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Set Identified Dynamic Economies and Robustness to Misspecification
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Incorporating frictions in dynamic macroeconomic models is undoubtedly essential, both in terms of their theoretical and empirical implications. They are being routinely used to explain both the observed persistence of macroeconomic time series and several stylized facts obtained from firm and household level data. However, at the macroeconomic level, the selection between different types of frictions and the specification of their corresponding mechanism is a complicated process featuring arbitrary aspects. This arbitrariness implies that different studies may find support for alternative types of frictions, depending on other assumptions made. For example, the choice of nominal rigidities might depend on which real rigidities are included in the model or whether there is variable capital utilization or not\footnote{ See the analysis of christiano2005nominal and their comparison to the results of 2000.}. More importantly, unless frictions are micro-founded, policy conclusions may become whimsical as different mechanisms imply different relationships between policy parameters and economic outcomes.
In this paper we propose a new inferential methodology that is robust to misspecification of the mechanism generating frictions in a dynamic stochastic economy. The approach treats economies with frictions as perturbations of a frictionless economy, which are not uniquely pinned down, and are consistent with different structural specifications. From a hypothesis testing point of view, the set of alternatives considered are no longer arbitrary.
The paper makes several contributions. First, we derive a characterization for the law of motion of an economy with frictions that imposes identifying restrictions on the solution of the model. ECTA:ECTA768, CKM hereafter, identify wedges in the optimality conditions of a frictionless model that produce the same equilibrium allocations in economies with specific parametric choices for the frictions. We also take a frictionless model as a benchmark but contrary to CKM we construct a general representation for the wedge in the decision rules. We illustrate through examples that the sign of the conditional mean of the decision rule wedge is typically known, even when the exact mechanism generating the friction is unknown. We utilize knowledge of the sign of the conditional mean to set-identify the parameters of the benchmark model.
Moment inequality restrictions have been used to characterize frictions in specific markets, see for example 1996 and ECTA:ECTA1214. To our knowledge, we are the first to characterize such restrictions in dynamic stochastic macroeconomic models and to show their relationship with the literature on wedges in equilibrium models. We are also the first to characterize set identification in this class of models. The econometric theory we develop can accommodate inequality restrictions of general form.
Due to set identification, many models with frictions are likely to be consistent with the robust identifying restrictions. Thus, additional data other than macroeconomic time series can be potentially useful in order to further constrain the set of admissible models. As a second contribution, our methodology permits the introduction of additional restrictions. We show how qualitative survey data can be useful for this purpose, since they provide distributional information which is highly relevant in models featuring frictions and heterogeneity. More specifically, qualitative survey data are linked to current and future beliefs of agents in the model. If survey data reflect subjective conditional expectations, they contain important indications about agents' actions through the behavioral equations. In the literature\footnote{ We mainly refer to the treatment of survey data in the most recent "modern" DSGE literature. The treatment of survey data in Rational Expectations models date back to the work of Pesaran and others.}, survey data are typically linked to model based random variables using additional observation equations with additive measurement error (see e.g. DelNegro2013). However, this paper follows a different approach as the model is incomplete and there is no well defined model quantity that can be linked to the data. This is where the law of motion representation we derive proves useful, as we can link the macroeconomic distortions to aggregated qualitative surveys through additional moment inequality restrictions.
Regarding possible applications of the methodology, we show how it contributes to model selection, where the object of interest is not the set-identified parameter vector itself, but the implied semi-parametric estimates of frictions. To that end we propose a novel Wald test that compares the estimates of distortions implied by a candidate model to the robust set obtained using our method. We also derive large sample theory for this type of test. Beyond restricting the set of observationally equivalent distortions, the use of additional data i.e. surveys prevents the test statistic from degenerating when the data generating process is such that the incomplete model nests the candidate model.
We apply the methodology to estimate the distortions present in the Spanish economy due to financial frictions. We estimate a small open economy version of the 10.1257/aer.97.3.586 model using qualitative survey data collected by the Commission on the financial constraints of firms and consumers. We contrast the implied estimates of distortions to macroeconomic aggregates due to these frictions to those identified using a complete model that incorporates the Bernanke_Gertler financial accelerator.
We have already referred to how our paper relates to some strands of literature, namely the literature on wedges and frictions (i.e. ECTA:ECTA768, christiano2005nominal, 2000) and the literature on including survey data in DSGE models (i.e. DelNegro2013). We contribute to the literature that deals with partial identification in structural macroeconomic models (e.g. lubik,coroneo2011testing) and the literature on applications of moment inequality models, see ECTA:ECTA1480 and references therein. We postpone illustrating more detailed connections to the vast partial identification literature for later sections, when we deal with estimation and inference using our methodology.
The rest of the paper is organized as follows. Section 2 provides a motivating example of the methodology in a partial equilibrium context, while section 3 presents the prototype economy which will be used as an experimental lab to illustrate our methodology. Section 4 examines the distortions present in the decision rules and their observable aggregate implications. Section 5 provides the formal treatment of identification without additional information while Section 6 analyzes the case with additional information, including qualitative survey data. Section 7 discusses inference issues and provides the test. Section 8 contains the application to Spanish data and Section 9 concludes and provides avenues for future work. Appendix A contains proofs and the empirical results. Appendix B (online) contains more examples of moment inequality restrictions, details on the relation to the model uncertainty literature and the algorithm to compute frictions, an example on identification using our method, an illustration of the validity of bootstrap for the proposed test and details on the survey data used in the application.
Throughout the paper we refer to three different conditional probability measures. The objective probability measure, $\mathbb{P}_{t}$, the probability measure determined by the frictionless model $M_{f}$, $\mathbb{P}^{M_{f}}_{t}(.|,.)$, and the subjective probability measure $\mathcal{P}_{t,i}$ where $i$ identifies agent $i$ and $t$ the timing of the conditioning set. All three are absolutely continuous with respect to Lebesgue measure. The corresponding conditional expectation operators are $\mathbb{E}_{t}(.)$, $ \mathbb{E}_{t}^{M_{f}}(.|.)$ and $\mathcal{E}_{i,t}(.)$. We distinguish between the first and the second conditional expectation as the model will be correctly specified only if the econometrician employs the right DGP. Moreover, $T$ denotes the length of both the aggregate and average survey data, and $L$ the number of agents. We denote by $\theta \in \Theta $ the parameters of interest with $\Theta^{CS}_{\alpha}$ the corresponding $1-\alpha$ level confidence set, and by $ q_{j}(;\theta )$ a measurable function. Bold capital letters e.g. $\mathbf{Y} _{t}$ denote a vector of length $\tau$ containing $\{Y_{j}\}_{j\leq \tau}$. The operator $\rightarrow _{p}$ signifies convergence in probability and the operator $\rightarrow _{d}$ convergence in distribution; $\mathcal{N}(.,.)$ is the Normal distribution whose cumulative is $\Phi(.,.)$; $||.||$ is the Euclidean norm unless otherwise stated; $\perp $ signifies the orthogonal complement and $\emptyset $ the empty set. We denote by $(\Omega ,\mathcal{S}_{y},\mathbb{P})$ the probability triple for the observables to the econometrician, where $ \mathcal{S}_{y}=\sigma (Y)$, is the sigma-field generated by $Y$ and $ (\Omega ,\mathcal{F},\{\mathcal{F}\}_{t\geq 0},\mathbb{P})$ the corresponding filtered probability space. Finally we denote by $\odot$ and $\oslash$ the Hadamard product and division respectively.
