text:
\section{Introduction}

Incomplete markets have appealing theoretical implications for dynamic macroeconomic models, as they can  generate heterogeneous consumption outcomes and different dynamics for macroeconomic aggregates.\footnote{For example, the standard  Aiyagari-Huggett-\.{I}mrohoro\u{g}lu setup features households which face uninsurable idiosyncratic earnings risk, and thus individual consumption is no longer solely driven by aggregate risk.} A strand of the literature that has studied the consequences of fitting aggregate models to data generated by heterogeneous agents economies has also shown that heterogeneity has important consequences for parameter identification, see for example \citet{GUVENEN20061451,RePEc:ucp:jpolec:v:104:y:1996:i:2:p:219-40} and \citet{JEEA:JEEA1098}.\footnote{More specifically, \citet{GUVENEN20061451} shows that estimates of the elasticity of intertemporal substitution using aggregate data capture the behavior of poorer agents while  \citet{RePEc:ucp:jpolec:v:104:y:1996:i:2:p:219-40} find that cross sectional heterogeneity is another source of bias in estimates of risk aversion and the discount factor. \citet{JEEA:JEEA1098} show that structural parameters are not invariant to policy when heterogeneity is ignored.} Accounting for incomplete markets is therefore deemed necessary when identifying key structural parameters such as agents' preferences.

Nevertheless, in heterogeneous agent economies the underlying mechanism and the resulting equilibrium leads to a complex mapping from the structural parameters to observable outcomes. Modeling choices determine the degree of frictions generated by the model and are therefore consequential for preference identification.

A natural question to ask is whether we can still obtain meaningful identification by accounting for market incompleteness and yet not relying on a particular mechanism. This can lead to more credible inference and the resulting  estimates being portable across models. Moreover, by knowing the extent to which parameters are identified without invoking a particular mechanism can aid our understanding in how specific assumptions may lead to particular empirical results.

Towards this goal, the paper employs identifying  restrictions that are consistent with a variety of mechanisms. More particularly, the paper seeks to identify key preference parameters from time series variation in aggregate data instead of variation at the individual level, obviating the need to model heterogeneity at the micro level. The latter is of course necessary if someone is interested in identifying particular aspects of  heterogeneity.

The proposed approach does not assume a particular equilibrium nor does it impose much structure on idiosyncratic earnings risk, while it accommodates for aggregate risk. The resulting identifying restrictions generate moment inequalities and therefore set identification.\footnote{In a working paper \citep{tryphonides2017set} I show that this approach can be extended to study other forms of frictions including departures from rational expectations and how this relates to perturbations of the measure implied by the frictionless model in an information theoretic framework.}  Using a second order approximation around the representative agent allocation, the paper exemplifies which data - other than standard time series of macroeconomic aggregates - are necessary for identification while it also shows that additional aggregated micro moments can be potentially useful in order to further constrain the set of admissible models without compromising the original set of assumptions. In particular, employing aggregated 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 can provide additional information that can tighten the corresponding parameter bounds. The moment inequality approach is also shown to be robust to under-predicting this margin in the data. Moreover, since identification is based on time variation, aggregate risk is shown to be sufficient for the informativeness of the additional distributional information as it is a source of variation to the agents' distribution.

 The paper provides identification analysis based on aggregate time series as well as simulation experiments where the data generating process is the \citet{krusellsmith} economy. Analytical identification results for an extended example that utilizes individual household time series data can be also found in  Appendix B. The paper provides estimates using  Spanish macro and aggregated micro data, where the proportion of constrained consumers is identified by combining information from the Survey of Household Finances (SHF) and the Business and Consumer Survey (BCS). This is also relevant for empirical practice, as the methodology proposed can be applied to other countries as well since the BCS survey is harmonized across the European Union.

 The analytical, simulation and empirical results of the paper on the importance of the extensive margin for sharper identification have several implications. First,
 it contributes to our understanding that since these models are highly non-linear, certain moments can provide information about preferences without necessarily having a first order predictive power for the level of macroeconomic time series. This is indeed the case for the borrowing constrained, as these individuals might represent a small proportion of the population, so their consumption choices  may not substantially affect the \textit{level} of aggregate consumption, yet fluctuations in this proportion are informative for preferences. Second, and perhaps more substantial is that had we not accounted for the extensive margin (and the growth rate of cross sectional dispersion in real consumption), the neglected variation would be absorbed by the structural parameters. This can in fact explain the gap between
 between micro and macro estimates of elasticities.   This result speaks to both the equity premium puzzle \citep{MEHRA1985145}, in the sense that a lower degree of relative risk aversion is needed to rationalize the co-movement of consumption growth and returns,
 and the "Frisch elasticity puzzle" \citep{10.1257/aer.101.3.471}, which concerns the gap between labor supply elasticities estimated using individual and aggregate data.

 Given that identification is tightly linked to model predictions, the paper investigates further the asset pricing implications of borrowing constraints and heterogeneity. In a counterfactual experiment, I show that shutting down the estimated distortions renders the  representative agent model unable to rationalize the observed equity premium since 2001. Given that the paper does not specify the mechanism that generates frictions in financial markets, this constitutes robust evidence that incomplete markets can indeed be a potential factor contributing to the equity risk premium. An accounting exercise in the spirit of \citet{ECTA:ECTA768} reveals that distortions in capital markets are quantitatively ambiguous when compared to distortions in household borrowing. Nevertheless, there is some evidence that trading constraints e.g. restrictions on short selling seem to dominate transaction costs. Importantly, ignoring the extensive margin of constrained consumers when identifying preferences may lead to misjudging the relative importance of these distortions.

\subsection{Contacts with the literature}Moment inequality restrictions
have been used to characterize frictions in specific markets, see for
example \citet{1996} and \citet{ECTA:ECTA1214}. To my knowledge, this is the first paper that characterizes
such restrictions in dynamic stochastic macroeconomic models with heterogeneity and thus contributes to the literature that deals with partial identification in
structural macroeconomic models (e.g. \citet{lubik,coroneo2011testing}) and the literature on applications of moment inequality models.\footnote{See \citet{ECTA:ECTA1480} and references therein. Partial identification in structural micro models has been applied in a variety of settings, from discrete choice in single agent models, games with multiple equilibria to auctions and networks. For a more comprehensive list of applications of partial identification see \citet{RePEc:ifs:cemmap:25/19} and \citet{ ho_rosen_2017}.}
Moreover, the restrictions imposed are different from the ones used in the strand of papers that seek to obtain identification  using panel data i.e. \citet{10.1257/aer.104.7.2075} and \citet{10.1257/aer.20121549}. The motivation is nevertheless similar in the sense that this paper fully shares the view that identification should be disentangled from the particular formulation of risk sharing possibilities.\footnote{In order to obtain analytical tractability and point identification, \citet{10.1257/aer.104.7.2075} construct an equilibrium in which agents are either perfectly insured against some risks or not insured at all, leading to an inoperative bond market, extending the work of \cite{RePEc:ucp:jpolec:v:104:y:1996:i:2:p:219-40}. This paper differs by not imposing a particular equilibrium while its exploits aggregate information for identification. Similarly, this paper differs from  \citet{10.1257/aer.20121549} as it abstracts from family composition while it accounts for the possibility of binding liquidity constraints. }

The paper's methodology speaks also to the debate on estimating Euler equations.  \citet{ DeathtotheLogLinearizedConsumptionEulerEquationAndVeryPoorHealthtotheSecondOrderApproximation} posed a challenge as to whether the log-linerized Euler equation can be consistently estimated at all, while \citet{ATTANASIO2004406} argued that using time series variation (at the individual level), one can obtain good estimates as long as the growth of consumption variance is unrelated to the instruments used. The latter paper also finds that the estimation of non-linear Euler equations is as bad as in the case of the log-linearized Euler equation when the discount rate is too low, that is, when agents become impatient such that they hit their borrowing constraints. The methodology built in this paper contributes to tackling this issue by recognizing that the presence of borrowing constraints leads to moment inequalities instead of equalities, and that additional data can aid identification.

The rest of the paper is organized as follows. Section 2 presents the class of models in consideration and the identification conditions. Section 3 provides identification analysis, Section 4 presents the empirical study and Section 5 concludes. Appendix A contains the following material: Proofs, a diagrammatic exposition of the identifying power of the extensive margin, additional simulation evidence on the performance of the proposed approach, the mixed frequency model used to extract the proportion of constrained consumers and details on the data. Appendix B contains an extended analytical identification example.

\section{The Model Class}
\subsection{Household Optimization}
The theoretical model features households maximizing expected discounted lifetime utility subject to a general intertemporal budget constraint. Each household receives total income $y_{i,t}(l_{i,t})$ which may depend on labor hours supplied ($l_{i,t}$)\footnote{Notice also that we do not specify a specific process for labor income risk. It can therefore include any kind of individual specific risk to income as well as any kind of transfers.} and makes consumption $(c_{i,t})$, labor supply and investment decisions. In particular, the household chooses to store wealth in $J$ assets  $\left\{a^{j}_{i,t}\right\}_{j=1..J}$ which earn gross returns equal to $R^{j}_{i,t}=1+r^{j}(a^{j}_{i,t},X_{t})$,  while $\psi(a^{j}_{i,t+1},a^{j}_{i,t})$ are the transaction costs for the $j^{th}$ asset.
Furthermore, there is a general constraint technology $g(.)$ that restricts trades between these assets which is increasing in wealth\footnote{This includes popular restrictions e.g. on short selling and non-collateralized borrowing ($a^{j}_{t+1}\geq -\underline{a}$) and collateralized borrowing $(a^{j}_{t+1}\geq - \sum^{J}_{k\neq j}w_{k} a^{k}_{t/t+1})$ where $w_{k}$ summarize restrictions on quantities across assets.}.

Let $\beta\in (0,1)$ be the discount factor and $U(c_{i,t},l_{i,t};\omega,X_{t})$ the instantaneous utility function, where $\omega$ signifies preference related parameters and $X_{t}$ an aggregate state vector\footnote{For example,  with external habits past aggregate consumption, $C_{t-1}$, belongs to $ X_{t}$}. The  household problem is as follows:
\begin{eqnarray*}
	&&\max_{\{c_{i,t},l_{i,t},\left\{a^{j}_{i,t+1}\right\}_{j=1..J}\}_{t=0}^{\infty }}\mathbb{E}_{0}\sum_{t=0}^{\infty }\beta
	^{t}U(c_{i,t},l_{i,t};\omega,X_{t})\\
	s.t. &  c_{i,t} -  y_{i,t}(l_{i,t})   &= \sum \limits^{J}_{j=1}\left(1+r_{j}(a^{j}_{i,t},X_{t})\right)a^{j}_{i,t} - \displaystyle \sum \limits^{J}_{j=1} a^{j}_{i,t+1} - \sum \limits^{J}_{j=1}\psi(a^{j}_{i,t+1},a^{j}_{i,t}) \\
	& c_{i,t}>0&\text{, } g_{i}\left(\left\{a^{j}_{i,t},a^{j}_{i,t+1}\right\}_{j=1}^{J}\right)\geq 0   \label{eq:model}
\end{eqnarray*}
Denoting the marginal utility of consumption by $U'(c_{i,t};l_{i,t},\omega,X_{t})$ and marginal transaction costs for $(a^{j}_{i,t},a^{j}_{i,t+1})$ by  $\psi^{j}_{1}$ and $\psi^{j}_{2}$ respectively, the Euler equation for any asset $j$ is distorted by the non-negative Lagrange multipliers on the
occasionally binding constraint $(\mu_{i,t})$ which are positive when the constraint is binding:
\small
\begin{eqnarray}
&& U'_{1}(c_{i,t},l_{i,t};\omega,X_{t})(1+\psi^{j}_{1,i,t+1})\nonumber\\
 & = & \beta \mathbb{E}_{t}\left(1+r_{j}(a^{j}_{i,t+1},X_{t+1})+\frac{\partial r_{j}(a^{j}_{i,t+1},X_{t+1})}{\partial a^{j}_{i,t+1}}a^{j}_{i,t+1}-\psi^{j}_{2,i,t+1}\right)U'_{1}(c_{i,t+1},l_{i,t+1};\omega,X_{t+1}) \nonumber\\  && +\mu_{i,t}\frac{\partial g_{i}\left(\left\{a^{j}_{i,t},a^{j}_{i,t+1}\right\}_{j=1}^{J}\right)}{\partial a^{j}_{i,t+1}}+\beta\mathbb{E}_{t}\mu_{i,t+1}\frac{\partial g_{i}\left(\left\{a^{j}_{i,t+1},a^{j}_{i,t+2}\right\}_{j=1}^{J}\right)}{\partial a^{j}_{i,t+1}}  \nonumber
\end{eqnarray}
which can be simplified to
\begin{eqnarray}
\label{eq:genzeldes}
&& U'_{1}(c_{i,t},l_{i,t};\omega,X_{t})(1+\psi^{j}_{1,i,t+1}) \\
& = &
\beta \mathbb{E}_{t}\left(1+r_{j}(a^{j}_{i,t+1}, X_{t+1})+\frac{\partial r_{j}(a^{j}_{i,t+1},X_{t+1})}{\partial a^{j}_{i,t+1}}a^{j}_{i,t+1}-\psi^{j}_{2,i,t+1}\right)U'_{1}(c_{i,t+1},l_{i,t+1};\omega,X_{t})\nonumber \\
&&+ \lambda^{j}_{i,t} \nonumber
\end{eqnarray}
\normalsize

\normalsize where $\lambda_{i,t}$ summarizes the distortions, and is positive as $\mu_{i,t}$ is positive and $g_{i}$ is increasing in wealth.\footnote{The optimality condition for hours is $\frac{\partial y_{i,t}(l_{i,t})}{\partial l_{i,t}}U'_{1}(c_{i,t},l_{i,t};\omega,X_{t})=U'_{2}(c_{i,t},l_{i,t};\omega,X_{t})$.}

The above conditions characterize optimality for every choice of asset $j$ in the portfolio of
an individual $i$ and are general enough to accommodate a large class of models.\footnote{Similar restrictions also arise in the context of limited commitment and endogenous solvency constraints as in \citet{10.1093/rfs/14.4.1117}.} For example, abstracting from transaction costs, in a one asset economy with a risk free government bond $(a^{1}_{i,t}:=b_{i,t})$ equation  \eqref{eq:genzeldes} becomes:
\[U'_{1}(c_{i,t};l_{i,t},\omega,X_{t})=
\beta \mathbb{E}_{t}(1+r)U'_{1}(c_{i,t+1};l_{i,t+1},\omega,X_{t+1})\nonumber + \lambda^{1}_{i,t} \] while including productive capital leads to an additional Euler  equation for $a^{2}_{i,t}:=k_{i,t}$ as follows:
\[U'_{1}(c_{i,t};l_{i,t},\omega,X_{t})=
\beta \mathbb{E}_{t}\left(1+\frac{\partial y_{i,t+1}}{\partial k_{i,t+1}}-\delta\right)U'_{1}(c_{i,t+1};l_{i,t+1},\omega,X_{t})\nonumber + \lambda^{2}_{i,t} \]

\subsection{Identifying Preferences}
 Given this setup, I next focus on what can be learned from the data in terms of the preference parameters $(\beta,\omega)$.

