p λ i,t ą qď Φ i,t p v ´ p ρ ´ q c i,t q P t p λ i,t “ q ` Φ i,t p v ´ ˜ ρ IV c i,t q P t p λ i,t ą q (6) where we use the only information available: that is λ i,t ą
0, and that the IV regressionestimate of ρ ´ ρ IV , is a lower bound to the true value .The simplest way to show that (6) provides additional information is to show that thereexists parameter value in ρ P ρ ID, that is not consistent with (6). In the Appendix, it isshown that ˜ ρ IV , which is consistent with (4), does not satisfy (6) when the probability ofbeing constrained is higher than a certain threshold which depends on the true model: P t p λ i,t ą q P p g i,t , q (7) See Appendix for the formal derivation. To see this, notice that ˜ ρ IV “ ρ ´ ` ` C ov p c i,t , y i,t q ´ C ov p y i,t , λ i,t | λ i,t ą q P p λ i,t ą q ˘ ă ρ ´
1. The biasterm is negative as higher income relaxes the constraint and can be verified by the solution of the model, as λ “ ´ ρ . igure 1. Range for v “ , c i,t “ , r “ . , σ (cid:15) “ . , ¯ y “ where g i,t : “ Φ i,t p v ´ p ρ ´ q c i,t q ´ Φ i,t p v ´ ˜ ρ IV c i,t q Φ i,t p v ´ p ρ ´ q c i,t q ´ Φ i,t p v ´ λ ,i ´ p ρ ´ ` λ q c i,t q To gain intuition, consider the extreme cases where the probability of being constrained iszero or one. Condition (7) becomes invalid as p , q and p , q do not contain P t p λ i,t ą q .In either case, additional survey information is redundant. When liquidity constraints areoccasionally binding, survey information matters.Figure 1 provides an example when Φ i,t is the cumulative standard Normal distribution.For higher values of the intensive distortion λ , e.g. 2 λ the admissible range (grey shade)becomes larger. The more severe the distortions, the more likely is that additional in-formation matters. Conversely, in the frictionless limit the threshold contradicts (7) as g i,t Ñ λ i Ñ P t p λ i,t ą q .In general, surveys that report households that have applied for a loan and have been rejectedare robust to alternative assumptions about the economic environment. In fact, as we willsee in the application, such questions also ask to choose the reason for which these householdshave been rejected, and potential answers cover a wide spectrum. This is not a claim that his is the only survey question that matters, but the information that is key here is the onethat identifies when the individual is likely to be constrained or not, as this data providesmore information than the moment inequality that only captures average behavior and isconsistent with being constrained for e.g. 99% or 1% of the time. Such data will also beinformative when we consider aggregate fluctuations in the next section, as this probabilitywill describe the mass of financially constrained agents over time and will be related tovariations in consumption heterogeneity and precautionary effects.4. Aggregate Restrictions
Having shown how survey responses of a household can convey information about preferences,this section characterizes the corresponding aggregate restrictions. Assuming CRRA prefer-ences and separability between consumption and leisure as well as external habit formatione.g. "catching up with the Joneses" effects, implies the following utility function: u p c i,t , l i,t , ω, η, C t ´ q : “ p c i,t ´ hC t ´ q ´ ω ´ ´ ω ´ l ` ηi,t ` η where ω is the coefficient of relative risk aversion, η the (inverse) Frisch elasticity of laborsupply and h the habit parameter.The first order conditions from (1) become as follows: p c i,t ´ hC t ´ q ´ ω “ β E t p c i,t ` ´ hC t q ´ ω ˜ ` r j p a ji,t ` , X t q ` B r j p a ji,t ` , X t qB a ji,t ` a ji,t ` ¸ ` λ i,t l ηi,t “ W t P t c ´ ωi,t where I have used that B y i,t p l i,t qB l i,t “ W t P t , that is, the derivative of total individual incomewith respect to hours worked is the real wage. The parameters of interest are therefore p ω, β, η, h q . The assumption of separability can be relaxed without any bearing on the main point of the paper. Withnon-separability, aggregation will involve identifying an additional moment from micro data, that of thecross sectional covariation of consumption and hours worked, while it wil introduce an additional parameterto be estimated, that of the degree of separability. Hence, separability does not buy as much in terms of theorder condition for identification. .1. Aggregation.
Researchers have recently employed a second order approximation to theheterogeneous agent model around the representative agent (RA) allocation as a tractableway of analyzing its aggregate implications . This paper will also follow this route. Inparticular, looking at marginal utility of consumption in the CRRA case: p c i,t ´ hC t ´ q ´ ω « p C t ´ hC t ´ q ´ ω ´ ω p C t ´ hC t ´ q ´ ω ´ p c i,t ´ C t q` ω p ω ` q p C t ´ hC t ´ q ´ ω ´ p c i,t ´ C t q “ p C t ´ hC t ´ q ´ ω r ´ ω p C t ´ hC t ´ q ´ p c i,t ´ C t q` ω p ω ` q p C t ´ hC t ´ q ´ p c i,t ´ C t q s : “ p C t ´ hC t ´ q ´ ω Ξ t,i p h q Therefore, p C t ´ hC t ´ q ´ ω “ β E t p C t ` ´ hC t q ´ ω Ξ i,t ` p h q Ξ i,t p h q ˜ ` r j p a ji,t ` , X t q ` B r j p a ji,t ` , X t qB a ji,t ` a ji,t ` ¸ ` λ i,t (8) In general, aggregation involves estimating the cross sectional covariance of asset returnsand their marginal dependence on the asset level with the growth rate in the aggregationresidual Ξ t ` ,i p h q Ξ t,i p h q .4.1.1. Which assets to use?