We first motivate our approach by illustrating the special case of observing a single household receiving a random exogenous endowment $y_{i,t}$ and making consumption-savings decisions $(c_{i,t},s_{i,t})$ in a partial equilibrium context. As in Zeldes1989, wealth $w_{i,t}$ earns a riskless return $R$ and there is a borrowing limit at zero:
The corresponding consumption Euler equation is distorted by the non-negative Lagrange multiplier on the occasionally binding liquidity constraint, denoted by $\lambda_{i,t}$:
Whether the borrowing constraint is binding depends on the household's expectations about future income, and therefore on its information set. Consequently, condition (ref) is a conditional moment inequality, and for any conformable variable $z_{i,t}$ that belongs to the household's information set, the following unconditional moment inequality holds:
Given the inequality, there is no unique vector $(\theta,\beta)$ that satisfies it, and we therefore no longer have point identification. In order to derive explicit identification regions, we adopt the approximation of Hall1978, which for CRRA utility and constant interest rate implies the following law of motion for consumption:
where $\tilde{\lambda}_{i,t}\equiv-(U''(c_{i,t+1}))^{-1}\lambda_{i,t}$. Since $\mu$ may be exceed one if $\beta R>1$ and vice verca\footnote{In fact, $\mu$ is equal to $\left(\beta(1+r)\right)^{-\frac{u'(c)}{cu''(c)}}>0$, and it will be constant for CRRA utility.}, we re-express (ref) in terms of consumption growth, $\Delta c_{i,t+1}=\tilde{\mu}c_{i,t}+\epsilon_{t+1}+\tilde{\lambda}_{i,t}$.
Since the sign of $\mathbb{E}_{t}\tilde{\lambda}_{t}$ is still positive, the identified set for $\mu_{0}$ is :
As we can readily see from (ref), the inherent unobservability of $\tilde{\lambda}_{i,t}$ is the root of the loss of point identification of $\mu_{0}$. An interesting question is whether additional data can be informative about, or even point identify, $\mu_{0}$. The answer in this case is affirmative.
Suppose 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) financially constrained. An honest household will answer positively whenever $\lambda_{i,t}>0$. What this implies is that the time series of responses $\{\tilde{\chi}_{i,t}\}_{t\leq N}$ for $\tilde{\chi}_{i,t}\equiv\mathbf{1}('\lambda_{i,t}>0')$ can be used to estimate $\mathbb{P}_{t}(\lambda_{i,t}>0)$. The conditional probability distribution of $\Delta c_{i,t+1}$, $\mathbb{P}_{t}(\Delta c_{i,t+1}<u)$ for any $u\in \mathbb{R}$ is equal to
For $\epsilon_{t+1}\sim N(0,\sigma_{\epsilon}^{2})$, (ref) simplifies further to,
A key observation is that the distortion $\lambda_{i,t}$ is positive only when the state variables $(w_{i,t},y_{i,t})$ lie in a particular area of their domain $\mathcal{A}$ which is determined by an unknown endogenous threshold $(w^{*}_{i},y_{i}^{*})$. A complete model would pin down, analytically or numerically this domain. Observing $\tilde{\chi}_{i,t}$ dispenses us with the need to characterize $\mathcal{A}$. Once the threshold condition is satisfied, $\lambda_{i,t}$ will be a linear(ized) function of $(w_{i,t},y_{i,t})$.
Moreover, it is also reasonable to assume that $\lambda_{i,t}$ is a function of the unobserved exogenous shocks $\epsilon_{t+1}$. We can therefore substitute $\lambda_{i,t}$ for $\lambda_{1}c_{i,t}+\lambda_{2}\epsilon_{i,t+1}$, in (ref), which yields the following quantile restriction:
where $\tilde{\mu}^{c}_{ols}$ is equivalent to the least squares estimate in the case of borrowing constraints with probability one. The inequality is derived for a class of functions $\mathbb{P}_{t}(\lambda_{i,t}>0)$.
How does this additional inequality refine the identified set in (ref)? Denote by $\mu_{ID,2}$ the set of $\mu$ consistent with the unconditional moment inequality constructed using $z_{t}$ and (ref). The simplest way to show the refinement of (ref) is to show that there exists a $\mu\in\mu_{ID,1}$ that does not belong to $\mu_{ID,2}$. For simplicity, let $\tilde{\mu}_{0}=0$, which implies that $\Delta c_{i,t+1}= \tilde{\chi}_{i,t}(\lambda_1c_{i,t}+\lambda_2\epsilon_{i,t+1})+(1-\tilde{\chi}_{i,t})\epsilon_{i,t+1}$. This is equivalent to considering an economy in which $\beta(1+r)<1$ and the consumer has very high risk aversion ($\omega\to \infty$). In Appendix A we show that using $\mu_{ols}$ as the test point, (ref) is not satisfied for $p_t=1(c_{i,t}<0)$, and therefore $\mu_{ols}\notin \mu_{ID,2}$. The upper bound in $\mu_{ID}$ should thus be lower than $\mu_{ols}$. In words, as long as the agent has some positive unconditional probability of not being constrained (here $\frac{1}{2}$), observing her responses provides additional information.
The identified set for the reduced form, $\mu_{ID}=\mu_{ID,1}\cap\mu_{ID,2}$, is informative in different ways. First, as already mentioned, given $\mu_{ID}$, we can recover the implied identified set for the risk aversion parameter, $\omega_{ID}$. As is typical in the empirical macro literature, $(r,\beta)$ are not identified from the dynamic properties of the model but rather from other information, i.e steady states. In the case when $\beta(1+r)=1$, risk aversion is unidentified, $\omega_{ID} =\mathbb{R}^{+}$, as consumption follows a random walk. For $\beta(1+r)\neq 1$,
This interval has a very intuitive interpretation, and actually reflects restrictions on preferences implied by the presence of non-diversifiable income risk and observed behavior. For the sake of illustration let us focus on the identified set implied by (ref). If $\beta(1+r)<1$, the household is impatient and does not accumulate wealth indefinitely. Using $z_{i,t}=y_{i,t}$, since income and consumption growth are negatively correlated, which is usually the case when liquidity constraints are present\footnote{Deaton}, $\omega_{ID,1}= \left\{\omega \in \mathbb{R}^{+}: \omega < \frac{\|\log(\beta(1+r))\|}{\log \mathbb{E}y_{i,t}c_{i,t}-\log\left({\mathbb{E}y_{i,t}c_{i,t}-{\|\mathbb{E}y_{i,t}\Delta c_{i,t+1}\|}{}}\right)}\right\}$. The stronger the negative correlation, the lower the upper bound on risk aversion, indicating the fact that the less risk averse household is not accumulating enough wealth to fully insure against income risk. On the contrary, weak negative correlation permits higher estimates of risk aversion. Similar to ARELLANO2012256, the degree of under-identification therefore depends on this correlation, which is itself a function of $(\omega,\beta,r)$.
Second, the upper and lower bounds for $\mu_{ID}$ can be used to do inference on other interesting quantities, including the model itself. For example, we can construct set estimates of distortions in consumption implied by liquidity constraints by plugging-in the identified set and averaging over the observations. In population, this produces an identified set for distortions:
Another key observation is that although we have motivated the sign of $\mathbb{E}_{t}\lambda_{i,t}$ by looking at the exogenous constraint of no borrowing, the sign is robust to different mechanisms, as long as they prevent the household from smoothing consumption. For example, a non zero constraint on next period wealth, i.e. $w_{i,t+1}\geq \underbar{b}$ or a constraint that is endogenous, i.e. $w_{i,t+1}\geq \underbar{b}(y_{i,t})$ where $y_{i,t}$ is the relevant state variable, lead to an Euler equation which is distorted in the same direction. Therefore, the identified set of distortions will be a priori consistent with different mechanisms generating liquidity constraints.
A mechanism will not be rejected as long as it generates distortions that will statistically belong to the identifed set for $\mathbb{E}\lambda_{t}$. Moreover, we will see that in general equilibrium, survey data will generate additional moment restrictions which can potentially add information. The identified set is expected to be refined, and this refinement can potentially provide power to reject alternative models that cannot be rejected using bound (ref).
In the rest of the paper we will gradually characterize the case for general equilibrium models and what information aggregated surveys can provide, based on similar assumptions to those made in the simple example. To do so, we will provide below the benchmark general equilibrium model.