There are alternative ways of achieving identification. For example, if cross section or panel data is available, one can split the sample into unconstrained and constrained households using some criterion. In his seminal paper, \citet{Zeldes1989} splits the sample using a cut-off rule on the end  of period nonhuman wealth and discusses different rules that could have been used, as well as liquidity constraints arising from credit market imperfections and illiquid assets as e.g. in \citet{doi:10.2307/1884163}. The importance of illiquid assets has been recently revived by the heterogeneous agent literature, see for example \citet{RePEc:bin:bpeajo:v:45:y:2014:i:2014-01:p:77-153} and references therein, where they also identify groups of wealthy and poor hand to mouth consumers using different rules.

However, this approach cannot be readily applied to time series data, as is the case for aggregate data. More importantly, alternative choices can lead to sample selection and thus whimsical conclusions on whether liquidity constraints are present.

An alternative way is a structural treatment which amounts to employing an estimated income process and computing optimal consumption. One would therefore place much more faith on the specific technology of the borrowing constraint and the nature of idiosyncratic income risk.

A more credible way to identify $(\beta,\omega)$ is to exploit that $\lambda^{j}_{i,t}$ is positive, that is, there is a positive discrepancy between current and next period marginal utility of consumption, as households cannot smooth consumption as much as they desire. As we already alluded to, this is a  general implication of the trading constraints which is valid across models with different types of heterogeneity and insurance mechanisms.
Conditions in \eqref{eq:genzeldes} therefore generate conditional moment inequalities, one of which, for example, corresponds to the Euler equation for a riskless government bond:\vspace{-0.1 in} \[U'(c_{i,t};l_{i,t},\omega,X_{t})  \geq  \beta (1+r)\mathbb{E}_{t}U'(c_{i,t+1};l_{i,t+1},\omega, X_{t+1})
\]
Using a positive random variable as an instrument e.g. $y_{i,t}$,
the following unconditional moment inequality holds, which  does not pin down a unique vector of values for $(\beta,\omega)$. However, it implicitly defines bounds on admissible values for these parameters :
\begin{eqnarray}
	\mathbb{E}\left[(U'(c_{i,t};l_{i,t},\omega,X_{t})-\beta(1+r)U'(c_{i,t+1};l_{i,t+1},\omega, X_{t+1})) y_{i,t}\right] & \geq & 0 \label{eq:unczeldes}
\end{eqnarray}

For illustration purposes, in Appendix B I derive explicit identification regions for $\omega$ using time series observations for household $i$'s consumption choices, and I show how we could obtain sharper identification using additional individual data on the prevalence of liquidity constraints.

Since the focus of the paper is on the use of aggregate time series data, I next derive the  aggregate moment inequality restrictions that correspond to the general case in \eqref{eq:genzeldes} . To do so I employ a second order approximation.
Researchers have recently employed the second order approximation to the heterogeneous agent model around the representative agent (RA) allocation as a tractable way of analyzing its aggregate implications. For example \cite{GaliDeb} show that a simple two-agent New Keynesian (TANK) model which abstracts completely from heterogeneity within unconstrained agents, captures reasonably well the implications of a baseline HANK model regarding the effects of aggregate shocks on aggregate variables.\footnote{\citet{Tryphonides2019} evaluates the quality of this approximation and utilizes it to identify how heterogeneity and the extensive and intensive margins of adjustment depend on aggregate shocks. } This paper will also follow this route.

Without loss of generality with respect to the friction generating mechanism, we make parametric assumptions on the utility function.\footnote{Although interesting, identification without parametric assumptions on the utility function is beyond the scope of the paper as the aim is to be robust with respect to the mechanism generating $\lambda_{i,t}$ and not the primitives. In a frictionless setting, nonparametric global point identification has been proved by \citet{escanciano_hoderlein_lewbel_linton_srisuma_2020} within the class of positive marginal utility functions. For a more general class of utility functions they show that both the discount factor and the marginal utility functions are set identified. Set identification in our context has to do with the fact that the parametric Euler equation is not satisfied with equality.}
 Assuming CRRA preferences and separability between consumption and leisure as well as external habit formation e.g. "catching up with the Joneses" effects, implies the following utility function:
\[u(c_{i,t},l_{i,t},\omega,\eta,C_{t-1}):= \frac{\left(c_{i,t}-hC_{t-1}\right)^{1-\omega }-1}{1-\omega }- \frac{l_{i,t}^{1+\eta}}{1+\eta}   \] where $\omega$ is the coefficient of relative risk aversion, $\eta$ the (inverse) Frisch elasticity of labor supply and $h$ the habit parameter. Note also that the assumption of separability can be relaxed without any bearing on the main point of the paper. With non-separability, aggregation will involve identifying an additional moment from micro data, that of the cross sectional covariation of consumption and hours worked, while it wil introduce an additional parameter to be estimated, that of the degree of separability. Hence, separability does not buy as much in terms of the order condition for identification.

The first order conditions from \eqref{eq:genzeldes} become as follows:\small
\begin{eqnarray*}
	\left(c_{i,t}-hC_{t-1}\right)^{-\omega}&=&\beta\mathbb{E}_{t}\left(c_{i,t+1}-hC_{t}\right)^{-\omega}\left(1+r_{j}(a^{j}_{i,t+1},X_{t})+\frac{\partial r_{j}(a^{j}_{i,t+1},X_{t})}{\partial a^{j}_{i,t+1}}a^{j}_{i,t+1}\right) + \lambda_{i,t}
\end{eqnarray*} \normalsize
and the intratemporal labor supply condition is
\begin{eqnarray*}
	l^{\eta}_{i,t} &=& \frac{W_{t}}{P_{t}}	\left(c_{i,t}-hC_{t-1}\right)^{-\omega}
\end{eqnarray*}\normalsize where I have used that $\frac{\partial y_{i,t}(l_{i,t})}{\partial l_{i,t}}=\frac{W_{t}}{P_{t}}$, that is, the derivative of total individual income with respect to hours worked is the real wage.
The parameters of interest are therefore $(\omega,\beta,\eta,h)$.

 A second order approximation to marginal utility around aggregate consumption\footnote{Aggregate consumption is individual consumption aggregated using the agents' distribution over the relevant states. This will be made precise shortly.} yields:
 \small
 \begin{eqnarray*}(c_{i,t}-hC_{t-1})^{-\omega} &\approx&  (C_{t}-hC_{t-1})^{-\omega} -\omega (C_{t}-hC_{t-1})^{-\omega-1}(c_{i,t}-C_{t})\\
 	 &&+ \frac{\omega(\omega+1)}{2} (C_{t}-hC_{t-1})^{-\omega-2} (c_{i,t}-C_{t})^{2}\\
  &=&  (C_{t}-hC_{t-1})^{-\omega}[1 -\omega (C_{t}-hC_{t-1})^{-1}(c_{i,t}-C_{t})\\
  &&+ \frac{\omega(\omega+1)}{2} (C_{t}-hC_{t-1})^{-2} (c_{i,t}-C_{t})^{2}]\\
   &:=&  (C_{t}-hC_{t-1})^{-\omega}\Xi_{t,i}(h)
\end{eqnarray*}\normalsize
 and therefore \small
   \begin{eqnarray}
  	\left(C_{t}-hC_{t-1}\right)^{-\omega}&=&\beta\mathbb{E}_{t}\left(C_{t+1}-hC_{t}\right)^{-\omega}\frac{\Xi_{i,t+1}(h)}{\Xi_{i,t}(h)}\left(1+r_{j}(a^{j}_{i,t+1},X_{t})+\frac{\partial r_{j}(a^{j}_{i,t+1},X_{t})}{\partial a^{j}_{i,t+1}}a^{j}_{i,t+1}\right)\nonumber \\ && +\lambda_{i,t}  \label{eq:cov}
  \end{eqnarray}
  \normalsize

  An advantage of the second order approximation is that it  makes clear which moments are likely to be important.
Aggregating the last expression involves estimating the cross sectional covariance of asset returns and their marginal dependence on the asset level with the growth rate in the aggregation residual  $\frac{\Xi_{t+1,i}(h)}{\Xi_{t,i}(h)}$. A natural question to ask next is which assets should we then employ. We can in principle use any asset class, conditional on being able to measure the relevant components of \eqref{eq:cov}. This is nevertheless a non-trivial task as (i) it requires assumptions on the structure of the return of the specific asset and (ii) it increases the dimensionality of the cross sectional moments that will have to be identified from the data\footnote{This line of research is pursued in current work.}. With an eye towards the application, and to facilitate aggregation, I focus on assets whose individual returns depend only on aggregate risk e.g. government bonds or savings accounts whose benchmark returns depend on the policy rate. I thus set $r(a^{j}_{i,t+1},X_{t})=r(X_{t})$.\footnote{This is also consistent with a wider asset class whose returns are characterized by $r(a^{j}_{i,t+1},X_{t})=r(X_{t}) + \frac{\partial r_{j}(a^{j}_{i,t+1})}{\partial a^{j}_{i,t+1}}a^{j}_{i,t+1}$ where the second idiosyncratic component is unobserved. This is true for e.g. a credit line which involves higher interest rates at higher levels of unsecured borrowing and can be thought of as a softer borrowing constraint which has similar properties to $\lambda_{i,t}$. }

In order to be explicit about aggregation, let $s_{i,t}\equiv(\nu_{i,t},y_{i,t},\left\{a^{j}_{i,t},a^{j}_{i,t+1}\right\}_{j=1}^{J})$ be the vector that includes all idiosyncratic states and $S_{t}$ the corresponding vector that includes all aggregate states, including $X_{t}$ and the distribution of $s_{i,t}$.

Aggregating the Euler equation using $p(s_{i,t}|S_{t})$, and using that agent expectations are formed using $p(s_{i,t+1},S_{t+1}|s_{i,t},S_{t})$,  the approximate aggregated first order condition is equal to:\small
\begin{eqnarray}
		 (C_{t}-hC_{t-1})^{-\omega}&=&\beta\mathbb{E}_{t}(C_{t+1}-hC_{t})^{-\omega}\frac{\Xi_{t+1}(h)}{\Xi_{t}(h)}(1+r(X_{t+1})) + \int \lambda_{i,t} p(s_{i,t}|S_{t})d s_{i,t}\nonumber\\  &:=&\beta\mathbb{E}_{t}(C_{t}-hC_{t-1})^{-\omega}\frac{\Xi_{t+1}(h)}{\Xi_{t}(h)}R_{t+1} + \mu_{t}  \label{eq:distR}
\end{eqnarray}\normalsize where \vspace{-0.1 in}  \small \[C_{t+1}\equiv \int c_{i,t+1 }p(s_{i,t+1},s_{i,t}|S_{t+1},S_{t})d(s_{i,t+1},s_{i,t}),\quad C_{t}\equiv \int c_{i,t }p(s_{i,t}|S_{t})ds_{i,t}\] \[\Xi_{t}(h) \equiv 1+\frac{\omega(\omega+1)}{2}\left(1-h\left(\frac{C_{t-1}}{C_{t}}\right)\right)^{-2}Var\left(\frac{c_{i,t}}{C_{t}}\right)\] \[\quad \Xi_{t+1}(h) \equiv 1+ \frac{\omega(\omega+1)}{2}\left(1-h\left(\frac{C_{t}}{C_{t+1}}\right)\right)^{-2}Var\left(\frac{c_{i,t+1}}{C_{t+1}}\right)\]\vspace{-0.1 in} \normalsize
Furthermore, the intratemporal condition is:
\[L_{t}=(W_{t}^{\frac{1}{\eta}}\left(C_{t}-hC_{t-1}\right)^{-\frac{\omega}{\eta}}\Xi^{lab}_{t}(h)\] \vspace{-0.1 in}
where    \[\Xi^{lab}_{t}(h)=1+\frac{\omega(\eta+\omega)}{2\eta^{2}}\left(1-h\left(\frac{C_{t-1}}{C_{t}}\right)\right)^{-2}Var\left(\frac{c_{i,t}}{C_{t}}\right)\]
Correspondingly, consumption variance is defined as follows: \begin{eqnarray}
	Var(c_{i,t}) &=&  \int (c_{i,t}-C_{t})^2p(s_{i,t}|S_{t})d s_{i,t}\\ \label{var}
	Var(c_{i,t+1}) &=&  \int (c_{i,t+1}-C_{t+1})^2p(s_{i,t+1},s_{i,t}|S_{t+1},S_{t})d  \nonumber (s_{i,t+1},s_{i,t})\end{eqnarray}
From the above expressions, it is clear that there are two main channels through which heterogeneity matters for aggregate fluctuations to first order. The first is fluctuations in the variance of the consumption share, which distorts both the intertemporal allocation of aggregate consumption as well as the intratemporal consumption - labor tradeoff even if all agents are "on their Euler equations". The variance of the consumption share is essentially a sufficient statistic for the underlying moments of the wealth distribution as consumption is a nonlinear function of wealth.   Notice also that unless $\eta=1$, that is the Frisch elasticity is equal to one, the way heterogeneity matters for hours is different than consumption (the aggregation residuals are different), which implies that the dynamics of hours will also be affected by heterogeneity.
The second channel is the additional aggregate distortion $\mu_{t}$ which is positive whenever there is a subset of agents who are constrained.

 Since $\lambda_{i,t}$ is weakly positive by construction, the aggregate distortion to the bond Euler equation 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 $Y_{t-1}$- gives rise to a moment inequality, which is also consistent with a general equilibrium response in $R_{t}$\footnote{In a closed economy, negative distortions to consumption will put downward pressure on the interest rate which then relaxes the borrowing constraint for some of the agents. Even if all the agents become unconstrained then the weak inequality is still valid.}.
 Therefore, the aggregated moment inequalities that are relevant for identification are the following, where the second moment inequality arises from incorporating the aggregate intratemporal condition in the Euler equation:
 	\begin{eqnarray}
 		\mathbb{E}\left(1-\beta \left(\frac{C_{t+1}-hC_{t}}{C_{t}-hC_{t-1}}\right)^{-\omega}\left(\frac{\Xi_{t+1}}{\Xi_{t}}\right)R_{t+1}\right) Y_{t-1}&\geq& 0 \\
 		\mathbb{E}\left(1-\beta\left(\frac{\Xi_{t+1}}{\Xi_{t}}\right)\left(\frac{\Xi^{lab}_{t}}{\Xi^{lab}_{t+1}}\right)^{\eta}\left(\frac{L_{t+1}}{L_{t}}\right)^{\eta}\frac{W_{t}}{W_{t+1}}R_{t+1}\right) Y_{t-1}&\geq& 0
 	\end{eqnarray}
 	\normalsize

Notice that the micro moment that is necessary for identification is  the cross sectional variance of consumption.  The variance measures derived in \eqref{var}  do not condition on any individual information 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 feature without taking a stance on the exact dependence structure. This is, perhaps surprisingly, an advantage of using aggregate measures and is in contrast to models of panel data that have specific implications for the law of motion of the cross sectional variance (see for example the critique in \citet{CHAMBERLAIN19841247}).