The methodology is flexible enough as we can in principle useany asset class, conditional on being able to measure the relevant components of (8). Whilethis is interesting, this is a non-trivial task as (i) it requires assumptions on the structure ofthe return of the specific asset and (ii) it increases the dimensionality of the cross sectionalmoments that will have to be identified from the data . For example Debortoli and Gali (2018) show that a simple two-agent New Keynesian (TANK) modelwhich abstracts completely from heterogeneity within unconstrained agents, captures reasonably well theimplications of a baseline HANK model regarding the effects of aggregate shocks on aggregate variables.Tryphonides (2019) evaluates the quality of this approximation and utilizes it to identify how heterogeneityand the extensive and intensive margins of adjustment depend on aggregate shocks. This line of research is pursued in current work. ith an eye towards the application, and to facilitate aggregation, I focus on assets whoseindividual returns depend only on aggregate risk e.g. government bonds or savings accountswhose benchmark returns depend on the policy rate. I thus set r p a ji,t ` , X t q “ r p X t q .This is also consistent with a wider asset class whose returns are characterized by r p a ji,t ` , X t q “ r p X t q ` B r j p a ji,t ` qB a ji,t ` a ji,t ` where the second idiosyncratic component is unobserved. This is truefor e.g. a credit line which involves higher interest rates at higher levels of unsecured bor-rowing and can be thought of as a softer borrowing constraint which has similar propertiesto λ i,t .Let s i,t ” p ν i,t , y i,t , (cid:32) a ji,t , a ji,t ` ( Jj “ q and S t be the vector that includes all aggregate states,including X t and the distribution of s i,t . Aggregating the Euler equation using p p s i,t | S t q ,and using that agent expectations are formed using p p s i,t ` , S t ` | s i,t , S t q , the approximateaggregated first order condition is equal to: p C t ´ hC t ´ q ´ ω “ β E t p C t ` ´ hC t q ´ ω Ξ t ` p h q Ξ t p h q p ` r p X t ` qq ` ˆ λ i,t p p s i,t | S t q ds i,t : “ β E t p C t ´ hC t ´ q ´ ω Ξ t ` p h q R t ` ` µ t (9) where C t ` ” ˆ c i,t ` p p s i,t ` , s i,t | S t ` , S t q d p s i,t ` , s i,t q , C t ” ˆ c i,t p p s i,t | S t q ds i,t Ξ t p h q ” ` ω p ω ` q ˆ ´ h ˆ C t ´ C t ˙˙ ´ V ar t ˆ c i,t C t ˙ Ξ t ` p h q ” ` ω p ω ` q ˆ ´ h ˆ C t ´ C t ˙˙ ´ V ar t ˆ c i,t C t ˙ Following exactly the same approach, one can derive the corresponding intratemporal con-dition : L t “ p W η t p C t ´ hC t ´ q ´ ωη Ξ labt where Ξ labt “ ` ω p η ` ω q η ˆ ´ h ˆ C t ´ C t ˙˙ ´ V ar t ˆ c i,t C t ˙ Cross Sectional Variance.
As is evident from above, in order to measure the aggrega-tion residuals Ξ t and Ξ labt , one has to identify the cross sectional dispersion of consumption, See the Appendix for the derivations. here consumption variance is defined as follows: V ar p c i,t q “ ˆ p c i,t ´ C t q p p s i,t | S t q ds i,t V ar p c i,t ` q “ ˆ p c i,t ` ´ C t ` q p p s i,t ` , s i,t | S t ` , S t q d p s i,t ` , s i,t q What is important here is that these quantities do not condition on any individual informa-tion and thus consumption can be arbitrarily related to individual states. Correspondingly,they do depend on aggregate states and the methodology of this paper allows for this featurewithout taking a stance on the exact dependence structure. This is, perhaps surprisingly, anadvantage of using aggregate measures and is in contrast to models of panel data that havespecific implications for the law of motion of the cross sectional variance, see e.g. Chamber-lain (1984).Furthermore, it is clear that there are two main ways in which heterogeneity matters foraggregate fluctuations to first order. The first is fluctuations in the variance of the consump-tion share, which distorts both the intertemporal allocation of aggregate consumption aswell as the intratemporal consumption - labor tradeoff even if all agents are "on their Eulerequations". The variance of the consumption share is essentially a sufficient statistic forthe underlying moments of the wealth distribution. The second is the additional aggregatedistortion µ t which is positive whenever there is a subset of agents who are constrained.Since λ i,t is weakly positive by construction, the aggregate distortion to the bond Eulerequation will be a weakly positive random variable as well. It immediately follows that -using any instrument that is positive e.g. an observed aggregate random variable Y t ´ - givesrise to a moment inequality, which is also consistent with a general equilibrium response in R t . In a closed economy, negative distortions to consumption will put downward pressure on the interestrate which then relaxes the borrowing constraint for some of the agents. Even if all the agents becomeunconstrained then the weak inequality is still valid. herefore, the aggregated moment inequalities that are relevant for identification are thefollowing, where the second moment inequality arises from incorporating the aggregate in-tratemporal condition in the Euler equation: E ˜ ´ β ˆ C t ` ´ hC t C t ´ hC t ´ ˙ ´ ω ˆ Ξ t ` Ξ t ˙ R t ` ¸ Y t ´ ě E ˆ ´ β ˆ Ξ t ` Ξ t ˙ ˆ Ξ labt Ξ labt ` ˙ η ˆ L t ` L t ˙ η W t W t ` R t ` ˙ Y t ´ ě The next section analyzes identification from combining macroeconomic and aggregatedsurvey data, while the Appendix contains the more formal results.5.
Identification with Aggregate Information
As illustrated in the previous section, the HA model can be approximated around the RAallocation, which results in distortions that are partially observable. Denoting by Y ft therepresentative agent (frictionless) model prediction, Y t “ Y ft ` λ ot ` λ unt where λ ot is the observable component (up to the parameter vector θ ) of the aggregatedistortion and λ unt the unobservable component. For example, in the previous section, Y t “ C ´ ωt , Y ft ” β E t C ´ ωt ` R t ` , λ unt ” µ t and λ ot ” β E t C ´ ωt ` ´ Ξ t,t ` Ξ t ´ ¯ R t ` .The only theoretical restriction on µ t is its positive sign. In the absence of any additionalinformation, theoretical or empirical, and assuming additive separability in λ t , we couldutilize the following restriction to estimate the structural parameters, where X t ´ is anypositive random variable which is Y t ´ measurable: E p g p Y t , Y t ` , θ q ´ λ ot p Y t , Y t ` , θ qq X t ´ ´ µ “ g p Y t , Y t ` , θ q : “ Y t ´ Y ft and µ : “ E p µ t X t ´ q P R ` Y t`8u .This restriction is unlikely to point identify θ . Yet, µ t is a quite complicated object; abstain-ing from imposing further restrictions results in more credible inference. The question is owever whether we could do better using additional information that does not underminethe credibility of the assumptions we have already made. One way of achieving this is to no-tice that the latent µ t , while it depends on a lot of information, its evolution over time can beexplained by either changes in the proportion of agents whose behavior is distorted, the ex-tensive margin, and by how much they distort their behavior, on average. More particularly,we can decompose µ t into two multiplicative components: µ t “ E t p µ i,t | µ i,t ‰ q P t p µ i,t ‰ q ” κ t B t where κ t is the intensive margin and B t the extensive margin.These margins can always be defined for a broad range of models. If we could find a wayto measure either margin, e.g. B t , then this could be utilized to sharpen inference as itgenerates a new set of moment inequalities. Using that B t and κ t have the same sign : E p g p Y t , Y t ` , θ q ´ λ ot p Y t , Y t ` , θ qq X t ´ B ´ t ´ κ “ κ : “ E p κ t X t ´ q P R ` Y t`8u .When B t is zero for all t , then (13) is trivially satisfied for all θ and (12) becomes a momentequality p µ “ q .5.1. Identification Analysis.