In the last section we considered liquidity constraints for a single household in a partial equilibrium context. In this section, we consider additional types of frictions by including capital and a representative firm in the economy and set up the general equilibrium framework. Building up from the previous section, consider a simple Real Business Cycle (RBC) model, in which dynastic households with Constant Relative Risk Aversion (CRRA) preferences form expectations about key state variables and make consumption - savings decisions. We denote by $\Lambda_{t}(i)$ the distribution of agents at time $t$, and the corresponding aggregate variables by capital letters i.e. $X_{t}\equiv \int {x_{i,t}d\Lambda_{t}(i)}$. Households rent capital $(k_{i,t})$ to a representative firm, which is used for production, and receive a share $(\eta _{i,t})$ of profits made by the firm. Profits are simply the production of output using capital intensive technology with random productivity $(Z_{t}K_{t}^{\alpha })$ minus the rents paid to all households $(R_{t}K_{t})$, that is, $pr_{i,t}=\eta _{i,t}(Z_{t}K_{t}^{\alpha }-R_{t}K_{t})$. Household income is therefore $y_{i,t}=R_{t}k_{i,t}+pr_{i,t}$. Individual investment decisions $( \iota_{i,t})$ increase the availability of capital for next period, up to a certain level of depreciation. The household's optimization problem is therefore as follows:
Note that individual expectations $\mathcal{E}_{i,0}$ are not necessarily formed with respect to the objective probability measure, nor with respect to the same information set. Denote by $\xi _{t}$ the aggregation residual, and by $\tilde{X}$ the percentage deviations of any aggregate variable $X_{t}$ from aggregate steady state $x_{ss}$. Then the system of aggregate linearized\footnote{Note that here we follow the linear approximation to the first order conditions as widely used in the DSGE literature, while in the last section we followed Hall1978, where the variables are not in percentage deviation from steady state. The analysis is nevertheless largely unaffected.} equilibrium and market clearing conditions are as follows:
Under approximate linearity, that is, when $\xi _{t}$ is negligible \footnote{Absence of frictions is one of the reasons for which approximate linearity holds.}, we can obtain the aggregate decision rules for X$_{t}$, which will depend on predetermined capital $\tilde{k }_{i,t}$ and on aggregate conditional expectations on productivity for some $j$ periods ahead, $\bar{\mathcal{E}}_{t}\tilde{Z}_{t+j}$.
In the previous section, we focused on the consumption decision of agents by taking the interest rate as exogenous and fixed, that is $R_{t}=R$. In this section, $R_{t}$ is endogenous as it is determined by the marginal productivity of capital, and is subject to aggregate risk. With regard to the aggregate implications of the borrowing constraints introduced in section 2, the characterization of individual behavior is identical. In general equilibrium, what needs to be taken into account is the determination of $R_{t}$. We will formally deal with general equilibrium in the next section. Moreover, in this section the idiosyncratic risk to income is $\eta_{i,t}$ while there is also aggregate risk, $Z_{t}$. Compared to the last section, the introduction of aggregate risk does not alter the observable implications on individual consumption behavior. This is for the reason that although aggregate risk complicates forecasting for households as predicting $R_{t}$ requires tracking $\Lambda_{t}(i)$\footnote{krusellsmith}, the inequality in (ref) is valid irrespective of the presence of aggregate uncertainty.
In what follows we will analyze the equilibrium law of motion for investment , where we can introduce additional types of frictions, in particular those that arise from the production side of the economy. As in the case of consumption, we first analyze the aggregate investment decision rule in the frictionless case, which should satisfy the following second order difference equation:
where $\bar{s}_{0}\equiv\frac{I_{ss,0}}{Y_{ss,0}}$, the frictionless savings ratio.
A sufficient condition for a saddle point solution for the latter is $\alpha>\bar{s}$, that is, the steady state savings rate should be smaller than the elasticity of output with respect to capital. Denoting by $\rho_{1,0},\rho_{2,0}$ the two roots of the corresponding lag polynomial and letting $\rho_{2,0}$ be the unstable root, the aggregate frictionless investment has the following law of motion:
Consider now the case in which the true model has no real or nominal frictions and agents have rational expectations ($\mathcal{P}_{t,i} =\mathbb{P}_{t,i}$) where the corresponding information set is $\mathcal{F}_{i}\equiv(\eta_{i,t},k_{i,t},Z_{t})$. Following the literature, we assume that only a subset of $\left\{\mathcal{F}_{i}\right\}_{i\leq N}$ is observed by the econometrician, i.e. $\mathcal{F}_{t}=K_{t}$. The decision rule used by the econometrician can then be rewritten as:
where
The second equality holds due to the linearity of conditional expectations and aggregation. Then, for any $\tilde{K}_{t-j,j\geq0}$-measurable function $\phi (.)$, and using the law of total expectations, the following moment equality holds\footnote{Note that we have not used any knowledge for the exogeneity of $\tilde{Z}_{t}$ for this result to hold i.e. it would hold trivially in the case of an exogenous productivity shock.}:
The case of interest in this paper is therefore when $\int\mathcal{E}(.|\mathcal{F}_{i,t})d\Lambda_{t}(i)\neq \mathbb{E}^{M_{f}}(.|\mathcal{F}_{t})$, where $M_{f}$ is the frictionless probability model- or the benchmark model in the general case\footnote{To avoid confusion, I use frictionless and benchmark model interchangeably.}. Here the agents and the econometrician not only have different information, but they also have a different structure of the economy in mind. $\mathbb{E}^{M_{f}}(.|X_{t})$ is consistent with frictionless behavior and we thus implicitly assume that the frictionless part of the model is well specified. The mismatch between the agents' expectations and the econometrician's prediction could be due to differences in the models and/or information sets, which can lead to different decision rules, and these differences do not vanish on average.
Therefore, denoting aggregate optimal investment in the presence of frictions by $\tilde{I}_{con,t}$, which is the data generating process, and by $I_{t}^{\ast }$ the frictionless investment rule used by the econometrician, we can represent these differences in terms of the econometrician's observables as follows:
or equivalently, as
Similar to the case of consumption with liquidity constraints, real, nominal or informational frictions therefore generate a \textquotedblleft wedge\textquotedblright\ $\tilde{\lambda}_{t}$ with non zero conditional mean and therefore a moment inequality. We will consider a general example of constraints in adjusting capital, which can be rationalized by ad hoc adjustment costs in capital accumulation, occasionally binding constraints i.e. capital irreversibility and financial frictions. The specific examples together with an example on non-rational expectations can be found in the Appendix B (online). Below, we present the general example.
The preceding example dealt with distortions in the decision rules, which directly affect- and are therefore informative for- the transmission of shocks and welfare. Moreover, they can be easily linked to additional data. Nevertheless, frictions are always justified using the first order necessary conditions for agents' optimal decisions. In the next section, we present the general case together with a representation result, which proves useful in translating distortions to the first order conditions into observationally equivalent general equilibrium distortions to decision rules. One can therefore directly use the latter. Because we work with linearized models, second or higher order effects will be ignored. However, moment inequalities, would also appear in nonlinear models. The way we treat frictions does not depend on assumptions regarding the approximation error.
The general framework involves a system of expectational equations . Denote by $x_{i,t}$ the endogenous individual state, by $z_{i,t}$ the exogenous individual state, and by $X_{t}=\int x_{i,t}d\Lambda (i)$ and $ Z_{t}=\int z_{i,t}d\Lambda (i)$ the corresponding aggregate states. The optimality conditions characterizing the individual decisions are as follows:
where $\mathbb{E}(\epsilon _{i,t})=0$. We assume that the coefficients of the behavioural equations are common across agents. Relaxing this assumption would make the notation more complicated, but would not change the essence of our argument. We could also specify equilibrium conditions that involve past endogenous variables but this is unnecessary as we can always define dummy variables of the form $\tilde{x}_{i,t}\equiv x_{i,t-1}$ and enlarge the vector of endogenous variables to include $\tilde{x}_{i,t}$. Aggregating across individuals, the economy can be characterized by the following system:
We will refer to the economy with frictions as the triple $(H(\theta ),\Lambda ,\mathcal{E}_{i})$ where
We partition the vector $\theta $ into two subsets, $(\theta _{1},\theta _{2})$ where $\theta _{2}$ collects the parameters characterizing the presence and intensity of frictions. Thus, setting $\theta_2=0$, reduces the model to the frictionless economy and this is without loss of generality. Furthermore, in an economy with no frictions, prices efficiently aggregate all the information. Thus there is no need to distinguish between individual and aggregate information when predicting aggregate state variables.