 \section{Identification and the Extensive Margin}
The inequalities derived in the previous section deliver set identification, that is, the set of parameter values consistent with the data is not a singleton. Could we use additional data to sharpen estimates? This section shows that information on the extensive margin of adjustment for the constrained consumers provides additional useful information, while more formal identification results on which the analysis rests are in Appendix A. More particularly, the approximated HA model results in distortions that are partially observable. We generalize notation as follows: Denote by $Y^{f}_{t}$ the representative agent (frictionless) model prediction, \vspace{-0.1 in}
\[ Y_{t} = Y^{f}_{t}+\lambda^{o}_{t} +  \lambda^{un}_{t} \]
  where $ \lambda^{o}_{t}$ is the observable component (up to the parameter vector $\theta\in \Theta$) of the aggregate distortion and $\lambda^{un}_{t}$ the unobservable component. For example, in \eqref{eq:distR}, $Y_{t}=C^{-\omega}_{t}$, $Y^{f}_{t}\equiv  \beta \mathbb{E}_{t}C^{-\omega}_{t+1}R_{t+1}$, $\lambda^{un}_{t}\equiv \mu_{t}$ and $\lambda^{o}_{t}\equiv  \beta \mathbb{E}_{t}C^{-\omega}_{t+1}\left(\frac{\Xi_{t,t+1}}{\Xi_{t}}-1\right)R_{t+1}$ and $\theta=(\beta,\omega,h)$. Utilizing that $\mu_{t}$ is positive, and assuming additive separability in $\lambda^{0}_{t}$, the estimating equation becomes as follows, where  $X_{t-1}$ is any positive random variable  which is $Y_{t-1}$ measurable:	\begin{eqnarray}
  \mathbb{E}\left(g\left(Y_{t},Y_{t+1},\theta\right)-\lambda^{o}_{t}(Y_{t},Y_{t+1},\theta)\right)X_{t-1} - \mu& = & 0   \label{eq:no_surv_mom}
  \end{eqnarray} where $g(Y_{t},Y_{t+1},\theta):=Y_{t}-Y^{f}_{t}$ and $\mu:=\mathbb{E}(\mu_{t}X_{t-1})\in \mathbb{R}_{+}\cup \{+\infty\}$.
  This restriction is unlikely to point identify $\theta$ as $\mu_{t}$ is latent, and the only restriction we can impose on $\mu$ is its positive sign.
  Any improvements in identification should therefore come from additional information on $\mu$.   One way of achieving this is to notice that the evolution of $\mu_{t}$ over time can be explained by either changes in the proportion of agents whose behavior is  distorted, the extensive margin, and by how much they distort their behavior, on average. More particularly,  $\mu_{t}$ is by definition multiplicatively decomposed  as follows:
  \begin{eqnarray*}
  	\mu_{t}& =& \mathbb{E}_{t}(\lambda_{i,t}|\lambda_{i,t}\neq 0)\mathbb{P}_{t}(\lambda_{i,t}\neq 0)\equiv \kappa_{t}B_{t}
  \end{eqnarray*}
  where $\kappa_{t}$ 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 way to measure either margin, e.g. $B_{t}$\footnote{ As  I will illustrate in the empirical application, combining information from different surveys is one fruitful way.}, then this could be utilized to sharpen inference as it generates a new set of moment inequalities. Using that $B_{t}$ and $\kappa_{t}$ have the same sign\footnote{Appendix A  includes a more formal proof.}:\vspace{-0.1 in}
 	\begin{eqnarray}
	\mathbb{E}\left(g\left(Y_{t},Y_{t+1},\theta\right)-\lambda^{o}_{t}(Y_{t},Y_{t+1},\theta)\right)X_{t-1}B^{-1}_{t}- \kappa& = & 0   \label{eq:surv_mom}
	\end{eqnarray}
	where $\kappa:=\mathbb{E}(\kappa_{t}X_{t-1})\in \mathbb{R}_{+}\cup \{+\infty\}$. When $B_{t}$ is zero for all $t$, then \eqref{eq:surv_mom} is trivially satisfied for all $\theta$ and \eqref{eq:no_surv_mom} becomes a moment equality $(\mu=0)$.

For exposition, I briefly  lay out the identification framework based on moment inequalities, while the more technical material is in Appendix A.
First, conditions like \eqref{eq:no_surv_mom} and \eqref{eq:surv_mom} can be generally expressed as follows, where $\mathbf{Y}^{\tau}$ denotes a matrix with observations of vector ${Y}$ up to time $\tau$, $\{Y_{j}\}_{j\leq \tau}$,   $\phi (.)$ is an \textit{inequality preserving} function of $\mathbf{Y}^{\tau-1}$, $r$ is the number of conditions and $U$ is a vector of nuisance parameters that restore the moment equality:
\begin{equation}
\mathbb{E}\left[\left(\underset{r\times 1}{g}\left(Y_{t},Y_{t+1},\theta\right)-{\underset{r\times 1}{\lambda_{t}^{o}}}(Y_{t},Y_{t+1},\theta)\right)\otimes\phi
(\mathbf{Y}^{\tau-1})\right]=\mathbb{E}\left[\mu_{t}\otimes\phi
(\mathbf{Y}^{\tau-1})\right]\equiv U \in \left[ 0,\infty \right]^{r}  \label{mom_comp_gmm}
\end{equation

The identified set is thus defined as:
\begin{eqnarray*}
\Theta_{I}&:=&\left\{\theta\in \Theta: \exists U \in \mathbb{R}_{+}^{r}\cup \{\infty\} ; \mathbb{E}\left(m(Y_{t},Y_{t+1},\theta)\otimes\phi(\mathbf{Y_{t-1}})\right)-U=0\right\}
\end{eqnarray*} where
 $m(Y_{t},Y_{t+1},\theta):=\left(g\left(Y_{t},Y_{t+1},\theta\right)-\lambda^{o}_{t}(Y_{t},Y_{t+1},\theta)\right)$ or when the extensive margin is also utilized, $m(Y_{t},Y_{t+1},\theta)$ contains $\left(g\left(Y_{t},Y_{t+1},\theta\right)-\lambda^{o}_{t}(Y_{t},Y_{t+1},\theta)\right)B^{-1}_{t}$.

\normalsize

Identification based on the moment conditions can be analyzed by first looking at sufficient information that can identify $\theta$ for a fixed value of the wedge $U$, and then how additional information essentially imposes restrictions on the the properties of the wedge. Correspondingly, Lemmas \ref{nr} and \ref{over} in Appendix A characterize identification in the case in which the number of moment conditions used is the same as the dimension of $\theta$ and the case of using a larger number of conditions. Moment conditions with survey data can always be decomposed into the component that corresponds to the original set of conditions and the component that provides additional information. They can therefore be casted into this framework.

The next proposition  utilizes Lemmas \ref{nr} and \ref{over} and characterizes how informative $B_{t}$  can be. In a nutshell, if survey data constrain the stochastic properties of $\mu_{t}$ as implied by \eqref{eq:no_surv_mom}, then the size of $\Theta_{I}$ is likely to be refined, in population. A diagrammatic exposition can be also found in Figure \ref{id_fig} in  Appendix A.  The two main assumptions made by Lemmas \ref{nr} and \ref{over} are that the moment conditions used are correctly specified as equalities when nobody is constrained, and that conditional on a specific value for the distortion $U$, $\theta$ is point identified\footnote{For the formal definitions, please see  Appendix A.}.

\begin{prop}{Identification}. Define $\Theta_{I}$ as the identified set without using the extensive margin and $\Theta_{I}'$ as the identified set when \eqref{eq:surv_mom} is utilized. Given Time Series observations on $\{Y_{t},B_{t}\}_{t=1..T}$ and the moment conditions in \eqref{eq:no_surv_mom}- \eqref{eq:surv_mom},
	\begin{enumerate}
		\item If $B_{t}\in(0,1)$, then $\Theta_{I}'\subset\Theta_{I}$.
		\item If $Var(B_{t})=0$, then $\Theta_{I}'=\Theta_{I}$.
		\item When $B_{t}\neq 0$, $\Theta_{I}$ is not a singleton (Impossibility of point identification).
	\end{enumerate} \label{over_surv}
\end{prop}
\begin{proof}
	See Appendix A
\end{proof} \vspace{-0.1 in}
$B_{t}\in(0,1)$ is essential for shrinking the admissible set of predictions. Moreover, absence of time variation in the proportion of agents that are constrained i.e. if the economy reaches a stationary state, implies that $B_{t}$ has no additional information for the parameters of the model. What this also implies is that additional identification  really comes from the presence of aggregate risk.

Intuitively, $B_{t}$ adds information as it describes the mass of financially constrained agents over time, and is thus related to variations in consumption heterogeneity and precautionary effects. Absent of information on  $B_{t}$, moment inequalities only capture average behavior and are thus consistent with $99\%$ or $1\%$ of agents being constrained.

Finally, viewed from the perspective of an individual, $B_{t}$ describes the probability that an agent chose at random is constrained or not. The analytical example in Appendix B uses time series observations at the individual level and shows explicitly how the extensive margin provides additional identifying information for the risk aversion parameter.

\subsection{Illustration with Simulated Data}
This sub-section provides supportive simulation evidence on using moment inequalities to identify risk aversion and the discount factor in a controlled environment, where the data generating process is the \citet{krusellsmith} (KS hereafter) economy, the workhorse heterogeneous agent model with aggregate risk.

Adopting the formulation in \citet{MALIAR201042, denhaanjudd}, the KS economy  features a standard capital accumulation problem under incomplete markets, where the household first order conditions are as follows:
\begin{eqnarray*}
	c^{-\omega}_{i,t}&=& \beta\mathbb{E}_{t}(1-\delta + r^{k}_{t+1})c^{-\omega}_{i,t+1}+\lambda_{i,t}\\
	k_{i,t+1} &=& (1-\tau)w_{t}\bar{l}\epsilon_{i} + \nu_{t} w_{t} (1-\epsilon_{i})+(1-\delta+r^{k}_{t})k_{i,t}-c_{i,t}\\
	k_{i,t+1}&\geq& 0
\end{eqnarray*}
where $\epsilon_{i}\in\{0,1\}$ is the idiosyncratic employment shock, $\bar{l}$ the time endowment and $v_{t}$ the fraction of the wage that is received as unemployment benefit.

Perfectly competitive markets imply that the interest and wage rates are equal to the marginal products of capital and labor: \[r^{k}_{t}=Z_{t}\alpha\left(\frac{K_{t}}{\bar{l}L_{t}}\right)^{\alpha-1},  \quad w_{t}=Z_{t}(1-\alpha)\left(\frac{K_{t}}{\bar{l}L_{t}}\right)^{\alpha}\]
while the labor income tax rate is designed to be equal to $\tau = \frac{\nu_{t} u_{t}}{\bar{l}L_{t}}$.
The only source of aggregate risk is the aggregate productivity shock $Z_{t}$ which takes two values, $\left\{Z_{low},Z_{high}\right\}$. In turn, the unemployment rate $u_{t}$ depends on the productivity shock as well, and takes two possible values, $\left\{u_{low},u_{high}\right\}$, with transition matrix $P=(p_{ll},p_{lh};p_{hl},p_{hh})$. Finally, full employment is normalized to one, thus $L = 1-u$.

The aggregated Euler equation is similar to \eqref{eq:distR} (with $h=0$) but now the interest rate is pinned down by a productive (capital) asset. The following moment inequalities characterize the restrictions without and with the extensive margin respectively:
\begin{eqnarray}
	\mathbb{E}\left(C^{-\omega}_{t}-\beta C^{-\omega}_{t+1}\left(\frac{\Xi_{t+1}}{\Xi_{t}}\right)(1-\delta+ r^{k}_{t+1})\right) X_{t-1}&\geq& 0\\
	\mathbb{E}\left(C^{-\omega}_{t}-\beta C^{-\omega}_{t+1}\left(\frac{\Xi_{t+1}}{\Xi_{t}}\right)(1-\delta+ r^{k}_{t+1})\right) B^{-1}_{t}X_{t-1}&\geq& 0   \label{simul_exp1}
\end{eqnarray}

where  $X_{t}$ is constructed using standardized values of aggregate consumption lagged by one and two periods.   I adopt the same parameterization and solution method as in \citet{MALIAR201042, denhaanjudd} to generate the data by simulating 1400 periods and keeping the last 200 as the pseudo-sample. The parameter values are set as follows: $\beta=0.97,\gamma=1.5,\alpha=0.36,\delta=0.025,Z_{high}=1.01,Z_{low}=0.99,\nu = 0.15,\bar{l}=\frac{1}{0.9}, u_{h}=0.04,u_{l} = 0.1$. Fixing $\delta$, I employ the above moment inequalities to estimate $(\beta,\omega)$  using standardized values of aggregate consumption lagged by one and two periods as the instrument vector $X_{t}$.

Reparameterizing the inequalities to obtain equalities, (as in \eqref{mom_comp_gmm}) the parameter estimates for $(\theta,U)$ are defined by minimizing the following Continuously Updated - GMM criterion:\vspace{-0.1 in}
\begin{eqnarray}
	Q_{T}(\theta,U)&=& \frac{1}{2}(\bar{m}(Y_{t},Y_{t+1},\theta)-U)^{T}V^{-1}(\bar{m}(Y_{t},Y_{t+1},\theta)-U) \label{eq:cugmm}
\end{eqnarray}
where $\bar{m}(Y_{t},Y_{t+1},\theta)$ is the sample counterpart of the moment conditions above with $\theta=(\beta,\omega)$ and $V^{-1}$ is the generalized inverse of the long run variance of the moment conditions estimated using the \citet{10.2307/1913610} estimator, truncated at first order.

Inference is based on quantiles of 300000 draws from the quasi-posterior distribution based on the above objective function using a 1-Block Metropolis-Hastings Markov Chain Monte Carlo sampler. Similar to standard quasi-Bayesian algorithms, the quasi-posterior is proportional to $\pi(\theta)e^{-TQ_{T}(\theta,U)}$ where $\pi(\theta,U)$ is the prior distribution, see e.g. \citet{doi:10.3982/ECTA14525}. Moreover,  I do not use any prior information, apart from $U>0$ (elementwise).

Figure \ref{MC} presents the $95\%$ confidence sets for $(\beta,\omega)$, with and without using the extensive margin when $T=200$, where the dashed lines indicate the true parameter values. As evident, utilizing the extensive margin shrinks the confidence sets.
Similar results are obtained with a higher sample size, and are presented in Figure \ref{ls} in the Appendix.

\begin{figure}[H]
	\begin{center}
		\includegraphics[scale=0.53]{\string"MonteCarlo_final_N200\string".png}
		\par  	\label{MC}
	\end{center}  \vspace{-0.2 in}
	\protect\caption{Comparison of Confidence sets for the identified set for $T=200$ with and without using the extensive margin.}

\end{figure}

A more general point worth making is that despite that the proportion of constrained agents is not, quantitatively speaking,  a variable that influences the level of aggregate consumption (in this simulation it is $0.01\%$ on average!) its covariation with the moment conditions is very informative for the preference parameters.

\section{Estimating Preferences using Spanish Data}
 Spain is a country in which financial frictions are a  priori expected to have played a significant role post 1999, and especially during the crisis period, 2008-2014. Unrestrained credit growth in the past and the subsequent bust has resulted in firm bankruptcies and a large drop in investment. Similarly, household indebtedness before the crisis was succeeded by de-leveraging.

Figure \ref{spain} plots the cyclical components for consumption, investment and output after 1999.
As evident, there was a large slump in aggregate investment, followed by a relatively smaller slump in aggregate consumption.

\vspace{-0.1 in}
\begin{figure}[H]
	\begin{centering}
		\includegraphics[scale=0.25]{\string"Cyclical_Components_Spain\string".png}
		\par
	\end{centering}
\caption{}
	\label{spain}
\end{figure}

An advantage of the framework developed in this paper is that the identification of deep parameters is not tied to taking a stance on how severe financial frictions have been, nor the time period in which credit constraints have been binding. This allows us to use the entire time series in the post - Euro period, as well as survey data that are partially harmonized across countries of the European Area, making the analysis done here portable to other countries as well\footnote{Another interesting case to apply this methodology is the case of the US using e.g. data from the Survey 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 with macroeconomic variables, but this survey started much later than EU surveys (2013).}.