While I briefly lay out the identification framework basedon moment inequalities, in the Appendix I establish the general conditions under which ad-ditional moment inequalities are informative for the parameters of interest. First, conditionslike (12) and (13) can be generally expressed as follows:(14) E ˆ g r ˆ p Y t , Y t ` , θ q ´ λ otr ˆ p Y t , Y t ` , θ q ˙ φ p Y τ ´ q “ E µ t φ p Y τ ´ q ” U P r , r where φ p . q is an inequality preserving function of Y τ ´ , r is the number of conditions and U is a vector of nuisance parameters that restore the moment equality. The identified set is As I will illustrate in the empirical application, combining information from different surveys is one fruitfulway. The Appendix includes a more formal proof. hus defined as:Θ I : “ (cid:32) θ P Θ : D U P R r ` Y t8u ; E p m p Y t , Y t ` , θ q φ p Y t ´ qq ´ U “ ( where m p Y t , Y t ` , θ q : “ p g p Y t , Y t ` , θ q ´ λ ot p Y t , Y t ` , θ qq .Theorems 2 and 3 in the Appendix characterize identification in the case in which the numberof moment conditions used is the same as the dimension of θ and the case of using a largernumber of conditions. Moment conditions with survey data can always be decomposed intothe component that corresponds to the original set of conditions and the component thatprovides additional information. Therefore they can be casted into this framework. The nextproposition is a corollary of these theorems and characterizes how informative B t can be ( ˆ B t signifying the empirical counterpart). In a nutshell, if survey data constrain the stochasticproperties of µ t as implied by (12) , then the size of Θ I is likely to be refined, in population.A diagrammatic exposition can be also found in Figure 6 in the Appendix. Proposition 1.
Identification with Surveys. Given Theorems 2 and 3(1) If ˆ B t ÝÑ N Ñ8 B t P p , q , Θ I Ă Θ I .(2) If V ar p ˆ B t q ÝÑ N,T Ñ8 V ar p B t q “ , then Θ I “ Θ I .(3) Impossibility of point identification: When B t ‰ , Θ I is not a singleton.Proof. See Appendix (cid:3) B t P p , q is essential for shrinking the admissible set of predictions. Moreover, absence oftime variation in the proportion of agents that are constrained i.e. if the economy reachesthe stationary state, implies that B t has no additional information for the parameters ofthe model. What this also implies is that additional identification really comes from thepresence of aggregate risk. Other micro moments.
The advantage of the second order approximation to the HA modelis that it makes clear which moments are likely to be important. In this particular case, themicro moment that is necessary for identifying the structural parameters of interest is the ross sectional variance of consumption. Since consumption is a nonlinear function of wealth,this is equivalent to using the corresponding moment from the wealth distribution. On theother hand, sharpening inference without compromising credibility requires information on B t and κ t . In the application we will combine information from different sources to identify B t . But first, I investigate how this approach fairs in a simulation experiment.6. Illustration with Simulated Data
The data generating process is the workhorse heterogeneous agent model with aggregaterisk, the Krusell and Smith (1998) economy (KS hereafter). Adopting the formulation inMaliar, Maliar, and Valli (2010); Haan, Judd, and Juillard (2010), the KS economy featuresa standard capital accumulation problem under incomplete markets, where the first orderconditions for each household are as follows: c ´ ωi,t “ β E t p ´ δ ` r kt ` q c ´ ωi,t ` ` λ i,t k i,t ` “ p ´ τ q w t ¯ l(cid:15) i ` ν t w t p ´ (cid:15) i q ` p ´ δ ` r kt q k i,t ´ c i,t k i,t ` ě (cid:15) i P t , u is the idiosyncratic employment shock, ¯ l the time endowment and v t thefraction of the wage that is received as unemployment benefit.Perfectly competitive markets imply that the interest and wage rates are equal to the mar-ginal products of capital and labor: r kt “ Z t α ˆ K t ¯ lL t ˙ α ´ , w t “ Z t p ´ α q ˆ K t ¯ lL t ˙ α while the labor income tax rate is designed to be equal to τ “ ν t u t ¯ lL t . The only source ofaggregate risk is the aggregate productivity shock Z t which takes two values, t Z low , Z high u .In turn, the unemployment rate u t depends on the productivity shock as well, and takestwo possible values, t u low , u high u , with transition matrix P “ p p ll , p lh ; p hl , p hh q . Finally, fullemployment is normalized to one, thus L “ ´ u . he aggregated Euler equation is similar to (9) (with h “
0) but now the interest rate ispinned down by a productive (capital) asset. The following moment inequalities characterizethe restrictions without and with survey data respectively: E ˆ C ´ ωt ´ βC ´ ωt ` ˆ Ξ t ` Ξ t ˙ p ´ δ ` r kt ` q ˙ X t ´ ě E ˆ C ´ ωt ´ βC ´ ωt ` ˆ Ξ t ` Ξ t ˙ p ´ δ ` r kt ` q ˙ B ´ t X t ´ ě Experiment.
I adopt the same parameterization and solution method as in Maliar, Maliar,and Valli (2010); Haan, Judd, and Juillard (2010) to generate the data by simulating 1400periods and keeping the last 200 as the pseudo-sample. The parameter values are set asfollows: β “ . , γ “ . , α “ . , δ “ . , Z high “ . , Z low “ . , ν “ . , ¯ l “ . , u h “ . , u l “ .