For $w_{i,t}\equiv(x_{i,t},X_{i,t}, z_{i,t}, Z_{i,t})$, when agents are rational, and the model is linear ( or linearized) aggregate expectations for $(x_{i,t+1}^T,X_{t+1}^T)^T$ become as follows:
where $P_{j,j={1..6}}$ are the coefficients of the linear projection. By Rational expectations and since the coefficients $(G,F,L)$ are common across i, equilibrium consistency requires $P_{i,1}=P_1,P_{i,2}=P_2$ and therefore $P_1+P_2=P_5$ and $P_3+P_4=P_6$. Thus, as expected, aggregate conditional expectations collapse to $\mathbb{E}(X_{t+1}|X_t,Z_t)$, and the frictionless economy, $(H(\theta _{1},0),\Lambda ,\mathbb{E})$, can be summarized by the equilibrium conditions:
Using the decision rule in the expectational system (ref) and solving for the undetermined coefficients \footnote{See for example Marimon_Comp.}, a Rational Expectations equilibrium for $(H(\theta _{1},0),\Lambda ,\mathbb{E})$ holds under the following conditions:
Since we know the model up to $\theta _{2}=0$, we rearrange the equations of the economy with frictions into the known and the unknown part of the specification. Adding and subtracting the first order conditions of the frictionless economy we get that:
This system of equations cannot be solved without knowing $\mu _{t}$. Nevertheless, we characterize the relationship between $\mu _{t}$ and a set of candidate decision rules that depend on the endogenous states and some unobserved process, $\lambda _{t}$. Proposition 1 states sufficient conditions such that decision rules are consistent with $\mu _{t}$.
Proposition 1 states that if for an admissible parameter vector $\theta _{1}\in \Theta _{1}$ condition (24) is true, the decision rule $X_{t}^{\ast }=X_{t}^{f,RE}+\lambda _{t}$ generates the same restrictions as those implied using the perturbed (by $\mu _{t}$) first order equilibrium conditions. We focus on parameter vectors that yield determinate and stable equilibria in the frictionless economy, which implies a restriction on the stochastic behavior of $\lambda _{t}$.
While it is true that in many cases the sign of $\mathbb{E}_{t}\lambda_t$ can be directly deduced from the distortions to equilibrium conditions, Proposition 1 can also be useful in practice, as we can deduce the sign of the distortions to decision rules without solving for the otherwise unknown expectational system using (ref)\footnote{In more complicated cases where knowledge of $\Gamma$ is strictly required, condition (ref) can always be checked ex post.}. Moreover, Proposition 1 states conditions under which we can guarantee that the moment inequality restrictions are going to be consistent with the rest of the model, and therefore the implied reduced form.
It is useful to note that in certain situations one may be interested in characterizing frictions over time (and not just on average), and therefore we need to obtain a conditional model that generates $\lambda _{t}$. This requires imposing more restrictions on the stochastic behavior of $\lambda _{t}$. In the supplementary material, we adopt a model uncertainty approach in the spirit of HansenSargent2005,Hansen_nobel and we show that restricting the class of distributions for aggregate shocks is enough to obtain a unique conditional model for $\lambda _{t}$. For the rest of this paper we focus on the identification using a set of unconditional moment restrictions.
In this section we provide a formal treatment of identification in a linear(ized) Dynamic Stochastic General Equilibrium (DSGE) model based on moment inequality restrictions. First, we illustrate how our statistical representation of a DSGE model relates to the state space representation that is typically used for estimation. Building on ECTA1171, we will show necessary and sufficient conditions for partial identification of the model arising from the theoretical moment inequalities. We also show conditions under which conditionally over-identifying inequalities provide a more informative (smaller) identified set.
We base the analysis on the innovation representation of the solution to the linear(ized) DSGE model. This is the natural representation to use when there are differences in information between economic agents and the econometrician, as it takes into account that not all the state variables relevant the decision of agents are observable. We consider the following class of models:
where $K_{t}(\theta )$ is the Kalman gain and $a_{t}$ is the one-step ahead forecast error which could be derived from the corresponding state space representation:
where $\epsilon _{t}$ is the innovation to the shock vector $Z_{t}$.
By construction,
Therefore, the forecast error is a combination of the true aggregate innovations to the information sets of the agents, $\epsilon _{t}$, the estimation error of the state variable, ${X}_{t-1}-\hat{X}_{t-1,t-1}$, and the frictions, $ \lambda _{t}$. Let $N(\theta )\equiv vec(A(\theta )^{\prime },B(\theta )^{\prime },C(\theta )^{\prime })$, and assume that $\mathbb{E}(\epsilon _{t}|\sigma (\mathcal{F}_{t}))=0$, and $\mathbb{E}(\epsilon _{t}\epsilon _{s}^{\prime }|\sigma (\mathcal{F}_{t}))=\mathbf{1}(s=t)\Sigma _{\epsilon _{t}}$, where $\Sigma _{\epsilon _{t}}\succ 0$. Given $\mathbb{E}(\lambda _{t}|\sigma (\mathcal{F}_{t-1}))\geq 0$, we define the following conditional moment restriction:
For any inequality preserving function $\phi (.)$ of a random vector $\mathbf{Y}_{t-1}$ that belongs to the information set of the econometrician, the following holds:
for a random function $\mathcal{V}(\mathbf{Y}_{t-1})\in \lbrack 0,\infty ]$.
In order to study identification through estimating equations we need to make assumptions about the local identification of $\Theta _{0},$ given the value of $\mathbb{E}\mbox{\ensuremath{\mathcal{V}}(\ensuremath{ \mathbf{Y}_{t-1}})\ensuremath{\phi}(\ensuremath{\mathbf{Y}_{t-1}})}$. We resort to sufficient conditions that make the mapping from $\theta $ to the solution of the model regular, and thus assume away population identification problems (see, for example CanovaSala). We assume that $\Theta $ belongs to a compact subset of $\mathbb{R}^{n_{\theta }}$. Since certain parameters are naturally restricted, e.g. discount factors, persistence parameters or fractions of the population, and others cannot take excessively high or low values, assuming compactness is innocuous. We also need to acknowledge that due to \textsl{cross - equation restrictions}, which we denote by $\mathfrak{L}(\theta )=0$, the number of observables used in the estimation need not be equal to the cardinality of $\Theta $, i.e., $ n_{y}<n_{\theta }$ . ECTA1171 provide the necessary and sufficient conditions for local identification of the DSGE model from the auto-covariances of the data. We reproduce them below, with the minor modification that Assumption LCI-6 holds for any element of the identified set $\Theta _{0}$.
Given the maintained assumptions, we next characterize the identified set implied by the restrictions using macroeconomic data.
As already shown, the identified set is defined by the conditional moment inequalities in (ref),
In all of the identification results we assume the existence of appropriate random variables to construct unconditional moment restrictions from the conditional moment inequalities. Such instruments can be either past data or past state variables constructed with the Kalman filter. By construction, the latter are uncorrelated with current information, but they might be noisy. In the case of additional moment conditions, a.k.a supernumerary\footnote{We borrow this term from ECTA:ECTA1218.}, we need to show conditions under which they can further reduce the identified set, as $\Theta_{I}'$ will now satisfy multiple moment inequalities. Although the work of ECTA:ECTA1218 deals with linear models, we cannot use their results here for several reasons. First, the characterization of the identified set is done through the support function of $\Theta _{I}$\footnote{ The support function, which is $\sup (q^{T}\Theta ),\forall q\in \mathbb{R} ^{n_{\theta }}$, can fully characterize any convex set $\Theta $.}, which requires the identified set to be convex\footnote{One can alternatively work with the Aumann expectation of the non-convex random set, which is always convex. Nevertheless, this requires minimizing the support function of the random set for any value of $\theta$, which is costly in high dimensional settings like ours doi:10.1146/annurev-economics-063016-103658.}. In our case, since the stability conditions and the cross equation restrictions introduce on $\mathfrak{L} (\theta )$ and therefore nonlinear restrictions on $\Theta$, the identified set is not necessarily convex and its geometry is difficult to be known a priori. Second, we deal with moment conditions of general form and therefore additional conditions arising from more instruments in an IV setting is just a special case. We nevertheless also provide an adapted Sargan condition. Third, point identification cannot in general occur despite the moment inequalities as we deal with typically continuously distributed random variables.