\subsection{Measurements}
 As discussed earlier on, being able to measure $B_{t}$ improves identification and it is therefore useful to first discuss pertinent measurement issues and how they can be addressed. The exact triennial measure of $B_{t}$ is computed from the Survey of Household Finances (SHF) and is plotted in Figure \ref{mixedLCC}. The questionnaire asks households whether they have been denied a loan application during this period.  This question is consistent with a variety of reasons for which these households have been rejected, depending on individual characteristics, employment, guarantees, changes in the institution's credit policy and excessive debt. Moreover, accounting for consumers that have not applied for loans because they expect to be rejected (also coined as "discouraged borrowers" in \citet{10.2307/2077836} ) or consumers that were granted a fraction of the requested loan does not affect the estimate of this proportion.

 In order to obtain a quarterly measure, I utilize monthly data from the Business and Consumer Survey (BCS) of the European Commission\footnote{BCS data can be downloaded from \href{https://ec.europa.eu/info/business-economy-euro/indicators-statistics/economic-databases/business-and-consumer-surveys_en}{here}.}. In particular, the survey asks the households about their financial situation, and the possible answers range from "We are saving a lot" to "We are running into debt". In Figure \ref{mixedLCC}, I plot the proportion that corresponds to 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 of households 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 or in total.

 To infer the true proportion of constrained households for the periods in which it is not observed, so all periods between the triennial measures from SHF, I exploit the variation in the BCS data by employing a mixed frequency model, where the quarterly observations on consumers running into debt are linked to the triennial exact measure.

  More particularly, let  $B^{con}_{t}$ be the proportion of borrowing constrained agents and $\Pi_{t}$ be the proportion of consumers that are running into debt. The former are necessarily a share of the latter, as a fraction of the measure of indebted households are those  that would have liked to take on more debt within the quarter but could not do so for various reasons. In other words, $B^{con}_{t}$ is the share of $\Pi_{t}$ that hits their minimum level of permissible debt and are therefore not on their individual Euler equation.\footnote{Note also that those that for precautionary reasons are saving more in equilibrium in order to avoid the consumption loss associated with hitting the borrowing constraint satisfy their Euler equation.} This leads to the following measurement equation:
 \[B^{con}_{t}=\zeta_{t}\Pi_{t}\] where $\zeta_{t}\in(0,1)$,
 As shown in the appendix (Section \ref{meas}), this measurement equation is part of a state space system where $\tilde{B}^{con}_{t}$ and $\tilde{\zeta}_{t}$ are the latent states and $\hat{\Pi}^{o}_{t}$ and  $\hat{B}^{con,o}_{t}$ are corresponding observables at quarterly and triennial frequency respectively. It can be shown that when the exact measure was observed in period $t-1$ $(\tilde{B}^{o}_{t-1})$ but missing in $t$, the (Kalman) filtered measure $(\hat{\tilde{B}}_{t})$ is equal to
 \[\hat{\tilde{B}}^{con}_{t} = \rho \tilde{B}^{con}_{t-1} +  \frac{\sigma^{2}_{\zeta}+\sigma^{2}_{\Pi}}{2\sigma^{2}_{\zeta}+\sigma^{2}_{\Pi}}\left(v_{\tilde{B}^{con}_{t}}-v_{\tilde{\zeta}_{t}}\right)= \rho \tilde{B}^{con}_{t-1} +  \frac{\frac{\sigma^{2}_{\zeta}}{\sigma^{2}_{\Pi}}+1}{\frac{2\sigma^{2}_{\zeta}}{\sigma^{2}_{\Pi}}+1}v_{\Pi_{t}}\] \normalsize
 where $(\sigma^{2}_{\zeta},\sigma^{2}_{\Pi})$ are the variances of $\zeta_{t}$ and $\Pi_{t}$ correspondingly, and $\rho$ is the autoregressive parameter for $B_{t}$ in the state equation.

 If the proportion $\zeta_{t}$ is not varying much around its long run mean, that is $\sigma^{2}_{\zeta}$ is low, then the mass of constrained agents covaries with those who are getting indebted. Fluctuations in the latter therefore are very informative for measuring $B_{t}$, and thus the innovation to $\hat{\Pi}^{o}_{t}$ ($v_{\Pi_{t}}$) receives a weight close to one.
 Conversely, when $\sigma^{2}_{\zeta}$ is high enough, the variation in the mass of agents that are getting indebted is not as informative for $\tilde{B}^{con}_{t}$, and thus the innovation in $\hat{\Pi}^{o}_{t}$ receives a low weight. In this case the filtered measure is then largely based on a forecast from last period's observed proportion of constrained consumers.

 The MLE estimate of $\frac{\sigma^{2}_{\zeta}}{\sigma^{2}_{\Pi}}$ is $0.3051$, yielding a gain parameter equal to $0.8125$. Variation in $\Pi_{t}$ is then informative, albeit not perfectly. The extracted measure is plotted in Figure \ref{mixedLCC}.

  \begin{figure}[]
	\begin{centering}
		 		\includegraphics[scale=0.42]{\string"Extracted_B\string".png}
		\par
	\end{centering}   	\protect\caption{}
 	\label{mixedLCC}
 \end{figure}

Since this measure is primarily informed from the SHF survey, one might still be concerned that the latter  understates the true measure of constrained households. Notice nevertheless that even if ${B}^{o}_{t}\leq B^{con}_{t}$, the moment inequality restrictions are still valid:
               	\begin{eqnarray*}
               m_{t}({B}^{o}_{t})^{-1}&\geq&m_{t}({B}^{con}_{t})^{-1}\\
               \mathbb{E}(m_{t}({B}^{o}_{t})^{-1}X_{t-1})&\geq& \mathbb{E}(m_{t}({B}^{con}_{t})^{-1}X_{t-1})\geq 0
               \end{eqnarray*}

where $m_{t}:=\left(1-\beta \mathbb{E}_{t}\left(\frac{C_{t+1}-hC_{t}}{C_{t}-hC_{t-1}}\right)^{-\omega}\left(\frac{\Xi_{t,t+1}}{\Xi_{t}}\right)R_{t+1}\right)$.
Our approach therefore is robust to concerns about underpredicting the constrained households.

Finally, another variable that needs to be measured from micro data is the dispersion in the consumption share, which is important to construct the aggregation residuals $\Xi_{t}$ and  $\Xi^{lab}_{t}$. The measure is constructed using the Survey of Household Finances (SHF)\footnote{SHF data can be downloaded from \href{https://www.bde.es/bde/en/areas/estadis/estadisticas-por/encuestas-hogar/relacionados/Encuesta_Financi/}{here}. Please consult the appendix for a detailed explanation of how the measure is constructed.}. In order to impute the value of dispersion when it is latent, I make the simplifying assumption  that it is constant. This is not likely to be a detrimental assumption for this kind of exercise as consumption is smoother than income, and its distribution is a slowly moving variable see e.g. \citet{bover}.\footnote{See i.e. Figure 13 in their paper.}

\subsubsection*{Identifying Restrictions}
For completeness, I repeat the moments derived in Section 2, with the only difference being the introduction of the component-wise positive valued nuisance vector $U:=(\mu^{T},\kappa^{T})^{T}$:\vspace{-0.36 in}
\begin{spacing}{1.5}\small
	\begin{eqnarray*}
	\mathbb{E}\left(1-\beta \left(\frac{C_{t+1}-hC_{t}}{C_{t}-hC_{t-1}}\right)^{-\omega}\left(\frac{\Xi_{t+1}}{\Xi_{t}}\right)R_{x,t+1}\right) \phi(Y_{t-1})-\mu_{1}&=& 0 \\
	\mathbb{E}\left(1-\beta\left(\frac{\Xi_{t+1}}{\Xi_{t}}\right)\left(\frac{\Xi^{lab}_{t}}{\Xi^{lab}_{t+1}}\right)^{\eta}\left(\frac{L_{t+1}}{L_{t}}\right)^{\eta}\frac{W_{t}}{W_{t+1}}R_{x,t+1}\right) \phi(Y_{t-1})-\mu_{2}&=& 0
	\end{eqnarray*}
	\normalsize
\end{spacing}
where $R_{x}\in\{R_{g},R_{ib}\}$, the ex-post real 3-month Treasury bond rate and the Interbank rate. The corresponding moments using the extensive margin are \vspace{-0.2 in} \begin{spacing}{1.5}\small
	\begin{eqnarray*}
		\mathbb{E}\left(1-\beta \left(\frac{C_{t+1}-hC_{t}}{C_{t}-hC_{t-1}}\right)^{-\omega}\left(\frac{\Xi_{t+1}}{\Xi_{t}}\right)R_{x,t+1}\right)B^{-1}_{t}\phi(Y_{t-1})-\kappa_{1}&=& 0 \\
		\mathbb{E}\left(1-\beta\left(\frac{\Xi_{t+1}}{\Xi_{t}}\right)\left(\frac{\Xi^{lab}_{t}}{\Xi^{lab}_{t+1}}\right)^{\eta}\left(\frac{L_{t+1}}{L_{t}}\right)^{\eta}\frac{W_{t}}{W_{t+1}}R_{x,t+1}\right)B^{-1}_{t} \phi(Y_{t-1})-\kappa_{2}&=& 0
	\end{eqnarray*}
	\normalsize
\end{spacing}
 To construct $\phi(Y_{t})$, I use lagged values  of growth rates using combinations of sets of instruments among $(\Delta C_{t},\Delta Y_{t},\Delta W_{t},\Delta L_{t},\Delta I_{t})$, $I_{t}$ being aggregate investment, and $(R_{g,t},R_{ib,t})$.
 \subsubsection{Details on Inference}
As in the simulation exercise, the parameter estimates for $(\theta,U)$ are defined by minimizing the continuously updated GMM criterion in \eqref{eq:cugmm}  where $\bar{m}(Y_{t},Y_{t+1},\theta)$ contains the sample counterpart of the above moment conditions with $\theta=(\omega, \eta,h,\beta)$. For the benchmark results I employ the inverse of the covariance matrix of the moment conditions as a weighting matrix.

The algorithm for inference used here is different than the corresponding MCMC based procedure used in the simulation exercise as the latter can be more conservative.\footnote{See \citet{doi:10.3982/ECTA14525} for more details on the performance of both procedures.} Partitioning the parameter space into the space of interest and its complement,  $(\Theta^{j},\Theta\backslash \Theta^{j},\mathcal{U})\equiv (\Theta^{j},\tilde{\Theta})$, inference for each parameter $\theta^{j}\in\Theta^{j}$ is based on Procedure 3 in \citet{doi:10.3982/ECTA14525} by comparing the profile likelihood ratio  to critical values from the chi-squared distribution.

More particularly, this procedure involves first finding a minimizer to the criterion function, $(\hat{\theta}^{j},\hat{\tilde{\theta}})=\underset{(\theta^{j},\tilde{\theta})\in (\Theta,\mathcal{U})}{arg \inf }Q_{T}(\theta,U)$, and then use it to construct the (profile) likelihood ratio where the nuisance parameters have already been profiled out, $\sup_{\tilde{\theta}\in \tilde{\Theta}}2T(-Q_{T}(\hat{\theta}^{j},\hat{\tilde{\theta}})+Q_{T}(\theta^{j},\tilde{\theta}))$. Then any parameter combination within the prior bounds which yields a profile likelihood ratio that is smaller than $\chi^{2}_{95\%}(1)$ is considered to belong to the (conservative) $95\%$ confidence set for the parameters of interest.
The prior upper and lower bounds are $\omega\in [0.1,6]$, $\frac{1}{\eta}\in [0.1,100]$,  $\beta\in [0.95,0.999]$ and  $h\in [0.01,0.97]$.

The results that follow provide additional empirical support to Proposition  \ref{over_surv}.
Table \ref{Lim_info_est_1} presents the $95\%$ confidence sets using $R_{ib}$ and $R_{g}$, as well as the results when  utilizing the intratemporal condition as an additional moment condition.\footnote{\small
	\begin{eqnarray*}
	\mathbb{E}	\left(\left(\frac{C_{t+1}-hC_{t}}{C_{t}-hC_{t-1}}\right)^{-\omega}- \left(\frac{\Xi^{lab}_{t}}{\Xi^{lab}_{t+1}}\right)^{\eta}\left(\frac{L_{t+1}}{L_{t}}\right)^{\eta}\frac{W_{t}}{W_{t+1}}\right) \phi(Y_{t-1})&=&0
	\end{eqnarray*}
	}.

	\normalsize

Table \ref{Lim_info_est_2}  examines resulting estimates using alternative modifications to instruments and moment weighting.\footnote{In some cases the confidence sets do not completely overlap, yet this is attributed to the different weighting that moments receive in finite sample, as in all cases the parameters are (conditional on $U$) overidentified.}

\begin{table}[H]
	\tiny
	\begin{centering}
		\caption{Confidence Set for Preference Parameters}\bigskip
		\label{Lim_info_est_1}
		\resizebox{\columnwidth}{!}{
			\begin{tabular}{c c c c c c c }
				\hline
				Parameter& \multicolumn{2}{c}{without $B_{t}$}\vline &
				\multicolumn{2}{c}{with $B_{t}$}\vline& & \\
				\hline &$q_{2.5\%}$ & $q_{97.5\%}$& $q_{2.5\%}$ & $q_{97.5\%}$&Instruments & Return
				\\ \hline \hline
				$\omega$  &  03384  & 6 &  0.3384  & 3.6162 &   & \\
				$\frac{1}{\eta}$&  0.3406 & 0.8929 & 0.3948 & 0.8929 & $\Delta Y_{t-1},\Delta W_{t-1}$,$\Delta L_{t-1}$  & {$R_{ib}$} \\
				$\beta$ & 0.9866  & 0.9926 &  0.9520  &   0.9926  &$\Delta I_{t-1},R_{ib,t-1},R_{g,t-1}$&\\
				$    h$ & 0.2671  & 0.6112 & 0.3979	  &   0.6112 &  &\\\hline
				$\omega$   &  0.1   & 3.3778  &  0.4576  & 3.6758 &  & \\
				$\frac{1}{\eta}$& 0.4484&4.7214 & 1.008 & 1.0493 & $\Delta Y_{t-1},\Delta W_{t-1},\Delta L_{t-1}$ &{$R_{g}$}  \\
				$\beta$   &  0.9856&  0.9921  &   0.9530  & 0.9921 &  $\Delta I_{t-1},R_{ib,t-1},R_{g,t-1}$& \\
				$    h$   & 0.1748  & 0.6112  & 0.4173	  &   0.6209 &  &\\\hline
				\multicolumn{7}{c}{Adding Intratemporal Condition}\\  \hline
				$\omega$&  0.3384   &2.6030& 0.3980  &1.8283 &  &  \\
				$\frac{1}{\eta}$   & 0.1 & 0.8191 & 0.4697 & 0.7565&$\Delta Y_{t-1},\Delta W_{t-1},\Delta L_{t-1}$  &$R_{g}$  \\
				$\beta$  & 0.9856& 0.9911 &  0.9792 &   0.9916  & $\Delta I_{t-1},R_{ib,t-1},R_{g,t-1}$   & \\
				$    h$  & 0.4270  & 0.970  & 0.3591	  &   0.970 & \\\hline\hline
		\end{tabular}}
		\par\end{centering}
\end{table}
\normalsize

Comparing the sum of squared lengths of the confidence sets for all parameters in the case in which the extensive margin is utilized to the case in which it is not, we find that there is  a reduction of $61.39\%$ when using the the interbank rate Euler equation, and a $43.81\%$ reduction in the bond Euler equation case.
In all cases, had we neglected the extensive margin of constrained consumers higher values of risk aversion would be admissible. This is also consistent with the results of Section 3 and the analytical results in Appendix B. As long as there is a non-trivial probability that some of the agents become constrained, the extensive margin provides additional information.