1. Fixing δ , I employ the above moment inequalities to estimate p β, ω q using standardized values of aggregate consumption lagged by one and two periods as theinstrument vector X t .Reparameterizing the inequalities to obtain equalities, (as in (14)) the parameter estimatesfor p θ, U q are defined by minimizing the following CU-GMM criterion: Q T p θ, U q “ p ¯ m p Y t , Y t ` , θ q ´ U q T V ´ p ¯ m p Y t , Y t ` , θ q ´ U q where ¯ m p Y t , Y t ` , θ q is the sample counterpart of the moment conditions above with θ “p β, ω q and V ´ is the generalized inverse of the long run variance of the moment conditionsestimated using the Newey and West (1987) estimator, truncated at first order. Inferenceis based on quantiles of 300000 draws from the quasi-posterior distribution based on theabove objective function using a 1-Block Metropolis-Hastings Markov Chain Monte Carlosampler. Similar to standard quasi-Bayesian algorithms, the quasi-posterior is proportionalto π p θ q e ´ T Q T p θ,U q where π p θ, U q is the prior distribution, see e.g. Chen, Christensen, andTamer (2018). In this experiment I do not use any prior information, apart from U ą igure 2 presents the 95% confidence sets with and without using survey data when T “ T “ .
01% on average!) its covariation with the momentconditions is very informative for the preference parameters.
Figure 2 Empirical Application: The case of Spain
This section applies the above analysis to the case of Spain, as it is a country in whichfinancial frictions are a priori expected to have played a significant role post 1999, andespecially during the crisis period, 2008-2014. Unrestrained credit growth in the past and thesubsequent bust has resulted in firm bankruptcies and a large drop in investment. Similarly,household indebtedness before the crisis was succeeded by de-leveraging. Figure 3 plotsthe cyclical components for consumption, investment and output after 1999. As evident, here was a large slump in aggregate investment, followed by a relatively smaller slump inaggregate consumption. Figure 3
An advantage of the framework developed in this paper is that the identification of deepparameters is not tied to taking a stance on how severe financial frictions have been, nor thetime period in which credit constraints have been binding. This allows us to use the entiretime series in the post - Euro period, as well as survey data that are partially harmonizedacross countries of the European Area, making the analysis done here portable to othercountries as well .7.1. Measurement.
As discussed earlier on, being able to measure B t improves identifica-tion and it is therefore useful to first discuss pertinent measurement issues and how they canbe addressed. The exact triennial measure of B t is computed from the Survey of HouseholdFinances (SHF) and its corresponding 95% confidence bands are plotted in Figure 4. Thequestionnaire asks households whether they have been denied a loan application during thisperiod. This question is consistent with a variety of reasons for which these households havebeen rejected, depending on individual characteristics, employment, guarantees, changes in Another interesting case to apply this methodology is the case of the US using e.g. data from the Sur-vey of Consumer Finances (also available for every three years) and FRBNY microeconomic data centre(Survey of Consumer Expectations). Indeed the latter dataset is available at a high frequency, on par withmacroeconomic variables, but this survey started much later than EU surveys (2013). he institution’s credit policy and excessive debt. Moreover, accounting for consumers thathave not applied for loans because they expect to be rejected (also coined as "discouragedborrowers" in Cox and Jappelli (1993) ) or consumers that were granted a fraction of therequested loan does not affect the estimate of this proportion.In order to obtain a quarterly measure, I utilize monthly data from the Business and Con-sumer Survey (BCS) of the European Commission . In particular, the survey asks thehouseholds about their financial situation, and the possible answers range from "We are sav-ing a lot" to "We are running into debt". In Figure 4, I plot the proportion that correspondsto the latter category (those that claim to be getting indebted), which increased from 5%in 2008 to 10% by the end of 2016. This is an an upper bound to the true fraction ofhouseholds that are likely to face borrowing constraints within the quarter, as some of those(already into) getting into debt may (have been) be denied a loan application, partially orin total.To infer the true proportion of constrained households for the periods in which it is notobserved, so all periods between the triennial measure from SHF, I employ a mixed frequencymodel, where the quarterly observations on consumers running into debt are linked to thetriennial exact measure.Letting B t be the exact measure and Π t the proportion of consumers running into debt, then B t “ ζ t Π t where ζ t P p , q , as the agents that hit the minimum level of permissible debtare a fraction of the total measure of indebted households. As shown in the appendix, thisserves as a measurement equation in a mixed frequency state space model. The extractedmeasure is plotted in Figure 4.Since this measure is primarily informed from the SHF survey, one might still be concernedthat the latter might under-state the true measure of constrained households. While mea-surement error is already controlled for in the state-space formulation, notice that even if BCS data can be downloaded from here. igure 4 ˆ B t Ñ ˜ B t ď B t , the moment inequality restrictions are still valid: m t ˜ B ´ t ě m t B ´ t E p m t ˜ B ´ t X t ´ q ě E p m t B ´ t X t ´ q ě m t : “ ´ ´ β E t p C t ` ´ hC t C t ´ hC t ´ q ´ ω ´ Ξ t,t ` Ξ t ¯ R t ` ¯ . Our approach therefore is robust to con-cerns about underpredicting the constrained households.Finally, another variable that needs to be measured from micro data is the dispersion in theconsumption share, which is constructed using the Survey of Household Finances (SHF) . Inorder