Recall that the number of moment conditions we use for estimation depend on the number of observables. Assumption LCI-6 requires that there has to be enough (or the right kind) of observables such that a rank condition is satisfied. In our case, the number of observables used determines the number of first order conditions used for estimation. The minimum number ($r$) of observables required such that conditional identification is achieved ( Lemma 4 is satisfied) maps to the necessary first order conditions. For example, if we have $Y_{1}$ and $Y_{2}$ to estimate the model, and we only need $Y_{1}$ to conditionally identify $\theta $, then the $n_{\theta }\times 1$ first order conditions arising from $Y_{1}$ will be the necessary conditions. The rest of the conditions, i.e. those arising from $Y_{2}$ are then supernumerary.
Notice that given the structure of the class of models we consider in this paper, it is straightforward to find a re-parameterization that restores moment equalities, that is, for every observable $Y^{o}_t$, $\exists U_{t}:\mathbb{E}U_t\phi(\mathbf{Y_{t-1}})\equiv U\in\mathbb{R}^{+}$ that satisfies the restriction $\mathbb{E}(Y_{t}^{o}-{C(\theta )^{*}}\hat{X}_{t|t-1})\phi(\mathbf{Y_{t-1}})-U=0$. Then, for $\mathbf{U}\equiv(U_1,U_2..U_r)'$, the following set determines the map from $U_{I}$ to $\Theta_{I}$: $ $
Note that the proposition applies to the infeasible case of using optimal instruments. If the case of non-optimal but valid instruments, the set is again as sharp as possible, up to the information loss implied by the non-optimality of the instrument\footnote{Recent work in the literature proposes constructing instrument functions to avoid this information loss, see for example andrews2013inference. We nevertheless do not pursue this in this paper.}. It is also important to note that our characterization is based on the fact that the unobserved endogenous variables are integrated out using the Kalman filter. This implies that the only unobservables we need to deal with in the characterization of $\Theta_{I}$ is the aggregate shocks in the economy. Moreover, the set is robust to individual heterogeneity as long as it vanishes on aggregate. More importantly, given correct specification, $U$ is non zero if and only if there is a positive mass of agents that indeed face frictions.
Having established the determination of $\Theta_{I}$ using the minimum number of observables, we turn to the case of using a larger number of observables, including non-macroeconomic variables. Let $m_{\alpha ,t}(\theta )$ and $m_{\beta ,t}(\theta )$ denote the necessary and supernumerary moment functions for identifying $\Theta $, where $\mathbb{E}(m_{\alpha ,t}(\theta )|\mathbf{Y}_{t-1})=\mathcal{V} _{\alpha }(\mathbf{Y}_{t-1})\in \lbrack \underbar{\ensuremath{\mathcal{V}}} _{\alpha }(\mathbf{Y}_{t-1}),\bar{\mathcal{V}}_{\alpha }(\mathbf{Y}_{t-1}))$ and $\mathbb{E}(m_{\beta ,t}(\theta )|\mathbf{Y}_{t-1})=\mathcal{V}_{\beta }( \mathbf{Y}_{t-1})\in \lbrack \underbar{\ensuremath{\mathcal{V}}}_{\beta }( \mathbf{Y}_{t-1}),\bar{\mathcal{V}}_{\beta }(\mathbf{Y}_{t-1}))$. For notational brevity, we drop the dependence on $\mathbf{Y}_{t-1}$ hereafter.
Comparing these general bounds to the ones implied by the model equilibrium restrictions, $\underbar{\ensuremath{\mathcal{V}}}_{\alpha }=\underbar{\ensuremath{\mathcal{V}}}_{\beta }=0$ and $\bar{\mathcal{V}}_{\alpha }=\bar{\mathcal{V}}_{\beta }=\infty $. Due to the boundedness of $\Theta $ and the cross-equation and stability restrictions, the effective lower and upper bounds are likely to lie strictly within $[0,\infty)$ for every moment condition. Let $\phi (.)$ be any $\mathbf{Y_{t-1}}-$ measurable function for which $ \hat{m}_{\alpha ,t}(\theta ):=m_{\alpha ,t}(\theta )\phi_{t} $ and $\hat{m}_{\beta ,t}(\theta ):=m_{\beta ,t}(\theta )\phi_{t}$, $\mathbf{\hat{m}}_{\alpha }(\theta )$ and $\mathbf{\hat{m}}_{\beta }(\theta )$ the corresponding vectors, and $\mathbf{\bar{m}}_{\alpha }(\theta )$ and $\mathbf{\bar{m}}_{\beta }(\theta )$ the vector means.
Let $W$ be a real valued, possibly random weighting matrix, diagonal in $(W_{\alpha },W_{\beta })$. Furthermore, denote by $Q_{\alpha }$ the conditional expectation operator when conditioning on $W_{\alpha }^{\frac{1}{2}T}\mathbf{\hat{m}}_{\alpha }(\theta )$ and by $Q_{\alpha}^{\bot}$ the residual.
The following proposition specifies the additional restrictions that need to be satisfied by the additional moment conditions such that the size of the identified set becomes smaller. This is a general result, and it applies both to conventional cases where $n_{Y}>r$ or when data on non-macroeconomic variables is used.
The main argument behind Proposition 5 is the following. Suppose that the necessary moment conditions have no common information with the supernumerary conditions and that $W=I_{n_{\alpha }+n_{\beta }}$. From the minimization of $\mathbb{E}\frac{1}{2}(\mathbf{\bar{m}}-\bar{U}_{t})^{T}( \mathbf{\bar{m}}(\theta )-\bar{U}_{t})$ where $\mathbf{\bar{m}}(\theta )\equiv (\mathbf{\bar{m}}_{\alpha }(\theta ),\mathbf{\bar{m}}_{\beta }(\theta ))^{T}$, the first order condition is
which for $\bar{U}_{\alpha }\equiv\overline{Q_{\alpha }^{\bot }U_{t}}$ can be rewritten as $\mathbb{E}((\mathbf{\bar{m}}_{\alpha }(\theta )-\bar{U}_{\alpha })+(\mathbf{\bar{m}}_{\beta }(\theta )-\bar{U}^{\bot}_{\alpha }))=0$. By construction the two parts of the left hand side of the expression are independent, and therefore both have to be zero.
Notice that, by construction, the set of necessary moment conditions in 5.3. must have full rank, and this establishes a one-to-one mapping from $\Theta $ to the domain of variation of $U_{t}$, $[\underbar{U}(\mathbf{Y}_{t-1}),\bar{ U}(\mathbf{Y}_{t-1})]$. Thus, there exists an inverse operator $\mathcal{G} _{\alpha }$ such that $\theta =\mathcal{G}_{\alpha }(U_{t},\mathbb{P})$. Plugging this expression for $\theta $ in 5.4, we get that
This is a restriction on the values that $U_{t}$ can take in addition to the ones implied by the necessary conditions. A restriction on $U_{t}$ implies a restriction on the admissible $\Theta _{I}$ given the one-to-one relationship in 5.3. Notice that when the supernumerary conditions do not add any additional information, i.e. $m_{\alpha ,t}(\theta )\equiv m_{\beta ,t}(\theta )$, the restriction collapses to $Q_{\alpha }=Q_{\alpha }^{\bot }= \frac{1}{2}$. Given these identification results, we can analyze identification arising from any inequality restriction, and therefore any additional type of information can be potentially analyzed. Below, we discuss and formally show how qualitative survey data can provide additional restrictions that are informative about aggregative models of economies with frictions. Such information constrains further the stochastic properties of $\lambda_{t}$, and therefore the size of $\Theta_{I}$.