 Looking at the smallest interval compatible with the different estimates, risk aversion ranges from 0.34 to 1.83 which is close to estimates obtained in the microeconometric literature and much smaller than the typical estimates obtained using macroeconomic data in representative agent models. The confidence set for the Frisch elasticity is also refined, and results support values between 0.47 and 1, which are much closer to the values supported by microeconomic estimates. In fact, as \citet{10.1257/aer.101.3.471} suggest,  models that require an elasticity above 1 are inconsistent with micro evidence. \vspace{-0.1 in}
\begin{table}[H]
	\tiny
	\begin{centering}
		\caption{Alternative Choices}\bigskip
		\label{Lim_info_est_2}
		\resizebox{\columnwidth}{!}{
			\begin{tabular}{c c c c c c c }
				\hline
				Parameter& \multicolumn{2}{c}{without Survey}\vline &
				\multicolumn{2}{c}{with Survey}\vline& & \\
				\hline &$q_{2.5\%}$ & $q_{97.5\%}$& $q_{2.5\%}$ & $q_{97.5\%}$&Instruments & Return
				\\ \hline \hline
				\multicolumn{7}{c}{No lagged investment growth}\\ \hline
				$\omega$  &  0.3384  & 6 &  0.4576 & 5.4040 && \\
				$\frac{1}{\eta}$  & 0.2994 & 1.2235 & 0.4697 & 0.8929 & $\Delta Y_{t-1},\Delta W_{t-1}$  &$R_{g}$  \\
				$\beta$  &0.9564& 0.9926 &  0.9797  &   0.9921  &$\Delta L_{t-1},R_{ib,t-1},R_{g,t-1}$ & \\
				$    h$ & 0.1845  & 0.6306  & 0.4173	  &   0.6500 & &\\
				\hline
				\multicolumn{7}{c}{Newey-West}\\ \hline
				$\omega$&  0.5152   &4.1525& 0.3180  &3.8545 &  &  \\
				$\frac{1}{\eta}$   & 0.4482 & 1.3959 & 0.4697 & 0.8929 & $\Delta Y_{t-1},\Delta W_{t-1},\Delta L_{t-1}$  &$R_{g}$  \\
				$\beta$  & 0.9861& 0.9911 &  0.9817 &   0.9911  & $\Delta I_{t-1},R_{ib,t-1},R_{g,t-1}$ & \\
				$    h$  & 0.3785  & 0.6209  & 0.4270	  &   0.6015 & &\\\hline
		\end{tabular}}
		\par\end{centering}
\end{table}
Regarding further robustness checks, estimating long-run variance does not contribute much to obtaining tight sets. Moreover, including lagged investment growth seems to be an informative instrument as it produces tighter sets irrespective of whether survey data are used or not; Intuitively, lagged investment growth is an informative instrument as it is a strong predictor of consumption growth.\footnote{I have also experimented with including lagged values of the EONIA rate as an instrument, but due to its strong correlation with the rest of the interest rates, see Figure \ref{rates} in  Appendix A, it does not aid much to obtaining tight sets.}

\normalsize
\subsection{How does heterogeneity matter for identification?}

In order to gauge how  heterogeneity can affect the identification of preferences, I express parameter estimates in terms of potentially observable moments. For analytical convenience, looking at the log-linearized Euler equation around the aggregate steady state, risk aversion can be implicitly identified as follows, where all variables are expressed in log-deviation units and $\tilde{R}_{t}$ is used as an instrument:
\begin{eqnarray*} \omega^{IM}&=&\frac{Cov(\tilde{R}_{t+1},\tilde{R}_{t}) + Cov(\tilde{\Xi}_{t}(\omega),\tilde{R}_{t})+\frac{1-\beta \bar{R}}{\beta \bar{R}}\left(Cov(\tilde{B}_{t},\tilde{R}_{t})+Cov(\tilde{\kappa}_{t},\tilde{R}_{t})\right)}{Cov(\Delta\tilde{C}_{t+1},\tilde{R}_{t})}
\end{eqnarray*}
When markets are complete, $ \omega^{CM}=\frac{Cov(\tilde{R}_{t+1},\tilde{R}_{t}) }{Cov(\Delta\tilde{C}_{t+1},\tilde{R}_{t})}
$.
As long as heterogeneity and the extensive and intensive margins covary with the return, the identified risk aversion parameter will differ between the complete and incomplete market case, $\omega^{IM}$ and $\omega^{CM}$ respectively. Moreover, notice that in the complete markets case the steady state gross interest rate is be pinned down by $\frac{1}{\beta}$. When markets are incomplete, precautionary motives pull the equilibrium interest rate below the discount rate, making estimates of risk aversion more sensitive to variation in $B_{t}$ and $\kappa_{t}$. Since in this paper we measure $B_{t}$, the values of $\omega$ will vary with the admissible values of  $Cov(\tilde{\kappa}_{t},\tilde{R}_{t})$. On the other hand, precautionary savings may also alter how the intensive margin covaries with the interest rate.

The above observation can also give a partial justification on why representative agent models that are used in consumption based asset pricing may deliver implausibly high estimates of risk aversion. Positive distortions in the Euler equation for bonds imply that smooth consumption growth is consistent with the more volatile returns without requiring implausibly large estimates of risk aversion. This also implies that the  elasticity of intertemporal substitution is not too low, addressing the risk free rate puzzle as well, see e.g \cite{WEIL1989401}.

Turning attention to the Frisch elasticity of labor supply, we can correspondingly analyze identification by inspecting the intratemporal condition.
Denoting the marginal utility of consumption by $MU_{t}:=\left(C_{t}-hC_{t-1}\right)^{-\omega}$, the Frisch elasticity is potentially  identifiable using the elasticity of different macroeconomic aggregates with respect to  wage  i.e.  hours ($L$), aggregation residual ($\Xi^{lab}$), the undistorted marginal utility of wealth ($MU-\mu$) and the extensive and intensive margins of the distortion, $B$ and $\kappa$ respectively:
\begin{eqnarray*} \frac{1}{\eta}=\frac{\frac{\partial ln L_{t}}{\partial ln W^{}_{t}}-\frac{\partial ln \Xi^{lab}_{t}(\eta)}{\partial ln W^{}_{t}}}{1+\frac{\partial ln MU_{t}}{\partial ln W^{}_{t}}}&=&\frac{\epsilon_{L,W}-\epsilon_{\Xi^{lab}(\eta,\omega),W^{}}}{1+\epsilon_{MU-\mu,W^{}} +(\epsilon_{B,W^{}}+\epsilon_{\kappa,W^{}})\frac{\mu}{MU-\mu}}
\end{eqnarray*}
This is an implicit equation in $\eta$, which essentially identifies $\eta$ by adjusting  the raw elasticity $\epsilon_{L,W}$ for variation that is due to wealth effects in order to isolate the substitution effect.

The admissible values of $\eta$ will depend non-linearly in all of these elasticities and on the admissible values of $\omega$. The more responsive are dispersion and the two margins of the distortion to changes in the wage rate, the greater the divergence between the wage elasticity of aggregate hours from $\frac{\epsilon_{L,W}}{1+\epsilon_{MU,W}}$, which is the Frisch elasticity in the complete markets case. Being able to measure $B$ implies that the admissible estimates of $\frac{1}{\eta}$ will only vary with the admissible domain of the response of the intensive margin of adjustment to the wage, $\epsilon_{\kappa,W}$.

Consistent with the results of \citet{DOMEIJ2006242}, the adjustment to individual behavior in the presence of borrowing constraints can imply a higher elasticity of labor supply as $\epsilon_{\kappa,W}$ is expected to be negative. Moreover, the distortion to the extensive margin $B_{t}$ affects aggregate supply in the same way. This is also in line with the fact that the lower bound on the 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 definitely true as long as $\epsilon_{\Xi^{lab}(\eta,\omega),W^{t}}$ is negligible. Nevertheless, when $\epsilon_{\Xi^{lab}(\eta,\omega),W^{t}}$ is not negligible and as it depends on $\eta$,
identification depends on the relative magnitudes in the wage elasticity of consumption dispersion, $\epsilon_{V_{c,t},W_{t}}$ and  the wage elasticity of consumption growth, $\epsilon_{g_{c,t},W^{}}$ as \footnote{ \[\gamma_{1,t}(\eta,\omega):=\frac{\omega(\eta+\omega)Var_{t}\left(\frac{c_{i,t}}{C_{t}}\right)}{2\eta^{2}\left(1-h\left(\frac{C_{t-1}}{C_{t}}\right)\right)^{2}+\omega(\eta+\omega)Var_{t}\left(\frac{c_{i,t}}{C_{t}}\right)}\]\[ \gamma_{2,t}(\eta,\omega)=\frac{Var_{t}\left(\frac{c_{i,t}}{C_{t}}\right)\frac{C_{t-1}}{C_{t}-hC_{t-1}}}{2\eta^{2}\left(1-h\left(\frac{C_{t-1}}{C_{t}}\right)\right)^{2}+\omega(\eta+\omega)Var_{t}\left(\frac{c_{i,t}}{C_{t}}\right)}\]}
\[\epsilon_{\Xi^{lab}(\eta,\omega),W^{}}=\gamma_{1,t}\epsilon_{V_{c,t},W_{t}}-2h\gamma_{2,t}\epsilon_{g_{c,t},W_{t,t}}\]

For example, if consumption dispersion is relatively unresponsive to wage changes, and thus the first elasticity in the above expression is negligible, the mere presence of a non-zero consumption dispersion in $\gamma_{2,t}$ implies that $\epsilon_{\Xi^{lab}(\eta,\omega),W^{t}}$ is increasing in $\eta$ as $\epsilon_{g_{c,t},W^{}}$ is expected to be positive. When
$\epsilon_{V_{c,t},W_{t}}$ is different than zero, the final result depends in a more complicated way on the rest of the parameters e.g. risk aversion and the habit parameter as well as the other quantities in $\gamma_{1,t}$ and $\gamma_{2,t}$. Therefore, while the refinement on the lower bound can be rationalized without necessarily accounting for heterogeneity, the refinement 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 of inertia 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 in Appendix A, as $\beta$ appears to be point identified.
\subsection{Incomplete Markets and Asset Prices}
Having shown that incomplete markets do matter for preference identification, I next turn to analyzing in more detail how this is likely to affect inference on asset prices in the presence of heterogeneity.
Since we have not utilized equity returns to estimate preferences, it is instructive to see whether the estimated preferences can also explain the observed equity premium in the Spanish Stock Market. Using the IBEX35 Total returns index, which includes the 35 most liquid stocks in the Madrid Stock Exchange, I compute the mean risk premium over government bonds in the period 2001Q1-2018Q1 and the corresponding prediction of the risk premium as implied by the model, accounting for the two distortions as follows:
\[{ln\left(\frac{\mathbb{E}R^{e}}{\mathbb{E}R^{g}}\right)}\approx - \frac{Cov(\mathcal{M}(\theta),R_{e})+\mu^{e}(\theta)-\mu^{g}(\theta)}{1-\mu^{g}(\theta)}\]
where $\mathcal{M}(\theta):=\beta \left(\frac{C_{t+1}-hC_{t}}{C_{t}-hC_{t-1}}\right)^{-\omega}\left(\frac{\Xi_{t+1}}{\Xi_{t}}\right)$ is the heterogeneity adjusted stochastic discount factor, evaluated at all $\theta\in\Theta_{95\%}$ (Table \ref{Lim_info_est_1}, case 1), $R_{e}$ is the real equity return index and $(\mu^{e},\mu^{e})$ are the average distortions in the Equity and Bond Euler equations respectively, also evaluated at $\theta\in\Theta_{95\%}$.\footnote{Notice that these distortions can be easily computed by plugging in $\theta$ and computing the sample mean of the corresponding Euler Equations e.g. \[\hat{\mu}^{g}(\theta)=\frac{1}{T}\sum_{t=2..T}\left( (C_{t}-hC_{t-1})^{-\omega}-\beta (C_{t+1}-hC_{t})^{-\omega}\frac{\Xi_{t+1}(h)}{\Xi_{t}(h)}R_{g,t+1}\right)\]}

We observe that the data based prediction for the average risk premium lies within the set valued prediction. Moreover, shutting down frictions and heterogeneity delivers a counterfactual prediction for a weakly negative risk premium. What this implies is that had we not accounted for incomplete markets and the possibility of trading constraints across assets, in order to match the observed premium, parameter estimates would have to be severly distorted.
This consists of indirect evidence of the relevance of incomplete markets for the equity premium puzzle \citep{10.2307/1832056,MEHRA1985145,10.2307/1815722}.
Several mechanisms have been considered as potential resolutions to both the risk premium and the risk free puzzle e.g. uninsurable income risk and borrowing constraints, limited asset market participation and transaction costs among others (\citet{10.1162/003355302753399508,10.1086/340776,10.2307/2138860, AIYAGARI1991311})\footnote{See also the role of non-time separable preferences \citep{doi:10.1111/j.1540-6261.2004.00670.x, RePEc:eee:moneco:v:24:y:1989:i:3:p:401-421}, lower tail risk \citet{NBERw11310} and misperceptions in returns \citep{10.1257/000282803321947407}.}, and yet results are sensitive to the particular model setups, parameterizations and wider implications (see e.g. \citet{4137daa05d1449ddbf4171960bde2680,10.1257/jep.23.1.193} for a review.).

\begin{figure}[H]
	\begin{centering}
		\includegraphics[scale=0.48]{\string"Equity_Premium_Pred\string".png}
		\includegraphics[scale=0.48]{\string"Equity_Premium_Pred_NoF\string".png}
		\par
	\end{centering} \vspace{-0.2 in}  	\protect\caption{The top figure compares the predicted equity risk premium based on the covariance between the heterogeneity consistent stochastic discount factor and the IBEX35 total returns index, adjusted by the Euler equation distortions. The lower figure computes the counterfactual prediction for $\mu^{e}=\mu^{g}=0$ and $\Xi_{t}=1,\forall t$.}
	\label{eqp}
\end{figure}
The above results use the minimum amount of restrictions implied by incomplete markets and borrowing constraints and yet they show that these frictions are both qualitatively and quantitatively important.
Notice also that these results hold despite accounting for external habits in consumption, which have also been proposed as a potential resolution of the equity premium puzzle \citep{10.2307/2937698}.

Furthermore, we can utilize the estimates of the distortions to Euler equations to judge which trading restrictions are more severe. This exercise, which is similar in spirit to business cycle accounting (\citet{ECTA:ECTA768}), can shed light on the size of the distortions that fully specified structural models should be able to replicate for each asset class.\footnote{Corespondingly, these estimates may also be used as target statistics to validate structural models. In \citet{tryphonides2017set}, I develop a formal test.}

In contrast to typical business cycle accounting exercise where parameters are externally calibrated, the moment inequality framework has allowed us to estimate the structural parameters first while being consistent with the class of frictions under consideration. Thus, the estimated wedges will reflect both the in-sample information, as well as the model uncertainty that is already encoded by the identified set.
\\
Figure \ref{bca} plots the estimated distortions to the two Euler equations used in estimation, as well as the Equity Euler equation, where estimates are expressed as a share of marginal utility. Distortions in the bond market range from $0.2\%$ to $4.8\%$ of marginal utility, where distortions in equity trading  range from $-0.8\%$ to $3.8\%$.