to impute the value of dispersion when it is latent, I make the simplifying assumptionthat it is constant. This is not likely to be a detrimental assumption for this kind of exerciseas consumption is smoother than income, and its distribution is a slowly moving variablesee e.g. Anghel, Basso, Bover, and et al (2018) (Figure 13). SHF data can be downloaded from here. Please consult the appendix for a detailed explanation of howthe measure is constructed. stimation Details and Results. For completeness, I repeat the moments derived in Section5, with the only difference of introducing the component-wise positive valued nuisance vector U : “ p µ T , κ T q T : E ˜ ´ β ˆ C t ` ´ hC t C t ´ hC t ´ ˙ ´ ω ˆ Ξ t ` Ξ t ˙ R x,t ` ¸ φ p Y t ´ q ´ µ “ E ˆ ´ β ˆ Ξ t ` Ξ t ˙ ˆ Ξ labt Ξ labt ` ˙ η ˆ L t ` L t ˙ η W t W t ` R x,t ` ˙ φ p Y t ´ q ´ κ “ where R x P t R g , R ib u , the ex-post real 3-month Treasury bond rate and the Interbankrate.To construct φ p Y t q , I use lagged values of growth rates using combinations of sets of instru-ments among p ∆ C t , ∆ Y t , ∆ W t , ∆ L t , ∆ I t q , I t being aggregate investment, and p R g,t , R ib,t q .The criterion function used is the same as in the simulation experiment, the CU-GMM ob-jective function, while the inference procedure is modified in order to obtain less conservativeconfidence sets than what the Monte Carlo approach can deliver. More particularly, parti-tioning the parameter space into the space of interest and its complement, p Θ j , Θ z Θ j , U q ”p Θ j , ˜Θ q , inference for each parameter θ j P Θ j is based on Procedure 3 in Chen, Chris-tensen, and Tamer (2018) by comparing the profile likelihood ratio sup ˜ θ P ˜Θ T p´ Q T p ˆ θ j , ˆ˜ θ q ` Q T p θ j , ˜ θ qq to the χ p q critical value. The prior upper and lower bounds are ω P r . , s , η P r . , s , β P r . , . s and h P r . , . s .The results that follow provide additional empirical support to Proposition 1. Table 1presents the 95% confidence sets. The first two blocks present estimates using R ib and R g , while the rest blocks examine resulting estimates using alternative modifications toinstruments, moment weighting and additional moment conditions to evaluate the robustnessof the empirical results. Including lagged investment growth as an instrument producestighter sets irrespective of whether survey data are used or not . Correspondingly, p ˆ θ j , ˆ˜ θ q “ arg inf p θ j , ˜ θ qPp Θ , U q Q T p θ, U q . I have also experimented with including lagged values of the EONIA rate as an instrument, but due to itsstrong correlation with the rest of the interest rates, see Figure 7 in the Appendix, it does not aid much toobtaining tight sets. able 1. Robust Confidence Set for Preference Parameters
Parameter without Survey with Survey q . q . q . q . Instruments Return ω η Y t ´ , ∆ W t ´ ,∆ L t ´ R ib β I t ´ , R ib,t ´ , R g,t ´ h ω η Y t ´ , ∆ W t ´ , ∆ L t ´ R g β I t ´ , R ib,t ´ , R g,t ´ h ω η Y t ´ , ∆ W t ´ R g β L t ´ , R ib,t ´ , R g,t ´ h ω η Y t ´ , ∆ W t ´ , ∆ L t ´ R g β I t ´ , R ib,t ´ , R g,t ´ h ω η Y t ´ , ∆ W t ´ , ∆ L t ´ R g β I t ´ , R ib,t ´ , R g,t ´ h Estimating long-run variance does not contribute much to obtaining tight sets. More im-portantly, utilizing the information in the intratemporal condition as an additional momentequality can aid identification, and in particular when survey data is also utilized . More-over, in some cases the confidence sets do not completely overlap, yet this is attributed to ˜ˆ C t ` ´ hC t C t ´ hC t ´ ˙ ´ ω ´ ˆ Ξ labt Ξ labt ` ˙ η ˆ L t ` L t ˙ η W t W t ` ¸ φ p Y t ´ q “ he different weighting that moments receive in finite sample, as in all cases the parametersare (conditional on U ) overidentified.In most cases, had we neglected the extensive margin of constrained consumers higher valuesof risk aversion would be admissible. This is also consistent with the analytical result in theexample of Section 3 and the results of Section 5. As long as there is a non-trivial proba-bility that some of the agents become constrained, the extensive margin provides additionalinformation. Looking at the smallest interval compatible with the different estimates, riskaversion ranges from 0.46 to 1.83 which is close to estimates obtained in the microeconometricliterature and much smaller than the typical estimates obtained using macroeconomic datain representative agent models . As mentioned in the introduction, this provides evidencethat incomplete markets and borrowing constraints can account for variation that would bewrongly attributed to risk aversion.The confidence set for the Frisch elasticity is also refined, and results support values between0.47 and 1, which are much closer to the values supported by micro evidence .To understand which information is important for identification in this case, we can usethe intratemporal condition. Denoting the marginal utility of consumption by M U t : “p C t ´ hC t ´ q ´ ω , the Frisch elasticity is potentially identifiable using the elasticity of differ-ent macroeconomic aggregates with respect to wage i.e. hours ( L ), dispersion (Ξ lab ), theundistorted marginal utility of wealth ( M U ´ µ ) and the extensive and intensive margins ofthe distortion, B and κ respectively:1 η “ B lnL t B lnW t ´ B ln Ξ labt p η qB lnW t ` B lnMU t B lnW t “ (cid:15) L,W ´ (cid:15) Ξ lab p η,ω q ,W ` (cid:15) MU ´ µ,W ` p (cid:15) B,W ` (cid:15) κ,W q µMU ´ µ This is an implicit equation in η , which essentially identifies η by adjusting the raw elasticity (cid:15) L,W for variation that is due to wealth effects in order to isolate the substitution effect. In representative agent models, in order to rationalize the high equity premium over the risk free rate,very high estimates of risk aversion need to be assumed, which also generates a puzzle on why the risk freerate is so low (Weil, 1989), see e.g. Kocherlakota (1996). In fact, Chetty, Guren, Manoli, and Weber (2011) suggest that models that require an elasticity above 1are inconsistent with micro evidence. he admissible values of η will depend non-linearly in all of these elasticities and