Additional data can be potentially linked to dynamic equilibrium models by augmenting the observation equation with moment restrictions. The latter can be motivated by the fact that not all data are explicitly modeled using structural equations, but they can nevertheless be used to sharpen economic and econometric inference, especially for unobserved processes. An example of such a process is the distortion $\lambda_{t}$. We focus on qualitative survey data because, as we illustrate, they contain distributional information that can be linked to the aggregate model and are informative about $\lambda_{t}$. This is especially true for economies with frictions, as the proportion of agents whose behavior is distorted is an important statistic. An example of a distributional statistic that has been routinely used to judge the validity of complete models is the proportion of firms which cannot change prices in New Keynesian models featuring a Calvo adjustment mechanism. Our focus is on more general information that is contained in qualitative surveys, and given that we deal with incomplete models, this information is able to discard a set of economic mechanisms, and not just one. We thus view our method as a significant generalization of the treatment of microeconomic information in dynamic equilibrium models.
Qualitative survey data are usually available in the form of aggregate statistics, where aggregation is performed over categories of answers to particular questions. As in micro-econometric studies, the categorical variable is a function of a continuous latent variable. In a structural context, there is a measurable mapping from the categories of answers to the random variables relevant to the decision of each agent. For example, if the question is of the type "How do you expect your financial situation to change over the next quarter" and the answer is trichotomous, i.e "Better", "Same" and "Worse", then the answers map to a set of partitions of the end of period assets $a_{t+2}$: $a_{t+2}\in \lbrack a_{t+1}-\epsilon ,\infty )$, $a_{t+2}\in (a_{t+1}-\epsilon ,a_{t+1}+\epsilon )$ and $a_{t+2}\in (-\infty ,a_{t+1}-\epsilon ]$. For this interpretation to hold, we need to assume that agents report their states or beliefs truthfully. Furthermore, denote by $\{S_{i,k,t}\}_{i\leq N}$ the survey sample over a period of length $T,$ for the $i$th respondent. Let $C_{l}^{k}$ be the $l^{\text{th}} $ categorical answer to question $k$ and $\hat{\xi}_{i,t,k}$ the respondent's choice. Given some weights on each category $w_{l}$, the available statistics are of the form: $ \hat{\mathcal{B}}_{t}^{k}=\sum_{l\leq L}w_{l}\sum_{i\leq N}1(\hat{\xi} _{i,t}\in C^{k}) $. Given truth telling, we can map the answer to agent beliefs i.e. there exists a mapping $h:\hat{\xi}_{i,t}\rightarrow \xi _{i,t}\equiv \mathcal{E} _{i,t}(x_{i,t+1}|x_{i,t},X_{t})$. Moreover, since the conditional expectation is a function of the information set, let ${B}_{l}\equiv \{(x_{i,t},X_{t})\in \mathbb{R} ^{2n_{x}}:h(x_{i,t},X_{t})\in C_{l}^{k}\}$, that is, ${B}_{l}$ belongs to the partition of the support of individual $x_{i,t}$ (or aggregate $X_{t})$ that corresponds to category $C_{l}^{k}$. Consequently, the survey statistic has the following theoretical form, which can be linked to the model:
For every $Y_{t}^{o}$ with a corresponding model based conditional expectation $Y_{t}^{m}\equiv \mathbb{E}(Y_{t}^{o}|M,\mathcal{F}_{t})$, Proposition 6 illustrates the additional restrictions implied by $\hat{\mathcal{B}}_{t}^{k} $. For these to hold, we need to make the following assumptions:
The assumption of conditional linearity for $\lambda_{i,t}$ enables us to claim that once the distortion is activated, it is a linear function of the states and the shocks, e.g. $\lambda_{i,t}:=\lambda_{1}x_{i,t-1}+\lambda_{2}z_{i,t}$. We motivate assumption S-4 after presenting the following proposition:
Notice that in the proof, we define $(\mathcal{X}^{\star}_{t},\mathcal{Z}^{\star}_{t})$ as the time dependent subset of the support such that $\lambda_{i,t}\neq 0$. In models of heterogeneous agents these boundaries usually depend on aggregate states and shocks. Therefore, this inequality will in general be strict unless $B_{t}=0$.
Moreover, in the proof we establish two facts. First, since the equilibrium conditions of the model with frictions depend on subjective conditional expectations, and the model with frictions is a smooth perturbation of the frictionless model, any probability statement on the subjective expectations translates to a probability statement on $\mu_{i,t}$. Second, any probability statement on $\mu _{i,t}$ is a probability statement on the solution of the model, and therefore on $ \lambda _{i,t}$. Given representation (ref), qualitative survey data have information on the quantiles of subjective conditional expectations of the agents. Therefore, survey data relate directly to the conditional probability of observing a friction, $\mathbb{P}_{t}(\lambda _{i,t}\geq 0)$, which implies the above restrictions. The latter cannot in general deliver point identification, as the vector $\hat{\mathbf{R}}_{t}=(\hat{R}^{1}_{t},\hat{R}^{2}_{t}...\hat{R}^{n_{x}}_{t})$ is hard to pin down unless specific assumptions are made. Monotonicity, which is a mild assumption, provides a bound.
Given Proposition 6, we can also establish the following corollary result regarding identification using survey data.
Corollary 7 has several implications. First, the set of admissible structures becomes smaller, and we can therefore make more precise statements regarding parameters and conditional predictions. Second, using identified set and the definition of $\lambda (Y_{t},\theta )$, a plug-in set estimate of the average distortion (wedge) in a macroeconomic variable $Y_{t}$ is $\mathbb{E}\lambda(Y_{t},\theta _{I}(Y))$. These estimates of distortions can be used in several ways, which we explore in the next section.
We first discuss briefly how $\Theta_{I}$ and $\lambda(\Theta_{I})$ can be obtained in practice. Given the quasi-structural framework, performing inference for $\Theta_{I}$ rather than $\theta_{0}$ seems a natural thing to do. Many of the parameters in dynamic macroeconomic models are semi-structural, and therefore $\theta_{0}$ itself does not have a very specific economic interpretation. We therefore use methods appropriate for constructing consistent estimators for the identified set (IdS) and confidence sets (CS) for the IdS. In the frequentist literature, several methods have been proposed for a general criterion function like subsampling CHT,ECTA:ECTA998 or the bootstrap for moment inequality models ECTA:ECTA1025. Nevertheless, in macroeconomic models parameter dimensions are high and pointwise testing is a vastly inefficient way to construct confidence sets, so we focus on computationally tractable methods to perform statistical inference using Markov Chain Monte Carlo algorithms (MCMC).
When $\theta_{0}$ is point identified and root-$n$ estimable, Chernozhukov2003293 have proposed to use simulation methods to do inference on $\theta$ using the quasi-posterior distribution, which is constructed using the relevant loss function $L_{n}(\theta)$ that defines $\theta_{0}$. Generalized Method of Moment (GMM) class of estimators can be easily embedded in this case. Denoting by $\pi(\theta)$ the prior distribution, simulation draws $\left\{\theta^{j}\right\}_{j\leq L}$ are obtained from
and upper and lower $100(1-\alpha)/2$ quantiles are used to conduct inference that has valid frequentist properties. Nevertheless, once point identification fails, one has to consider adjusting the method to accommodate for this.
More particularly, let $L_{n}(\theta)=n^{\frac{1}{2}}q_{n}(\theta)_{+}'W_{n}n^{\frac{1}{2}}q_{n}(\theta)_{+}$ be the criterion function to be minimized, where $q_{n}(\theta)$ are the moment functions to be used and $q_{n}(\theta)_{+}\equiv\max(q_{n}(\theta),0)=\min(-q_{n}(\theta),0)$. If $L_{n}(\theta)$ is stochastically equicontinuous, that is, there exists a $\Delta_{n}(\theta_{0})$ and $J_{n}(\theta_{0})$ such that $L_{n}(\theta)$ admits a quadratic expansion \footnote{
} then this allows us to assume a Central Limit Theorem (CLT) on $\Delta_{n}(\theta_{0})$. Redefining $Q_{n}(\theta)\equiv q_{n}(\theta)-b$ where $b$ is the bias term and $\tilde{L}_{n}(\theta,b)=n^{\frac{1}{2}}Q_{n}(\theta)'W_{n}n^{\frac{1}{2}}Q_{n}(\theta)$, we assume that the following CLT holds
Estimating equations arising in DSGE models involve smooth functions. This is particularly true if we focus on determinate equilibria, so that model specification is uniform across the parameter space. Combining them with smooth instrument functions is enough to guarantee stochastic equicontinuity.