In order to further interpret these distortions, notice that the bond and equity pricing equations may be rewritten as follows:
\begin{eqnarray}
	\mathbb{E}_{t}\left(\mathcal{M}_{t+1}R^{g}_{t+1}-1\right)&=& - \frac{\mu^{g}_{t}}{U'(C_{t};\theta)}\\
	\mathbb{E}_{t}\left(\mathcal{M}_{t+1}\left(R^{e}_{t+1}-R^{g}_{t+1}\right)\right)&=&  \frac{\mu^{g}_{t}-\mu^{e}_{t}}{U'(C_{t};\theta)}
\end{eqnarray}
\begin{figure}[H]
	\begin{centering}
		\includegraphics[scale=0.52]{\string"Distortions_Assets_2\string".png}
		\par
	\end{centering}  	\protect\caption{Black lines are the distortions $(\mu^{ib},\mu^{g},\mu^{e})$ estimated using $\Theta_{95\%}$ from Table 1, Case 1 (with $B_{t}$) while gray lines are the estimates using  $\Theta_{95\%}$ from Table 1, Case 1 (without $B_{t}$). }
	\label{bca}
\end{figure}

The first expression shows the negative distortion to the risk free rate return due to borrowing constraints, that is, there is "excessive demand" for bonds which puts downward pressure on the return. This is consistent with the existing literature that seeks to rationalize the risk free rate puzzle through incomplete markets.

The second distortion is the weighted risk premium, where the weight is the stochastic discount factor. The average distortion $\mu^{e}$ takes both positive and negative values. This can be rationalized by the fact that with measured total returns, which include dividend payments but no transaction costs, the distortion to the equity Euler equation will inevitably soak up the aggregate unobserved transaction costs.

Utilizing \eqref{eq:genzeldes} in the case of equities that involve transaction costs, assuming that they are constant across agents, yields an aggregate distortion to the Euler equation as follows:\small
\begin{eqnarray}
	\frac{\mu^{e}_{t}}{U'(C_{t};\theta)}&\equiv&\frac{\mu_{t}}{U'(C_{t};\theta)}+ \mathbb{E}_{t}\mathcal{M}_{t+1}\left(  R^{e}_{t+1} \left(  \frac{1}{\psi^{j,1}_{t+1}}-1\right)-\frac{\psi^{j,2}_{t+1}}{\psi^{j,1}_{t+1}} \right)
\end{eqnarray}
\normalsize

Depending on the signs and magnitudes of the unobserved marginal transaction costs, the contribution to $\frac{\mu^{e}_{t}}{U'(C_{t};\theta)}$ can be negative, while the trading constraint e.g. the restriction in short selling has a positive contribution ($\mu_{t}$). The overall effect is therefore ambiguous.\footnote{As an example of transaction costs, consider the case in which $\psi$ is symmetric e.g. quadratic in $a^{j}_{i,t+1}-a^{j}_{i,t}$. This implies that $-\frac{\psi^{j,2}_{t+1}}{\psi^{j,1}_{t+1}}=1$ while $\psi^{j,1}_{t+1}$ can be large enough such that $ R^{e}_{t+1} \left(  \frac{1}{\psi^{j,1}_{t+1}}-1\right)-\frac{\psi^{j,2}_{t+1}}{\psi^{j,1}_{t+1}}$ becomes negative, making shares a costly asset to hold and thus increasing the required risk premium. Conversely, trading constraints such as short selling restrictions increase $\mu^{e}_{t}$, exerting downward pressure on the risk premium. See \citet{AIYAGARI1991311,10.2307/2138860} for quantitative results on the effect of transaction costs on the equity premium as well as  \citet{LAGOS2010913} for a liquidity interpretation of the puzzle.}
A negative value for  $\frac{\mu^{e}_{t}}{U'(C_{t};\theta)}$ provides indirect evidence for stronger net transaction costs, and it positively contributes to the risk premium, while a positive value is evidence of stronger trading constraints.
The identified distortion does not provide overwhelming evidence for one or the other channel as it takes both positive and negative values, but most of parameter values yield a positive distortion.  This result may of course be exacerbated by the fact that the IBEX35 index is based on the 35 most liquid stocks, and therefore trading in the stocks that comprise this index is less likely to be affected by difficulties in buying/selling due to low market liquidity. Repeating nevertheless the exercise with the total IGBM index (red line) yields very similar results, geared slightly more on the negative side, which is consistent with including less liquid stocks in the return.

Moreover, it is interesting to see whether the results change when we employ preference parameter estimates that do not incorporate information on the extensive margin. Results in grey colour indicate that had we not accounted for the borrowing constrained, parameter estimates (higher risk aversion in particular) would underpredict the level of distortions in the case of bonds, while they would give certainly more weight on the transaction cost/illiquidity interpretation of the risk premium.

  \section{Conclusion}
Identifying structural parameters such as the coefficient of relative risk aversion and the Frisch elasticity of labor supply is instrumental to understanding consumption growth, the response of hours to changes in wages, as well as how asset prices are determined. Nevertheless, taking the representative agent model to aggregate data has been historically challenging, in the sense that the parameter values required to fit aggregate moments are far away from what is considered reasonable based on micro-econometric estimates.  One possible reason for these discrepancies that has been identified in the literature is heterogeneity and incomplete markets.
Nevertheless, while accounting for these frictions is important,  modeling choices can be consequential for the degree of frictions generated by the model, and hence the corresponding preference parameter estimates.

This paper has developed a methodology to estimate preferences without resorting to a particular mechanism that generates agent heterogeneity. More particularly, time series variation of macroeconomic aggregates as well as consumption dispersion have been shown to be sufficient to set identify the model, while additional distributional information is useful in shrinking the set of admissible models. A statistic that incorporates such useful information is the measure of borrowing constrained households.

Using Spanish data, the paper has shown that accounting for market incompleteness and the possibility of financial constraints is also empirically relevant. Ignoring variation in these margins leads to a wider  set of estimates for relative risk aversion and the Frisch elasticity of labor supply. This has consequences for gauging the level of distortions in the market for bonds and the market for equities. Correctly identifying the level of these distortions leads to better predictions for the equity risk premium.

  \newpage
 \section{Appendix A}

 The Appendix contains the following material:  (1) Proofs of Lemmas 2 and 3 as well as Proposition \ref{over_surv} in the paper, (2) A diagrammatic exposition of the identifying power of the extensive margin, (3) Additional Simulation Evidence, (4) The mixed frequency model used to extract $B_{t}$, (5)  Details on Data, (6) Analytical Identification Example

\subsection{Proofs for results in main text}

\begin{proof} of {Results}  \eqref{eq:no_surv_mom} and \eqref{eq:surv_mom}
	\begin{enumerate}[leftmargin=*]
		\item Starting from the moment function which is separable in $\lambda^{o}_{t}$,  \vspace{-0.2 in} \[g\left(Y_{t},Y_{t+1},\theta\right) -\lambda^{o}_{t}(Y_{t},Y_{t+1},\theta)=\mu_{t} \vspace{-0.1 in}\] where $\mu_{t}$ is positive or negative, then multiplying by $X_{t-1}>0$ maintains the same sign. Taking unconditional expectations we conclude.
		\item Since $\mu_{t} = \kappa_{t}B_{t}$, dividing both sides by $B_{t}$ and taking unconditional expectations with $X_{t-1}$ we conclude.
	\end{enumerate}
\end{proof}

 \subsection{Identification Results}
 I first define two notions that will be helpful for the theorems that follow:
 \begin{defn} Denote $m(Y_{t},Y_{t+1},\theta)$ as \underline{correctly specified} if there exists $\theta_{0}\in \Theta$ such that if $U=0$, $\mathbb{E}m(Y_{t},Y_{t+1},\theta_{0})=0$.\label{correct}
 \end{defn}\vspace{-0.3 in}
 \begin{defn} For some $U\in \mathbb{R}^{r}_{+}$, $\theta_{1}(U)$ is \underline{conditionally identified} if there does not exist any other $\theta\in \Theta$, $\theta_{2}(U)$ such that $\mathbb{E}m(Y_{t},Y_{t+1},\theta_{1}(U))=\mathbb{E}m(Y_{t},Y_{t+1},\theta_{2}(U))$. \label{unique}
 \end{defn}\vspace{-0.2 in} The first ensures that the identified set is not empty, while the second guarantees that more information on the wedge $U$ leads to a smaller identified set for $\theta$. Given these definitions, Lemma \ref{nr} establishes that fixing a value for the wedge $U$, there exists a unique value for $\theta$, defined as $\theta(U)$, that satisfies \eqref{mom_comp_gmm}. Any additional information is likely to  make the identified set smaller.
 \begin{lem} {\textbf{$\Theta_{I}$ when $r=n_{\theta}$}}\\
 	Under correct specification of $m(.)$ and conditional identification,
 	\begin{center}$\exists!$ mapping $\mathcal{G}:\Theta_{I}=\mathcal{G}^{-1}(U)\cap \Theta$	\end{center} \label{nr}
 \end{lem}
 \begin{proof} of Lemma \ref{nr}\\
 	Fix $\underset{n_{\theta}\times 1}{U}\in  (\underline{U},\bar{U}]$. Assuming conditional identification, there is a unique value of $\theta^{*}$ such that $\mathbb{E}m(Y_{t},Y_{t+1},\theta^{*}) \equiv\mathcal{G}(\theta^{*}) = \underset{n_{\theta}\times 1}{U}$. Thus,
 	$\Theta_{I}=\Theta \cap \mathcal{G}^{-1}(U)$.
 \end{proof}
 \subsubsection{The use of additional conditions}
 I distinguish between moment functions $m_{\alpha}(\theta )$, the necessary moments functions for Lemma \ref{nr} to hold, and $m_{\beta }(\theta )$ the additional moment functions where for notational brevity, I have dropped the dependence on $\mathbf{Y}^{\tau-1}$. Let $W$ be a real
 valued, possibly data dependent matrix, diagonal in $(W_{\alpha },W_{\beta
 })$. Furthermore, denote by $Q_{\alpha }$ the projection operator when projecting onto the space spanned by $W_{\alpha
 }^{\frac{1}{2}T}{{m}}_{\alpha }(\theta )$ and by  $Q_{\alpha}^{\bot}$ the residual operator.
 The following proposition specifies that additional moments have to be less than perfectly correlated with the necessary conditions for the identified set to be smaller.
 \begin{lem}
 	For $\mathcal{U}= \left\{U\in [\underbar{U},\bar{U}]\right\}$ define  \[\mathcal{U}^{c}=\left\{U\in [\underbar{U},\bar{U}] : \mathbb{E}Q_{\alpha}^{\bot}\left(W_{\beta}^{\frac{1}{2}T}{m}_{\beta}(\theta(U))-U\right)  = 0 \right\}\]
 	Then,  $\Theta^{\prime }_{I}\subset\Theta_{I} \quad \text{iff} \quad \mathcal{U}^{c}\subset \mathcal{U} $
 	\label{over}
 \end{lem}  \vspace{-0.25 in}
 \begin{proof} of Lemma \ref{over}\\
 	Denote the moment conditions using macroeconomic aggregates  by $q_{1,t}(\theta)$ and the moment conditions that employ survey variables by $q_{2,t}(\theta)$ where for $r = n_{y}+k$:
 	\begin{eqnarray*}
 		\underset{n_{y}\times1}{q_{1,t}(\theta)} & = & \mathbb{\mathcal{V}}_{1,t}\in[\underbar{\ensuremath{\mathcal{V}_{1}}},\bar{\mathcal{V}}_{1}]\\
 		\underset{k\times1}{q_{2,t}(\theta)} & = & \mathbb{\mathcal{V}}_{2,t}\in[\underbar{\ensuremath{\mathcal{V}_{2}}},\bar{\mathcal{V}}_{2}]
 	\end{eqnarray*}
 	which imply the following unconditional moment equalities :
 	\begin{eqnarray*}
 		\mathbb{E}((q_{1,t}(\theta)-\mathbb{\mathcal{V}}_{1,t})\otimes \phi_{t}) & = & 0\\
 		\mathbb{E}((q_{2,t}(\theta)-\mathbb{\mathcal{V}}_{2,t})\otimes \phi_{t}) & = & 0
 	\end{eqnarray*}
 	\normalsize
 	Denote the vector of moment conditions $(q^{T}_{1},q^{T}_{2})^{T}$ by $m$, where we also drop dependence on $\theta$ and on $t$. Partition the vector $m$ into $(m_{\alpha}^{T},m_{\beta}^{T})^{T}$
 	where $m_{\alpha}$ contains the first $n_{\theta}$ moments. Since $r>n_{\theta}$,
 	and given a weighting matrix $W$, the first order conditions are as follows: \vspace{-0.1 in}
 	\begin{eqnarray*}
 		\mathbb{E}J^{T}(W_{\alpha}^{\frac{1}{2}T}m_{\alpha}+W_{\beta}^{\frac{1}{2}T}m_{\beta}-\mathcal{V})\otimes \phi & = & 0
 	\end{eqnarray*}
 	where the Jacobian $\underset{r\times n_{\theta}}{J}$ has full rank. To economize on notation, redefine the weights after multiplication with the Jacobian terms and the instrument $\phi$, which implies that: \vspace{-0.1 in}
 	\[\mathbb{E}(W_{\alpha}^{\frac{1}{2}T}m_{\alpha}+W_{\beta}^{\frac{1}{2}T}m_{\beta}) -\underset{n_{\theta\times1}}{U} =  0\]
 	This is a projection of $m$ on a lower dimensional subspace. Since
 	$W$ is an arbitrary matrix, and $(m_{\alpha},m_{\beta})$ are possibly
 	correlated in population, we reproject the sum onto the space spanned by $W_{\alpha}^{\frac{1}{2}T}{m}_{\alpha}$.

 	Since the original sum satisfies the moment condition, then the two
 	orthogonal complements will also satisfy it: \vspace{-0.2 in}
 	\begin{eqnarray*}
 		\mathbb{E}\left(Q_{\alpha}\left(W_{\alpha}^{\frac{1}{2}T}{m}_{\alpha}+W_{\beta}^{\frac{1}{2}T}{m}_{\beta}-U\right)\right) & = & \mathbb{E}\left(W_{\alpha}^{\frac{1}{2}T}{m}_{\alpha}+Q_{\alpha}\left(W_{\beta}^{\frac{1}{2}T}{m}_{\beta}-U\right)\right)=0\\
 		\mathbb{E}\left(Q_{\alpha}^{\bot}\left(W_{\alpha}^{\frac{1}{2}T}{m}_{\alpha}+W_{\beta}^{\frac{1}{2}T}{m}_{\beta}-U\right)\right) & = & \mathbb{E}\left(Q_{\alpha}^{\bot}\left(W_{\beta}^{\frac{1}{2}T}{m}_{\beta}-U\right)\right)=0
 	\end{eqnarray*}
 	where
 	$U\in\left[W_{\alpha}^{\frac{1}{2}T}\underbar{\ensuremath{\mathcal{V}}}_{\alpha}+W_{\beta}^{\frac{1}{2}T}
 	\underbar{\ensuremath{\mathcal{V}}}_{\beta},\quad W_{\alpha}^{\frac{1}{2}T}\bar{\mathcal{V}}_{\alpha}
 	+W_{\beta}^{\frac{1}{2}T}\bar{\mathcal{V}}_{\beta}\right]$\newline
 	As in Lemma \ref{nr} the first
 	set of restrictions identifies a one to one mapping from $U$ to $\Theta_{I}$, and therefore $\theta^{*}(U)\equiv \mathcal{G}^{-1}(U)$. Plugging this in the second set of restrictions eliminates dependence on  $m_{\alpha}$ and imposes further restrictions on the domain of variation of $U$. The admissible set for $U$ is now characterized by \[\mathcal{U}^{c}:=\left\{U \in [\underbar{U},\bar{U}]: \mathbb{E}\left(Q_{\alpha}^{\bot}\left(W_{\beta}^{\frac{1}{2}T}{m}_{\beta}(U)-U\right)\right)=0\right\}\] Therefore,  $\exists \theta\in \theta(U): \theta\notin \Theta_{I}(U^{c})$ and consequently $\Theta_{I}'\subset \Theta_{I}$.