on theadmissible values of ω . The more responsive are dispersion and the two margins of thedistortion to changes in the wage rate, the greater the divergence between the wage elasticityof aggregate hours from (cid:15) L,W ` (cid:15) MU,W , which is the Frisch elasticity in the complete markets case.Being able to measure B implies that the admissible estimates of η will only vary withthe admissible domain of the response of the intensive margin of adjustment to the wage, (cid:15) κ,W .Consistent with the results of Domeij and Flodén (2006), the adjustment to individual be-havior in the presence of borrowing constraints can imply a higher elasticity of labor supplyas (cid:15) κ,W is expected to be negative. Moreover, the distortion to the extensive margin B t affectsaggregate supply in the same way. This is also in line with the fact that the lower bound onthe Frisch elasticity is refined upwards in all the cases in which B t was employed.When macro aggregates are utilized for identification, the above effects would be defintelytrue as long as (cid:15) Ξ lab p η,ω q ,W t is negligible. Nevertheless, when (cid:15) Ξ lab p η,ω q ,W t is not negligible andas it depends on η , identification depends on the relative magitudes in the wage elasticityof consumption dispersion, (cid:15) V c,t ,W t and the wage elasticity of consumption growth, (cid:15) g c,t ,W as (cid:15) Ξ lab p η,ω q ,W “ γ ,t (cid:15) V c,t ,W t ´ hγ ,t (cid:15) g c,t ,W t,t For example, if consumption dispersion is relatively unresponsive to wage changes, andthus the first elasticity in the above expression is negligible, the mere presence of a non-zero consumption dispersion in γ ,t implies that (cid:15) Ξ lab p η,ω q ,W t is increasing in η as (cid:15) g c,t ,W isexpected to be positive. When (cid:15) V c,t ,W t is different than zero, the final result depends ina more complicated way on the rest of the parameters e.g. risk aversion and the habitparameter as well as the other quantities in γ ,t and γ ,t . Therefore, while the refinement γ ,t p η, ω q : “ ω p η ` ω q V ar t ´ c i,t C t ¯ η ´ ´ h ´ C t ´ C t ¯¯ ` ω p η ` ω q V ar t ´ c i,t C t ¯ , γ ,t p η, ω q “ V ar t ´ c i,t C t ¯ C t ´ C t ´ hC t ´ η ´ ´ h ´ C t ´ C t ¯¯ ` ω p η ` ω q V ar t ´ c i,t C t ¯ n the lower bound can be rationalized without necessarily accounting for heterogeneity, therefinement of the upper bound is possible by accounting for consumption dispersion.Regarding the habit parameter, survey data provide more information for the lower bound.The evidence points towards values between 0.43 and 0.6, which is a non-negligible degree ofinertia in consumption. Finally, there are no major information gains for the discount factor,which ranges from 0.98 to 0.993. This is also consistent with the simulation experiment, as β appears to be point identified. 8. Conclusion
This paper studied identification in heterogeneous agent economies with aggregate risk usingrestrictions that are consistent with alternative underlying mechanisms. While this approachresults in set identification, theory and simulation evidence show that additional distribu-tional information can be useful in shrinking the set of admissible models. One such statisticis the measure of the borrowing constrained. The paper has also shown that this is empir-ically relevant and that it can potentially help in credibly bridging the gap between microand macro elasticities. . Appendix
Constrained versus Unconstrained Consumption.
Notice that for large values of x t , C inct and C per.fort become parallel, as expected. Figure 5.
Example of Optimal Consumption in Incomplete Markets9.2.
Deriving Consumption Growth.
When the household is constrained, c coni,t “ x i,t while unconstrained consumption is equal to c unci,t “ p ` r ´ ρ q ` r x i,t . The latter is parallel to theperfect foresight solution, c pfi,t “ p ` r ´ ρ q a i,t ` ` r ´ ρ ` r p y i,t ` ¯ y i r q “ p ` r ´ ρ q ` r p x i,t ` ¯ y i r q where p ` r ´ ρ q ` r ¯ y i r is interpreted as the share of permanent income (¯ y i ” E y i,t ) that is channeledto precautionary savings . Using the constrained agent’s Euler equation, it has to be thecase that ρx i,t “ E p c i,t ` | λ i,t ą q ´ λ i,t . Expectations include the possibilities of beingconstrained or not in t `
1, thus: ρx i,t “ E p c i,t ` | λ i,t ą q ´ λ i,t “ E p x i,t ` | λ i,t ą q P p λ i,t ` ą | λ i,t ą q` ˆ ´ ρ ` r ˙ E p x i,t ` | λ i,t ą q P p λ i,t ` “ | λ i,t ą q ´ λ i,t See Carroll (2001) and Figure 5 for an example of a numerical solution that verifies this. E p x i,t ` | λ i,t ą q „ ´ ρ ` r P p λ i,t ` “ | λ i,t ą q ´ λ i,t “ ¯ y i „ ´ ρ ` r P p λ i,t ` “ | λ i,t ą q ´ λ i,t “ ¯ y i „ ´ ρ ` r p ´ F y i p x ‹ i qq ´ λ i,t where in the last two lines we use that cash on hand in t ` t is equalto y i,t ` whose expected value is ¯ y i , while the probability of being constrained in t ` F y i p x ‹ i q , x ‹ being the cash on hand threshold. Thus, λ i,t “ ´ ρx i,t ` ¯ y “ ´ ρ ` r p ´ F y i p x ‹ i qq ‰ and ∆ c inci,t ` “ ˜ ρc i,t ` (cid:15) t ` ` p λ i,t ą q ˆ ´ ρc i,t ` ¯ y i ˆ ´ ρ ` r p ´ F y i p x ‹ i qq ˙˙ : “ ˜ ρc i,t ` (cid:15) t ` ` p λ i,t ą q p λ c i,t ` λ ,i q (17)9.3. Alternative Derivation to Hall (1978).
Following Commault (2019), consider theEuler equation: U ´ p β p ` r qq ω c i,t ¯ “ E t U p c i,t ` q ` p β p ` r qq ´ λ i,t p β p ` r qq ω c i,t “ U ´ ` U p E t c i,t ` ´ φ i,t q ` p β p ` r qq ´ λ i,t ˘ where φ i,t ą λ i,t “ p β p ` r qq ω c i,t « E t c i,t ` ´ φ i,t ´ ˜ λ i,t where ˜ λ i,t “ ´ p β p ` r qq ´ λ i,t U pp β p ` r qq ω c i,t q ą
0. The distortion to random walk comes from both theprecautionary savings effect and occasionally binding constraint. Repeating the analysis in7.2 yields that: λ i,t ` φ i,t “ ¯ y i „ ´ ρ ` r p ´ F i,y p x ‹ i qq ´ ρx i,t , and ∆ c inci,t ` “ ˜ ρc i,t ` (cid:15) t ` ` p λ i,t ą q ˆ ´ ρc i,t ` ¯ y i ˆ ´ ρ ` r p ´ F y i p x ‹ qq ˙˙ ` p λ i,t “ q φ i,t inally, the bound derived in Section 3 is identical because φ i,t and Cov p φ i,t , y i,t q have thesame sign as λ i,t and Cov p λ i,t , y i,t q respectively.9.4. Derivations with external Habits in Consumption.