Given (ref), liao2010 show that when using a cutoff rate $\nu_{n}$ such that $ 1\prec\nu_{n} \prec n$, and defining $\mathcal{A}_{n}:=\left\{ \max_{\vartheta} ln(\Pi_{n}(\vartheta|\mathbf{Y}))- ln(\Pi_{n}(\theta|\mathbf{Y})) \leq \nu_{n} \right\}$ then $d_{H}(\mathcal{A}_{n},\Theta_{I})\to_{p} 0$ where $d_{H}(.,.)$ is the Hausdorff distance between two sets \footnote{$d_{H}(.,.)=\max\left[\sup_{a\in A}\inf_{b\in B}||b-a||,\sup_{b\in B}\inf_{a\in A}||a-b||\right]$}. Moreover, the rate of convergence of the pseudo-posterior density outside the identified region is exponential. As is evident, in finite samples the identified will depend on $\nu_{n}$, so some robustness checks are required. In addition, since our criterion function vanishes when $\theta\in\Theta_{I}$, we do not need to adjust $\nu_{n} $ as suggested in CHT.
With regards to constructing confidence sets for the identified set, which is what we do in the empirical application, recent work by mcmcids provides a computationally attractive procedure to construct CS for the IdS and functions of it that have correct coverage from a frequentist perspective. Compared to Chernozhukov2003293, liao2010, cutoff values are based on quantiles of draws of the loss function $L_{n}(\theta)$ rather than draws from the quasi-posterior of the parameter vector and there is no maximization involved. Also, results extend to models with singularities, that is models in which a local quadratic approximation involves a non vanishing singular component, and parameters are not root-$n$ estimable. This is useful in case one wants to derive bounds based on extensions to non-parametric treatments of the survey based bounds. Although we find the method suggested very appealing, we apply it only in the case of estimating frictions. Proving validity for the testing procedure we will propose in this section is not trivial as it involves combining two independent MCMC chains, and we leave this interesting research for the immediate future.
We next motivate the proposed test that utilizes $\lambda_{Y}(\Theta_{I})$. When a complete model is estimated, the pseudo true vector $\theta_{0}$ is point identified. Given its value, we can estimate the predicted level of friction in each of the macroeconomic variable, $\mathbb{E}\lambda_{Y}(\theta_{0})$. Misspecification of this model may imply that the predicted friction does not lie in the identified set of distortions. Therefore, the distance from $\mathbb{E}\lambda_{Y}(\Theta_{I})$ becomes a sufficient statistic to judge whether the suggested model is properly specified. We propose a Wald statistic that tests whether the expected distance from the point estimate for the wedge from the parametric model to the identified set of wedges is different than zero for all (or some of) the observables. Since survey data provide more information on the wedge, this increases power to reject non-local alternatives. In addition, survey data regularizes the test, as in the absence of additional data the distribution degenerates and requires non-standard inference.
Letting $H_{0}:\theta_{p}\in\Theta_{s}$ and $H_{1}:\theta_{p}\notin\Theta_{s}$, the proposed statistic is as follows:
where $\lambda $ is the estimated friction obtained using either the identified point in the parametric model case, $\lambda _{p}$, or the identified set in the robust case, $\lambda _{s}$. Individual frictions are weighted by their respective estimate of standard deviation. The statistic measures the Euclidean distance between the wedge that arises in the parametric model, and the set of admissible wedges, adjusted for estimation uncertainty. Under the following conditions, the test is consistent and has asymptotic power equal to one against fixed alternatives.
In the supplemental material we show that using the non-parametric block bootstrap to compute $c_{\alpha}$ is valid, and we illustrate through a measurement error example that the bootstrap distribution coincides with the asymptotic distribution. Thorough examination of the performance of this test in small samples is a very interesting avenue of research, which deserves separate treatment. Another equally interesting topic for future research is to investigate which criterion function would deliver robustness of the testing procedure to possible misspecification of the benchmark model, which we have assumed away. See ECTJ:ECTJ332 for the issue of misspecification in moment inequality models.
Finally, we briefly discuss the refutability of the candidate model, in the sense of Breusch. The Null hypothesis, tests whether the point identifying restrictions on $\lambda_{Y}$ implied by the candidate model are satisfied on $\lambda_{Y,s}$, where the latter is only set identified. Since $(\lambda_{Y,s},\lambda_{Y,p})$ is constructed using $\theta_{1}$ as identified using the frictionless and complete model respectively (See Proposition 1 for the definition of $\theta_{1}$), $\lambda_{Y,p}$ is a function of the additional parameters indexing the CM, $\theta_{2}$. In the absence of cross parameter (equation) restrictions on $\Theta_{1}\times\Theta_{2}$, $\lambda_{Y,p}$ would necessarily lie within $\lambda_{Y,s}$ as the latter would be consistent with any $\theta_{2}\in\Theta_{2}$. However, in the presence of cross equation restrictions, $(\theta_{1},\theta_{2})$ lie in a strict subset of $\Theta_{1}\times\Theta_{2}$. Thus, the completion $\lambda_{Y,p}$ may not necessarily lie within the family of completions, $\lambda_{Y,s}$. Equivalently, $\exists \lambda\in\lambda_{s}$ that is not observationally equivalent to $\lambda_{p}$. In this sense, the candidate model is refutable only when it imposes restrictions on $\lambda_{y}:=U_{y}(\theta_1,\theta_2)$ for any observable $Y$, which implies restrictions on the reduced form. This is indeed true in the context of incomplete general equilibrium models; taking the unconditional expectation in condition (ref) of Proposition 1, implies that ($\Theta_{1},\Theta_{2}$) is not variation free and therefore $U_{y}(\theta_1,\theta_2)$ is restricted. With additional information, the set on which $H_{0}$ and $H_{1}$ are defined is reduced, and therefore the set of alternatives is narrower.
We apply the methodology to the case of Spain, where financial frictions have arguably played a significant role during the last decade. The benchmark economy we use features some frictions and since it is standard, we will directly introduce the log-linearized conditions. We consider a small open economy with capital accumulation, along the lines of 10.1257/aer.97.3.586 and Gali01072005. There are households, intermediate good firms, final good firms, government expenditure, and a foreign sector which is composed by infinitesimal symmetric economies.
The type of frictions we allow in the baseline model are those we do not have sufficiently informative survey data to implement our methodology. Thus, we keep the parametric Calvo type of friction in the wage setting by labor unions and in the price setting behavior of firms. All other frictions are going to be semi-parametrically characterized. We thus remove capital adjustment costs, and therefore Tobin's q becomes constant. This implies that the arbitrage condition between capital and bonds has no dynamics in the benchmark model.