 \end{proof}

 \begin{proof}
 	\textbf{of Proposition \ref{over_surv}}\\
 	(1): When $B_{t}=0$, $\mathbb{E}\left(g\left(Y_{t},Y_{t+1},\theta\right) -\lambda^{o}_{t}(Y_{t},Y_{t+1},\theta)\right)X_{t-1}= 0$
 	which trivially restores point identification.
 	When  $B_{t}=1$, \eqref{eq:surv_mom} collapses to  \eqref{eq:no_surv_mom}. \[
 	\textit{}\mathbb{E}\left(g\left(Y_{t},Y_{t+1},\theta\right) -\lambda^{o}_{t}(Y_{t},Y_{t+1},\theta)\right)X_{t-1}= \textit{}\mathbb{E}(\kappa_{t}X_{t-1}) = \textit{}\mathbb{E}(\mu_{t}X_{t-1}) \]
 	For $B_{t}\in(0,1)$, the moment conditions in  \eqref{eq:surv_mom} and \eqref{eq:no_surv_mom} are not perfectly correlated as $Corr(\kappa_{t},\mu_{t})=Corr(\kappa_{t},B_{t}\kappa_{t})\neq 1$, element by element. 	 By Lemma \ref{over} we conclude.\newline
 	(2): If $Var(B_{t})=0$, then $Corr(\kappa_{t},B_{t}\kappa_{t})=1$.\newline
 	(3): Suppose that $\Theta_{I}$ is a singleton. Then it must be that the RHS of \eqref{mom_comp_gmm} is zero: Either $B_{t}=0$ for all $t$, or $p(s_{i,t}|S_{t})$ has unit mass on one agent who is unconstrained, and thus $B_{t}=0$ as well.
 \end{proof}
 \newpage
 \subsection{Diagrammatic exposition for the Extensive Margin} Dropping dependence on $(Y_{t},Y_{t+1})$, the aggregate moment condition without survey is summarized by $\mathbb{E}m(\theta)=\mu$, while to first order, the condition with the extensive margin can be written as $\kappa:=\mathbb{E}(m(\theta)B^{-1})\approx 2\mathbb{E}(m(\theta))-\mathbb{E}(m(\theta)B)\equiv 2\mu -  v$,  where the approximation is around $B^{\star}=1$ and $v:=\mathbb{E}(m(\theta)B)$.

 If $B$ is a non-degenerate random variable (when $B$ is degenerate i.e. it takes a single value $\bar{B}$, $v=\mu \bar{B}\in (0,\mu)$), then the admissible range for $\kappa$ lies within the admissible range for $\mu$, leading to a refined identified set for $\theta$.
 \begin{figure}[!h]
 	\centering
 	\begin{tikzpicture}[scale = 1]
 		\draw[thick,->] (0,0) -- (5,0) node[anchor=north west] {$\mu$};
 		\draw[thick,->] (0,-6) -- (0,5) node[anchor=south east] {$\kappa$};
 		\draw[line width=0.5 mm,red] (0,0) -- (0,4) ;
 		\draw[line width=0.5 mm,red] (0,0) -- (4,0) ;
 		\draw[dashed] (0,0) -- (5,5);
 		\draw[dashed, blue] (4,0) -- (4,3);
 		\draw[dashed, blue] (0,3) -- (4,3);
 		\draw (0,-5) -- (5,5);
 		\draw[gray] (0,0) -- (2.5,5);
 		\draw (2.4,5.1) -- (2.4,5.1) node[anchor=east] {$v=0$};
 		\draw (4.7,5.0) -- (4.7,5.0) node[anchor=east] {$v=\mu$};
 		\draw[line width=0.5 mm,blue] (0,0) -- (0,3) ;
 		\draw[thick,blue] (1pt,3 cm) -- (-1pt,3 cm) node[anchor=east] {$\bar{\kappa}$};
 		\draw[thick,red] (1pt,4 cm) -- (-1pt,4 cm) node[anchor=east] {$\bar{\mu}$};
 		\draw[thick,red] (4 cm, 1 pt) -- (4 cm, -6 pt) node[anchor=west ] {$\bar{\mu}$};
 		\draw (1pt,-5 cm) -- (-1pt,-5 cm) node[anchor=east] {$-v$};
 	\end{tikzpicture}
 	\caption{Identified Set for Nuisance Parameter with Survey Data (in Blue)}\label{id_fig}
 \end{figure}

 \newpage

 \subsection{Additional Evidence with Simulated Data} $ $
 \begin{figure}[H]
 	\begin{center}
 		\includegraphics[scale=0.60]{\string"MonteCarlo_final_N400\string".png}
 		\par
 	\end{center}  \vspace{-0.2 in} \label{ls}
 	\protect\caption{$T=400$}
 \end{figure}

\newpage
 \subsection{Mixed Frequency Model for Quarterly $B_{t}$}
 \label{meas}
 Using  $\tilde{B}^{con}_{t} = \tilde{\Pi}^{o}_{t}+\tilde{\zeta}_{t}$ as a measurement equation, the true (log) quarterly proportion of liquidity constrained consumers is extracted using the following mixed frequency Gaussian linear state space model, where $t=4(j-1)+q$, $t$ is the quarterly observation at year $j$ and $q=\{1,2,3,4\}$ the within year quarter index:\newline
 State Equation (ignoring identities):
 \begin{eqnarray*}
 	\left( \begin{array}{c}
 		\tilde{b}_{4(j)}\\
 		\tilde{\zeta}_{4(j)}\\
 	\end{array}\right) &=&  \left(\begin{array}{cc}
 		\rho_{b} & 0 \\
 		0 & \rho_{\zeta}
 	\end{array}\right)  \left( \begin{array}{c}
 		\tilde{b}_{4(j-1)+3}\\
 		\tilde{\zeta}_{4(j-1)+3}\\
 	\end{array}\right) + \left(\begin{array}{c}
 		\nu_{\tilde{b},4(j)}\\
 		\nu_{\tilde{\zeta},4(j)}
 	\end{array}\right)\end{eqnarray*}
 Observation equation:
 \begin{eqnarray*}
 	\left(\begin{array}{c}
 		\tilde{\Pi}^{o}_{4(j)}\\
 		\tilde{B}^{con,o}_{j}\end{array}\right)&=& \left( \begin{array}{cccccccccccc}
 		1 & 0 & 0 &0 & 0 & 0 &0 & 0 &1& 0 &... & 0\\
 		\frac{1}{8} &\frac{1}{8} & \frac{1}{8} &\frac{1}{8} & \frac{1}{8} & \frac{1}{8} &\frac{1}{8} &\frac{1}{8}&0&0&...&0 \\
 	\end{array} \right)\left( \begin{array}{c}
 		\tilde{b}_{4(j)}\\
 		\tilde{b}_{4(j-1)+3}\\
 		\tilde{b}_{4(j-1)+2}\\
 		\tilde{b}_{4(j-1)+1}\\
 		\tilde{b}_{4(j-1)}\\
 		\tilde{b}_{4(j-2)+3}\\
 		\tilde{b}_{4(j-2)+2}\\
 		\tilde{b}_{4(j-2)+1}\\
 		\tilde{\zeta}_{4(j-1)+3}\\
 		\tilde{\zeta}_{4(j-1)+2}\\
 		\tilde{\zeta}_{4(j-1)+1}\\
 		\tilde{\zeta}_{4(j-2)}\\
 		\tilde{\zeta}_{4(j-2)+3}\\
 		\tilde{\zeta}_{4(j-2)+2}\\
 		\tilde{\zeta}_{4(j-2)+1}\\
 		\tilde{\zeta}_{4(j-2)}
 	\end{array}\right) + v_{4(j)}
 \end{eqnarray*}
 where  $(\nu_{\tilde{b},t},\nu_{\tilde{\zeta},t})\sim N(0,diag(\Sigma_{\nu}))$ and $v_{t}\sim N(0,diag(\Sigma_{v}))$. The last diagonal component of $\Sigma_{v}$ is calibrated to the standard error from the estimation of $B_{t}$ in the SHF survey. The rest of the components of the diagonal are calibrated to $1\%$ of the variance of the BCS measure ($\tilde{\Pi}_{t}$).

 \newpage
 \subsection{Data} \vspace{-0.2 in}
 \begin{flushleft}
 	\begin{table}[H]
 		\begin{centering}
 			\caption{Sources and Transformations}\vspace{0.3 in}
 			\label{table:data}
 			\resizebox{\columnwidth}{!}{
 				\begin{tabular}{c|c|c|c|}
 					\hline
 					Variable& Data& Source & Final Transformation \\ \hline
 					$C_{t}$& Final Consumption Expenditures, S.A ($C^{tot}$) & FRED (OECD) & $\frac{C^{tot}}{Pop*P}*100$\\ 				$Pop_{t}$ & Active Population: Aged 15-74, S.A.& FRED (OECD)& \\
 					$L_{t}$ & Total Hours Worked , S.A. ($L^{tot}$)& Datastream (National Accounts) & $\frac{H^{tot}}{Pop}$\\
 					$P_{t}$ & GDP Implicit Price Deflator& FRED (OECD) &\\
 					$\pi_{t}$ & --& -- & $ln(\frac{P_{t}}{P_{t-1}})$\\

 					$W_{t}$ & Total Employee Compensation , S.A. ($W^{tot}$)&  Thomson Reuters (OECD) & $\frac{W^{tot}}{H}$\\

 					$R_{t+1}$ & EONIA (Quarterly rate, $i_{t}$)&  European Central Bank & $ i_{t} - \pi_{t+1} $\\
 					$R_{g,t+1}$ & Treasury Bills (Quarterly rate, $i_{g,t}$)&  FRED & $ i_{g,t} - \pi_{t+1} $\\
 					$R_{ib,t+1}$ & 3-month Interbank loans  (Quarterly rate, $i_{ib,t}$)&  FRED & $ i_{ib,t} -\pi_{t+1} $\\
 					$R_{e,t+1}$ & IBEX35/IGBM Indices (Quarterly Total return, $i_{e,t}$ ))&  Thomson Reuters & $ i_{e,t} -\pi_{t+1} $\\
 					$Var(c_{i,t})$& Consumption Expenditures& Survey of Household Finances& See 9.8.1\\
 					\hline
 			\end{tabular}}
 			\par\end{centering}

 	\end{table}
 	Excluding durables from aggregate consumption does not seem a reasonable thing to do since we consider liquidity constraints, which are more binding in the case of large purchases e.g. automobiles. As far as housing is concerned, aggregate consumption measures include imputed rents, which can be thought as the service value from housing.
 \end{flushleft}
 \subsubsection{Computing Consumption Variance using the Survey of Household Finances}
 Consistent with measurement of the aggregate, the variance of annual nominal consumption is computed using survey weights as follows: \begin{eqnarray*}
 	c^{nom,annual}_{i,t} &=& 12*(\textit{Imputed Monthly Rent} [code: 2.31]\\
 	&& + \textit{Car Purchases}/12 [code: 2.74] + \textit{House Durables}/12 [code:p2.70]\\
 	&& +\textit{monthly non durable consumption} [code:p9.1])\end{eqnarray*}

 \newpage
 \subsection{Interest Rates} Ex-Post Real interest rates:

 \begin{figure}[H]
 	\begin{centering}
 		\includegraphics[scale=0.55]{\string"Interest_Rates_Nosmooth\string".png} \label{rates}
 		\par
 	\end{centering}
 	\protect\caption{$ln(R_{x,t+1}) := ln(1+i_{x,t}) - ln(1+\pi_{t+1})$}
 	\label{rates}
 \end{figure}

\newpage

 \section{Appendix B-I}

\subsection{Analytical Example}
Recall that conditions in \eqref{eq:genzeldes} and instrument $y_{i,t}(\geq 0)$ generate
the following unconditional moment inequality holds, which  does not pin down a unique vector of values for $(\beta,\omega)$ : \vspace{-0.1 in}
\begin{eqnarray}
\mathbb{E}\left[(U'(c_{i,t};l_{i,t},\omega,X_{t})-\beta(1+r)U'(c_{i,t+1};l_{i,t+1},\omega, X_{t+1})) y_{i,t}\right] & \geq & 0
\end{eqnarray}

 In order to derive analytical identification regions I simplify the model by assuming exogenous income (so labour supply is not relevant) and eliminate $X_{t}$. I adopt the approximation of \citet{Hall1978} using CRRA utility $U(c_{i,t};\omega):= \frac{c_{i,t}^{1-\omega }-1}{1-\omega }$, which implies
the following law of motion for consumption: \vspace{-0.1 in}
\begin{eqnarray}
c_{i,t+1}&=&\rho c_{i,t}+\tilde{\lambda}_{i,t}+\phi_{i,t} +\epsilon_{i,t+1}\label{eq:HallZeldes}
\end{eqnarray}
where $\tilde{\lambda}_{i,t}\equiv-(U''(c_{i,t};\omega))^{-1}\lambda_{i,t}$, $U''(c_{i,t};\omega)$ is the second derivative of the utility function, $\rho=\left(\beta(1+r)\right)^{-\frac{U'(c_{i,t};\omega)}{c_{i,t}U''(c_{i,t};\omega)}}=\left(\beta(1+r)\right)^{\frac{1}{\omega}}$and $\phi_{i,t}$ is the residual of this approximation which is set to zero for simplicity\footnote{In a recent paper \citet{commault2019} challenged \citet{Hall1978}'s random walk result for consumption in the case of isoelastic utility. In Section \ref{alt} in Appendix B  clarify why the identification analysis is unaffected by this simplification; in a nutshell, $\phi_{i,t}$ has very similar properties to $\tilde{\lambda}_{i,t}$.}. Moreover, $\epsilon_{i,t+1}$ is the rational forecast error such that $\mathbb{E}_{t}\epsilon_{i,t+1}=0$ and summarizes the uncertainty faced by the otherwise rational individual.  Equivalently, consumption growth is equal to  \vspace{-0.15 in} \[\Delta c_{i,t+1}=(\rho-1)c_{i,t}+\tilde{\lambda}_{i,t}+\epsilon_{i,t+1}\] Since higher income relaxes the budget constraint ( $Cov(\tilde{\lambda}_{i,t},y_{i,t})<0$), the set of admissible values for $\rho$ is as follows: \vspace{-0.1 in}
\begin{eqnarray}
\rho_{{ID,1}}&:=& \left(0,1+\frac{Cov(y_{i,t},\Delta c_{i,t+1})}{Cov(y_{i,t}c_{i,t})}\right] = \left(0,1+\rho_{IV}\right] \label{eq:bound1}
\end{eqnarray}
Avoiding to take a stance on the exact nature of $\lambda_{i,}$, which can involve a vast amount of unobserved information, results in set identification.
 Nevertheless, we are still able to infer certain facts about the household's preferences and economic behavior and conclusions are valid across alternative environments.
\subsection{Bounds on Risk Aversion}
Given the set of admissible values for the reduced form coefficient $\rho$, we can recover the implied bounds for the risk aversion parameter, $\omega_{ID}$. Treating $(r,\beta)$ as known, if $\beta(1+r)=1$, risk aversion is unidentified ($\omega_{ID} =\mathbb{R}_{+}$) as consumption follows a random walk. If $\beta(1+r)\neq 1$, then risk aversion is set identified and has a very intuitive interpretation, as it reflects restrictions on preferences implied by the presence of non-diversifiable income risk.