With external habit forma-tion, the utility of the household becomes as follows: u p c i,t , l i,t , C t q : “ ˜ p c i,t ´ hC t ´ q ´ ω ´ ´ ω ´ l ` ηi,t ` η ¸ and the corresponding conditions are: p c i,t ´ hC t ´ q ´ ω “ β E t p c i,t ` ´ hC t q ´ ω π ´ t ` R t ` ` µ i,t l i,t “ W η t p c i,t ´ hC t ´ q ´ ωη Correspondingly, the aggregate intratemporal condition becomes: ˆ l i,t p p s i,t | S t q ds i,t “ W η t p C t ´ hC t ´ q ´ ωη ˆ ` ω p ω ` η q η p C t ´ hC t ´ q ´ V ar t p c i,t q ˙ ” W η t p C t ´ hC t ´ q ´ ωη Ξ labt .5. Proofs for results in main text.
Proof. of Result (7)Let p i,t : “ P t p λ i,t ą q . If ˜ ρ IV is admissible, evaluating the LHS of (6) at the true ˜ ρ “ ρ ´ ρ IV , and solving for p i,t : p i,t ď Φ i,t p v ´ ˜ ρc i,t q ´ Φ i,t p v ´ ˜ ρ IV c i,t q Φ i,t p v ´ ˜ ρc i,t q ´ Φ i,t p v ´ λ ´ p ˜ ρ ` λ q c i,t q : “ g p p i,t q The function g p p i,t q satisfies g p q “ g p q “ y “ (cid:3) Proof. of Results (12) and (13)(1) Starting from the moment function which is separable in λ ot , g p Y t , Y t ` , θ q ´ λ ot p Y t , Y t ` , θ q “ µ t where µ t is positive or negative, then multiplying by X t ´ ą µ t “ κ t B t , dividing both sides by B t and taking unconditional expectations with X t ´ we conclude. (cid:3) Identification with moment conditions.
I first define two notions that will be help-ful for the proposition that follows:
Definition 1.
Denote m p Y t , Y t ` , θ q as correctly specified if there exists θ P Θ such that if U “ E m p Y t , Y t ` , θ q “ Definition 2.
For some U P R r ` , θ p U q is conditionally identified if there does not exist anyother θ P Θ, θ p U q such that E m p Y t , Y t ` , θ p U qq “ E m p Y t , Y t ` , θ p U qq .The first ensures that the identified set is not empty, while the second guarantees that moreinformation on the wedge U leads to a smaller identified set for θ . Given these definitions,Theorem 2 establishes that fixing a value for the wedge U , there exists a unique value for θ , defined as θ p U q , that satisfies (14). Any additional information is likely to make theidentified set smaller. heorem 2. Θ I when r “ n θ Under correct specification of m p . q and conditional identification, D ! mapping G : Θ I “ G ´ p U q X Θ Proof.
Fix U n θ ˆ P p
U , ¯ U s . Assuming conditional identification, there is a unique value of θ ˚ such that E m p Y t , Y t ` , θ ˚ q ” G p θ ˚ q “ U n θ ˆ . Thus, Θ I “ Θ X G ´ p U q . (cid:3) The use of additional conditions.
I distinguish between moments m α,t p θ q , the nec-essary moments functions for Theorem 2 being true, and m β,t p θ q the additional momentfunctions where for notational brevity, I have dropped the dependence on Y τ ´ . In addi-tion, let ˆ m α,t p θ q : “ m α,t p θ q φ t and ˆ m β,t p θ q : “ m β,t p θ q φ t , ˆm α p θ q and ˆm β p θ q the correspondingvectors, and ¯m α p θ q and ¯m β p θ q the vector means. Let W be a real valued, possibly datadependent matrix, diagonal in p W α , W β q . Furthermore, denote by Q α the conditional expec-tation operator when conditioning on W Tα ˆm α p θ q and by Q K α the residual operator.The following proposition specifies that additional moments have to be less than perfectlycorrelated with the necessary conditions for the identified set to be smaller. Theorem 3.
For U t “ (cid:32) U t P r U , ¯ U s ( define U ct “ ! U t P r U , ¯ U s : E Q K α ´ W Tβ m β p θ p U qq ´ U t ¯ “ ) Then, Θ I Ă Θ I iff U ct Ă U t Proof.
Denote the moment conditions using n y macroeconomic variables by q p θ, Y τ ´ q andmoment conditions using k “ r ´ n y survey variables by q p θ, Y τ ´ q : q p θ, Y τ ´ q n y ˆ “ V p Y τ ´ q P r V p Y t q , ¯ V p Y t q s q p θ, Y τ ´ q p n y ´ k qˆ “ V p Y τ ´ q P r V p Y t q , ¯ V p Y t q s which imply the following unconditional moment equalities : E pp q p θ, Y τ ´ q ´ V p Y τ ´ q φ t q “ E pp q p θ, Y τ ´ q ´ V p Y τ ´ q φ t q “ enote this vector of moment conditions by m . Partition the vector m ” p m Tα , m Tβ q T where m α contains the first n θ moments. Since r ą n θ , and given a weighting matrix W , the firstorder conditions are as follows: E J p θ q T p W Tα m α ` W Tβ m β ´ V t q φ t “ J p θ q r ˆ n θ has full rank. To economize on notation, redefine the weights afterpre-multiplication with the Jacobian, which implies that: E p W Tα m α ` W Tβ m β q ´ U n θ ˆ “ m on a lower dimensional subspace. Since W is an arbitrary matrix,and p m α , m β q are possibly correlated, we reproject the sum onto the space spanned by W Tα m α . Denote the projection Q α : “ W Tα m α p m Tα W Tα m α q ´ m Tα W α and Q K α the orthogonalprojection. Since the original sum satisfies the moment condition, then the two orthogonalcomplements will also satisfy it: Q α ´ W Tα m α ` W Tβ m β ´ U ¯ “ W Tα m α ` Q α ´ W Tβ m β ´ U ¯ “ Q K α ´ W Tα m α ` W Tβ m β ´ U ¯ “ Q K α ´ W Tβ m β ´ U ¯ “ U P ” W Tα V p Y t q α ` W Tβ V p Y t q β , W Tα ¯ V p Y t q α ` W Tβ ¯ V p Y t q β ı b φ t As in Theorem 2 the first set of restrictions identifies a one to one mapping from U to Θ I ,and therefore θ ˚ p U q ” G ´ p U q . Plugging this in the second set of restrictions eliminatesdependence on m α and imposes further restrictions on the domain of variation of U . Theadmissible set for U is now ! U P p´ inf , ¯ U q : Q K α ´ W Tβ m β p U q ´ U ¯ “ ) Therefore, D θ P θ p U q : θ R Θ I p U q and consequently Θ I Ă Θ I . The same result carriesthrough if we replace the linear projection with conditional expectations. Letting Q α bethe conditional expectation operator implies that any integrable moment function m can bedecomposed as m “ Q α m ` Q K α m such that Q α p m ´ Q α m q “ (cid:3) roof. of Proposition 1 (1): When B t “ E p g p Y t , Y t ` , θ q ´ λ ot p Y t , Y t ` , θ qq X t ´ “ B t “
1, (13) collapses to (12). E p g p Y t , Y t ` , θ q ´ λ ot p Y t , Y t ` , θ qq X t ´ “ E p κ t X t ´ q “ E p µ t X t ´ q For B t P p , q , the moment conditions in (13) and (12) are not perfectly correlated as Corr p κ t , µ t q “ Corr p κ t , B t κ t q ‰
1. By Theorem 3 we conclude.(2): If
V ar p B t q “
0, then
Corr p κ t , B t κ t q “ I is a singleton. Then it must be that the RHS of (14) is zero: Either B t “ t , or p p s i,t | S t q has unit mass on one agent who is unconstrained, and thus B t “ (cid:3) .7. Diagrammatic exposition for the Extensive Margin.