Let $X_{1}^{o}$ denote the vector of variables that enter the moment equalities, $X_{2}^{o}$ the vector of variables used in the moment inequalities and $Z$ the vector of instruments. Model predictions are denoted with superscript '$m$'. The conditions we use are therefore:
The variables employed in estimation are Non Government Consumption expenditure (C), Hours (H), Inflation ($\pi $), Investment (I), Gross Domestic Product (Y), Wages (W), and the EONIA rate (R). Real variables are in per capita terms.W use lagged values of output and consumption as instruments, as both positively depend on the capital stock which is unobserved. We estimate the model using two different subsets of survey data from Spain, collected by the European Commission. Our sample period covers $1999Q_{1}$ to $2013Q_{4}$. Detailed information on this survey data can be found at: \url{http://ec.europa.eu/economy_finance/db_indicators/surveys/index_en.htm}
In the set of survey responses, we include responses to quarterly questions 1,2 and 11 from the Consumer Survey, that relate to the financial position of the household. We plot the time series in Appendix B. Credit constraints imply negative distortions to household consumption, and given that hours worked are complementary to consumption, they also imply negative distortions to labor supply and output. In addition, we use business survey data, in particular questions 8F4 and 8F6, relating to capital adjustment costs and financial constraints to production capacity. For the case of capital adjustment costs, we have restrictions on $Y$ and $I$ similar to those of the general example in Section 3. Financial constraints to productive capacity imply similar restrictions and lead to lower aggregate investment and output. We therefore choose $X_{1,t}^{o}\equiv (W,\pi ,R)$ and $ X_{2,t}^{o}\equiv (Y,I,H,C)$. We estimate the model using the 2-block RW-MCMC with uniform priors for all parameters\footnote{We keep the last 200000-300000 draws for inference.}. Since the survey based moment restrictions require estimating the nuisance parameters $\lambda^{1}$ (see Proposition 6), for a given draw of $\Theta_{1}$, we obtain $\hat{\lambda}^{1}$ by ordinary least squares. This approximation can, in theory, weaken the informativeness of the additional restrictions but we do not observe this in our results.
We obtain the $95\%$ confidence sets of wedges, that is, a range for the standardized wedge in each observable that is consistent with both macroeconomic and survey data, $\lambda_{y}\equiv (\mathbb{E}\mathbb{V}_{y|\hat{x}}\mathbb{E}\mathbb{V}_{\hat{x}})^{-1}\mathbb{E}\hat{X}_{t,t-1}(Y_{t}-C(\Theta_{I}))$, which under linearity is equivalent to $(\mathbb{E}\mathbb{V}_{y|\hat{x}})^{-1}\lambda_{y}$\footnote{We actually compute $\lambda_{t}$ using quadratic programming as in the second section of the online Appendix and then take the average. The result is equivalent to a standardized wedge.}. These estimates are the empirically relevant wedges that any model featuring financial frictions and adjustment costs should produce over the whole sample. We plot them (in red) in the left panel of Figures 1-6, in Appendix A.
We employ a complete model (CM) featuring Bernanke_Gertler (BGG) type of frictions (similar to DEGRAEVE20083415) and estimate it using the same macroeconomic aggregates as the IM using full information MCMC, where we also rely on mcmcids to obtain $\Theta^{cs}_{CM}$.
Despite that the model features financial frictions on the firm side (the "entrepreneurial sector"), and not borrowing constrained consumers, its implications can still be tested against the incomplete model (IM), which is robust to both types of frictions.
We plot (in blue) the corresponding confidence sets for the wedges that are consistent with the complete model. To facilitate the comparison, we also produce an alternative comparison based on QQ plots, which gives information on the support of each of these estimates. Note that in some of the cases the support is completely different.
In Table 1 (Appendix A), we display the corresponding confidence sets for $\Theta$ using survey data. Confidence sets for $\Theta_{IM}$ without using survey data and the corresponding wedges can be found in Appendix B.
A key observation is the fact that the sign of the wedge confidence sets that correspond to the restrictions imposed to identify the IM are in accordance with the sign of the distortions identified with the CM. While both the CM and IM identify negative distortions to output, the former identifies a significantly lower level of distortions. As is typical, empirical versions of accelerator type of models ignore the output costs (bankruptcy costs) as they are deemed to be numerically unimportant. Some of the neglected variation is indeed captured by the high estimates of $\phi_{p,CM}$ relative to $\phi_{p,IM}$.
Regarding inflation, using the CM we cannot reject zero frictions to $\pi_t$, while the IM identifies significantly negative distortions. Moreover, the CM identifies a much higher slope of the Phillips curve\footnote{ $\kappa:=\frac{(1-\iota_{p}\xi_{p})(1-\xi_{p})}{(1+\beta\iota_{p})(\xi_{p}(1+(\phi_{c}-1)\epsilon_{p}))}$ where $\epsilon_{p}$ (Kimball aggregator) is calibrated to 10. }, which is consistent with its "inability" to generate negative distortions to inflation.
With regard to the nominal interest rate, both the CM and the IM identify negative distortions (with no evidence of different magnitudes). Mechanically, this is due to the Taylor rule that pins down $r_t$. Nevertheless, there is also a deeper insight which comes from steady state reasoning. In an economy with uninsured idiosyncratic risk on both firms and consumers, the steady state interest rate is lower than in the case of complete markets, and as shown by ANGELETOS20071, this does not contradict negative distortions to capital accumulation due to the presence of a risk premium.
Furthermore, we cannot reject the hypothesis that the CM is consistent with zero distortions to $c_t$, while the IM identifies negative distortions. Moreover, the estimates of risk aversion are much lower in the IM compared to the CM. One of the reasons is the fact that the CM ignores credit constraints to consumers, and therefore requires much higher estimates in risk aversion to match the corresponding negative distortions to consumption.
With regard to hours worked, both the IM and the CM identify negative distortions which are statistically different. Since $\sigma_{c,IM}$ is much smaller than $\sigma_{c,CM}$, negative distortions to consumption are consistent with less distortions to hours in the IM. Moreover, the CM identifies a slightly higher (inverse) Frisch elasticity ($\sigma_{l,CM}$) and does not reject zero distortions to $w_t$, while the IM identifies positive distortions. The latter observation is not robust to excluding the survey data as shown in Table 2.
Finally, the IM and CM identify the same negative distortions to investment. Since no marginal $\Theta_{cs}$ has an extremely narrow support, this result can be possibly explained by the mapping from parameters to investment that is nearly not injective. This is not a problem per se, as the MCMC method we have used to obtain confidence sets in both the IM and the CM is robust to lack of identification.
The non overlapping standardized output, inflation and labour wedges provide enough evidence to reject the hypothesis that the SW-BGG model implies similar distortions to those identified by the IM. This finding also corroborates existing evidence of the lack of uniform fit for such a model over the entire business cycle ( DELNEGRO2016391 for the case of the US).
In this paper we propose a new inferential methodology which is robust to misspecification of the mechanism generating frictions in dynamic stochastic economies. We characterize wedges in equilibrium decision rules in a way which is consistent with a variety of generating mechanisms. We use the implied restrictions to partially identify the parameters of the benchmark model, and to obtain a set of admissible economic relationships. Moreover, we formally characterize set identification given the number of observables. Regarding the latter, beyond macroeconomic data, we show how qualitative survey data can provide additional distributional information, which is crucial in economies with ex post heterogeneous and frictions. Survey data provide a sufficient statistic for this information that would otherwise have to be generated by a model. We show how to exploit this additional information to test parametrically specified models of frictions. We apply our methodology to estimate the distortions in the Spanish economy due to financial frictions and adjustment costs using an small open economy version of 10.1257/aer.97.3.586 model and qualitative survey data collected by the European Commission. We investigate the adequacy of Bernanke_Gertler type of financial frictions.
In general, our work shows that adopting a robust approach to inference and using the information present in surveys is a fruitful way of dealing with lack of knowledge about the exact mechanisms generating frictions.
Our work is limited to dynamic linear economies and has focused on deviations of the representative agent approximation from the underlying heterogeneous agent economy. Possible extensions could study deviations based on a benchmark heterogeneous model. buera_moll have recently shown that the distortions generated in an aggregate equilibrium condition can depend on the type of heterogeneity present in the economy. Our approach is robust to this criticism. The methodology does not rest on observing residuals from representative agent frictionless economies - we just impose moment inequality restrictions theoretically motivated by deviations from the frictionless economy. In addition, if heterogeneity has additional implications for the form of these restrictions, they can be taken on board. In addition, we impose weak moment inequalities. This is important when heterogeneous distortions in decision rules cancel out. Finally, we impose restrictions implied by $\mu _{t}$ on all of the variables and thus take general equilibrium effects into account. Nevertheless, future work could focus on investigating the robustness of our methodology in environments where some heterogeneity is ignored when imposing the identifying restrictions.
Finally, we have assumed that aggregated survey data are not systematically biased and strategy proof. Analysing the consequence for inference would be another fruitful avenue of research.