In particular, for $\beta(1+r)<1$ the household is impatient and does not accumulate wealth indefinitely. Using \eqref{eq:bound1}, since income and consumption growth are negatively correlated {see e.g. \citet{Deaton}}, risk aversion is bounded by above and the set of values consistent with the data is: \vspace{-0.1 in}\small \[\omega_{ID,1}= \left\{\omega \in \mathbb{R}_{+}: \omega < \frac{\|\log(\beta(1+r))\|}{\log \left(\frac{Cov(y_{i,t}c_{i,t})}{Cov(y_{i,t}c_{i,t})-\|Cov(y_{i,t}\Delta c_{i,t+1})\|}\right)}\right\} \]\normalsize The stronger the negative correlation,  the lower the upper bound on risk aversion, indicating that the less risk averse household is not accumulating enough wealth to fully insure against income risk.

\subsection{\textbf{The extensive margin as additional information}} Suppose now that we observe the dichotomous response of the household over time to a survey question that asks whether the household is (or expects to be in the near future) financially constrained. An honest household will answer positively whenever $\lambda_{i,t}>0$.

It can be shown\footnote{See Appendix B-II for the formal derivation.} that had the data been generated by the model in \eqref{eq:zeldes} e.g. a restriction on borrowing, then consumption growth under incomplete markets becomes as follows:\vspace{-0.2 in}
\begin{eqnarray}\nonumber \Delta c_{i,t+1}& =& (\rho-1)c_{i,t}+\epsilon_{i,t+1}+\mathbf{1}(\lambda_{i,t}>0)\left(-{\rho}c_{i,t} + \bar{y}_{i}\left( 1- \frac{\rho}{1+r}(1-F_{y_{i}}(x^{\star}_{i}))\right)\right)\\
&:=& (\rho-1)c_{i,t}+\epsilon_{t+1}+\mathbf{1}(\lambda_{i,t}>0)\left(\lambda_{1}c_{i,t} + \lambda_{0,i}\right)\label{cons_growth}\end{eqnarray}
where $F_{y_{i}}(x^{\star}_{i})$ is the the probability of hitting the constraint, which depends on the distribution of income and cash on hand threshold $x^{\star}_{i}$. The higher this probability is, the higher the savings rate out of income, increasing thereby consumption growth.

The relevant question to ask is whether we can utilize the survey information without knowing the true data generating process i.e. the form of $\lambda_{i,t}$ in \eqref{cons_growth}, and whether this leads to even more precise estimates of risk aversion.
First, dichotomous responses to the survey can be used to estimate $\mathbb{P}_{t}(\lambda_{i,t}>0)$.
Therefore, for $v\in \mathbb{R}$ and $\epsilon_{i,t+1}$ with cumulative density $\Phi_{i,t}(.)$, the distribution function of consumption growth is: \small
\begin{eqnarray}
\mathbb{P}_{t}(\Delta c_{i,t+1}< v) & =& \Phi_{i,t}(v -({\rho}-1)c_{i,t})\mathbb{P}_{t}(\lambda_{i,t}=0)+\Phi_{i,t}(v -({\rho}-1)c_{i,t}-\lambda_{i,t})\mathbb{P}_{t}(\lambda_{i,t}>0) \nonumber \\
&\leq&\Phi_{i,t}(v -({\rho}-1)c_{i,t})\mathbb{P}_{t}(\lambda_{i,t}=0)+\Phi_{i,t}(v -\tilde{\rho}_{{IV}}c_{i,t})\mathbb{P}_{t}(\lambda_{i,t}>0)  \label{eq:probit4}
\end{eqnarray} \normalsize
where we use the only information available: that is $\lambda_{i,t}>0$, and that the IV regression estimate of  ${\rho}-1$ in \eqref{cons_growth}, $\tilde{\rho}_{{IV}}$, is a lower bound to the true value\footnote{To see this, notice that  $\tilde{\rho}_{{IV}}={\rho}-1 + {}{{\left({\mathbb{C}ov(c_{i,t},y_{i,t})^{-1}\mathbb{C}ov(y_{i,t},\lambda_{i,t}\mid \lambda_{i,t}>0)}{}\mathbb{P}(\lambda_{i,t}>0)\right)}}  < {\rho}-1
	$. The bias term is negative as higher income relaxes the constraint and can be verified by the solution of the model, as $\lambda_{1}=-\rho$.}.

The simplest way to show that \eqref{eq:probit4} provides additional information is to show that there exists parameter value in  $\rho\in\rho_{{ID,1}}$ that is not consistent with \eqref{eq:probit4}.
It can be shown that $\tilde{\rho}_{{IV}}$, which is consistent with \eqref{eq:bound1}, does not satisfy \eqref{eq:probit4} when the probability of being constrained is higher than a certain threshold  which depends on the true model\footnote{ The result is obtained as follows:
	Let $p_{i,t}:=\mathbb{P}_{t}(\lambda_{i,t}>0)$. If $\tilde{\rho}_{{IV}}$ is admissible, evaluating the LHS of \eqref{eq:probit4} at the true $\tilde{\rho}=\rho-1$ and the RHS at $\tilde{\rho}_{{IV}}$, and solving for $p_{i,t}$:  \begin{eqnarray*}
		p_{i,t}&\leq& \frac{\Phi_{i,t}(v-\tilde{\rho} c_{i,t})-\Phi_{i,t}(v-\tilde{\rho}_{IV}c_{i,t})}{\Phi_{i,t}(v-\tilde{\rho}c_{i,t})-\Phi_{i,t}(v-\lambda_{0}-(\tilde{\rho}+\lambda_{1})c_{i,t})}:=g(p_{i,t})
	\end{eqnarray*}
	The function  $g(p_{i,t})$ satisfies $g(0)=0$ (and $g(1)=1$ when $\bar{y}=1$).}: \begin{eqnarray}\mathbb{P}_{t}(\lambda_{i,t}>0)\in\left(g_{i,t},1\right) \label{eq:Probbound}\end{eqnarray}  where  \[g_{i,t}:=\frac{\Phi_{i,t}(v-({\rho}-1)c_{i,t})-\Phi_{i,t}(v-\tilde{\rho}_{IV}c_{i,t})}{\Phi_{i,t}(v-({\rho}-1)c_{i,t})-\Phi_{i,t}(v-\lambda_{0,i}-({\rho}-1+\lambda_{1})c_{i,t})}\]

  To gain intuition, consider the extreme cases where the probability of being constrained is zero or one. Condition \eqref{eq:Probbound} becomes invalid as  $(0,1)$ and $(1,1)$ do not contain $\mathbb{P}_{t}(\lambda_{i,t}>0)$. In either case, additional survey information is redundant. When liquidity constraints are occasionally binding, survey information matters.

  Figure \ref{refinement} provides an example when $\Phi_{i,t}$ is the cumulative standard Normal distribution.  For higher values of the intensive distortion $\lambda_{1}$, e.g. $2\lambda_{1}$ the admissible range (grey shade) becomes larger.   The more severe the distortions, the more likely is that additional information matters.    Conversely, in the frictionless limit the threshold contradicts \eqref{eq:Probbound} as  $g_{i,t}\underset{\lambda_{i}\to 0}{\to}\mathbb{P}_{t}(\lambda_{i,t}>0)$.
      \begin{figure}[]
  	\begin{center}
  		\includegraphics[scale=0.28]{\string"probrange_new\string".png}
  		\protect\caption{\small Range for $v=0,c_{i,t}=1,r=0.05,\sigma_{\epsilon}=0.1,\bar{y}=0$}
  	\end{center} \label{refinement}
  \end{figure}

\section{Appendix B-II}
This Appendix contains derivations relating  to Appendix B-I.
\subsection{Deriving Consumption Growth}

When the household is constrained, $c^{con}_{i,t}=x_{i,t}$ while unconstrained consumption is equal to $c^{unc}_{i,t}=\frac{(1+r-\rho)}{1+r}x_{i,t}$. The latter is parallel to the perfect foresight solution, $c^{pf}_{i,t}=(1+r-\rho)a_{i,t}+\frac{1+r-\rho}{1+r}(y_{i,t}+\frac{\bar{y}_{i}}{r})= \frac{(1+r-\rho)}{1+r}(x_{i,t}+\frac{\bar{y}_{i}}{r})$ where   $\frac{(1+r-\rho)}{1+r}\frac{\bar{y}_{i}}{r}$ is interpreted as the share of permanent income ($\bar{y}_{i}\equiv \mathbb{E}y_{i,t}$) that is channeled to  precautionary savings. An  example of a numerical solution that verifies this is in Figure \ref{conuncon} below.\footnote{See also \citet{10.1257/jep.15.3.23}.}

 Using the
constrained agent's Euler equation, it has to be the case that $\rho x_{i,t} = \mathbb{E}(c_{i,t+1}|\lambda_{i,t}>0)-\lambda_{i,t}$.  Expectations include the possibilities of being constrained or not in $t+1$, thus:
\begin{eqnarray*}
	\rho x_{i,t}& = &\mathbb{E}(c_{i,t+1}|\lambda_{i,t}>0)-\lambda_{i,t}\\
	& = &\mathbb{E}(x_{i,t+1}|\lambda_{i,t}>0)\mathbb{P}(\lambda_{i,t+1}>0|\lambda_{i,t}>0)\\
	&  & + \left(1-\frac{\rho}{1+r}\right)\mathbb{E}(x_{i,t+1}|\lambda_{i,t}>0)\mathbb{P}(\lambda_{i,t+1}=0|\lambda_{i,t}>0)-\lambda_{i,t}\\
	& = &\mathbb{E}(x_{i,t+1}|\lambda_{i,t}>0)\left[1-\frac{\rho}{1+r}\mathbb{P}(\lambda_{i,t+1}=0|\lambda_{i,t}>0)\right] -\lambda_{i,t}\\
	& = &\bar{y}_{i}\left[1-\frac{\rho }{1+r}\mathbb{P}(\lambda_{i,t+1}=0|\lambda_{i,t}>0)\right] -\lambda_{i,t}\\
	& = &\bar{y}_{i}\left[1-\frac{\rho }{1+r}(1-F_{y_{i}}(x^{\star}_{i}))\right] -\lambda_{i,t}
\end{eqnarray*}
where in the last two lines we use that cash on hand in $t+1$ if constrained at $t$ is equal to $y_{i,t+1}$ whose expected value is $\bar{y}_{i}$, while the probability of being constrained in $t+1$ is $F_{y_{i}}(x^{\star}_{i})$, $x^{\star}$ being the cash on hand threshold. Thus, $\lambda_{i,t}= - \rho x_{i,t} +\bar{y}\left[1-\frac{\rho }{1+r}(1-F_{y_{i}}(x^{\star}_{i}))\right]$ and
\begin{eqnarray}\Delta c^{inc}_{i,t+1}& =& \tilde{\rho} c_{i,t}+\epsilon_{t+1}+\mathbf{1}(\lambda_{i,t}>0)\left(-{\rho}c_{i,t} + \bar{y}_{i}\left( 1- \frac{\rho }{1+r}(1-F_{y_{i}}(x^{\star}_{i}))\right)\right)\nonumber\\
	&:=& \tilde{\rho} c_{i,t}+\epsilon_{t+1}+\mathbf{1}(\lambda_{i,t}>0)\left(\lambda_{1}c_{i,t} + \lambda_{0,i}\right)\label{eq:redc}\end{eqnarray}
\normalsize

\begin{figure}[H]
	\begin{centering}
		\includegraphics[scale=0.3]{\string"paper_incomplete_markets1\string".png} \vspace{-0.1 in}
		\protect\caption{Example of Optimal Consumption in Incomplete Markets: Constrained versus Unconstrained Consumption. Notice also that for large values of $x_{t}$, $C^{inc}_{t}$ and $C^{per.for}_{t}$ become parallel, as expected. }
	\end{centering} \label{conuncon}
\end{figure}

\subsection{Alternative Derivation to \citet{Hall1978}}
\label{alt}
Following \citet{commault2019}, consider the Euler equation:\vspace{-0.2 in}
\begin{eqnarray*}
	U'\left((\beta(1+r))^{\frac{1}{\omega}}c_{i,t}\right)&=& \mathbb{E}_{t}U'(c_{i,t+1})+(\beta(1+r))^{-1}\lambda_{i,t}\\
	(\beta(1+r))^{\frac{1}{\omega}}c_{i,t}&=&U^{'-1}\left(U'(\mathbb{E}_{t}c_{i,t+1}-\phi_{i,t})+(\beta(1+r))^{-1}\lambda_{i,t}\right)
\end{eqnarray*}
where $\phi_{i,t}>0$ if marginal utility is convex. A first order approximation of the RHS around $\lambda_{i,t}=0$ yields:
\vspace{-0.2 in}	\begin{eqnarray*}
	(\beta(1+r))^{\frac{1}{\omega}}c_{i,t}&\approx& \mathbb{E}_{t}c_{i,t+1}-\phi_{i,t} - \tilde{\lambda}_{i,t}
\end{eqnarray*}
where $\tilde{\lambda}_{i,t} = -\frac{(\beta(1+r))^{-1}\lambda_{i,t}}{U''((\beta(1+r))^{\frac{1}{\omega}}c_{i,t})}>0$. The distortion to random walk comes from both the precautionary savings effect and occasionally binding constraint. Repeating the analysis in 7.2 yields that:\vspace{-0.1 in}	\small
\begin{eqnarray*}
	\lambda_{i,t}+\phi_{i,t}& = &\bar{y}_{i}\left[1-\frac{\rho }{1+r}(1-F_{i,y}(x^{\star}_{i}))\right] -\rho x_{i,t}, \quad and
\end{eqnarray*}\vspace{-0.2 in}
\begin{eqnarray*}\Delta c^{inc}_{i,t+1}& =& \tilde{\rho} c_{i,t}+\epsilon_{t+1}+\mathbf{1}(\lambda_{i,t}>0)\left(-{\rho}c_{i,t} + \bar{y}_{i}\left( 1- \frac{\rho }{1+r}(1-F_{y_{i}}(x^{\star}))\right)\right)+ \mathbf{1}(\lambda_{i,t}=0)\phi_{i,t}
\end{eqnarray*} 	\normalsize
Finally, the bound derived in Section 3 is identical because $\phi_{i,t}$ and $Cov(\phi_{i,t},y_{i,t})$  have the same sign as $\lambda_{i,t}$  and $Cov(\lambda_{i,t},y_{i,t})$ respectively.

\bibliographystyle{econometrica}

redpajama_set_name: RedPajamaArXiv