Dropping dependence on p Y t , Y t ` q , the aggregate moment condition without survey is summarized by E m p θ q “ µ , while to first order, the condition with the extensive margin can be written as κ : “ E p m p θ q B ´ q « E p m p θ qq ´ E p m p θ q B q ” µ ´ v , where the approximation is around B ‹ “ v : “ E p m p θ q B q .If B is a non-degenerate random variable (when B is degenerate i.e. it takes a single value¯ B , v “ µ ¯ B P p , µ q ), then the admissible range for κ lies within the admissible range for µ ,leading to a refined identified set for θ . µκ v “ v “ µ ¯ κ ¯ µ ¯ µ ´ v Figure 6.
Identified Set for Nuisance Parameter with Survey Data (in Blue) .8. Mixed Frequency Model for Quarterly B t . Using ˜ b t “ ˜ π t ` ˜ ζ t as a measurementequation, the true (log) quarterly proportion of liquidity constrained consumers is extractedusing the following mixed frequency Gaussian linear state space model, where t “ p j ´ q` q , t is the quarterly observation at year j and q “ t , , , u the within year quarter index:State Equation (ignoring identities): ¨˝ ˜ b p j q ˜ ζ p j q ˛‚ “ ¨˝ ρ b ρ ζ ˛‚¨˝ ˜ b p j ´ q` ˜ ζ p j ´ q` ˛‚ ` ¨˝ ν ˜ b, p j q ν ˜ ζ, p j q ˛‚ Observation equation: ¨˝ ˜ π o p j q ˜ b oj ˛‚ “ ¨˝ ...
18 18 18 18 18 18 18 18 ... ˛‚¨˚˚˚˚˚˚˚˚˚˚˚˚˚˚˚˚˚˚˚˚˚˚˚˚˚˚˚˚˚˚˚˚˚˚˚˚˚˚˚˚˚˚˚˚˚˝ ˜ b p j q ˜ b p j ´ q` ˜ b p j ´ q` ˜ b p j ´ q` ˜ b p j ´ q ˜ b p j ´ q` ˜ b p j ´ q` ˜ b p j ´ q` ˜ ζ p j ´ q` ˜ ζ p j ´ q` ˜ ζ p j ´ q` ˜ ζ p j ´ q ˜ ζ p j ´ q` ˜ ζ p j ´ q` ˜ ζ p j ´ q` ˜ ζ p j ´ q ˛‹‹‹‹‹‹‹‹‹‹‹‹‹‹‹‹‹‹‹‹‹‹‹‹‹‹‹‹‹‹‹‹‹‹‹‹‹‹‹‹‹‹‹‹‹‚ ` v p j q where p ν ˜ b,t , ν ˜ ζ,t q „ N p , diag p Σ ν qq and v t „ N p , diag p Σ v qq . The last diagonal component ofΣ v is calibrated to the standard error from the estimation of B t in the SHF survey. The rest f the components of the diagonal are calibrated to 1% of the variance of the BCS measure(˜ π t ).9.9. Data.
Table 2.
Sources and Transformations
Variable Data Source Final Transformation C t Final Consumption Expenditures, S.A ( C tot ) FRED (OECD) C tot Pop ˚ P ˚ P op t Active Population: Aged 15-74, S.A. FRED (OECD) L t Total Hours Worked , S.A. ( L tot ) Datastream (National Accounts) H tot Pop P t GDP Implicit Price Deflator FRED (OECD) π t – – ln p P t P t ´ q W t Total Employee Compensation , S.A. ( W tot ) Thomson Reuters (OECD) W tot H R t ` EONIA (Quarterly rate, i t ) European Central Bank i t ´ π t ` R g,t ` Government Securities (Quarterly rate, i g,t ) FRED i g,t ´ π t ` R ib,t ` i ib,t ) FRED i ib,t ´ π t ` V ar p c i,t q Consumption Expenditures Survey of Household Finances See 9.8.1
Excluding durables from aggregate consumption does not seem a reasonable thing to dosince we consider liquidity constraints, which are more binding in the case of largepurchases e.g. automobiles. As far as housing is concerned, aggregate consumptionmeasures include imputed rents, which can be thought as the service value from housing.9.9.1.
Computing Consumption Variance using the Survey of Household Finances.
Consis-tent with measurement of the aggregate, the variance of annual nominal consumption iscomputed using survey weights as follows: c nom,annuali,t “ ˚ p Imputed Monthly Rent r code : 2 . s` Car Purchases { r code : 2 . s ` House Durables { r code : p . s` monthly non durable consumption r code : p . sq .10. Interest Rates.
Ex-Post Real interest rates:
Figure 7. ln p R x,t ` q : “ ln p ` i x,t q ´ ln p ` π t ` q .10.1. Simulation Results. p T “ q Figure 8.
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