The (Ir)Relevance of Rule-of-Thumb Consumers for U.S. Business Cycle Fluctuations

We estimate a medium-scale model with and without rule-of-thumb consumers over the pre-Volcker and the Great Moderation periods, allowing for indeterminacy. Passive monetary policy and sunspot fluctuations characterize the pre-Volcker period for both models. In both subsamples, the estimated fraction of rule-of-thumb consumers is low, such that the two models are empirically almost equivalent; they yield very similar impulse response functions, variance


INTRODUCTION
Most empirical papers investigating U.S. business cycle fluctuations rely on Representative Agent New Keynesian (RANK, henceforth) models where monetary policy is active and the so-called Taylor Principle holds.This is the case of Smets and Wouters (2007), for example, which has become the benchmark for estimated models for the U.S. economy.However, some seminal papers in the literature ascribe the occurrence of the Great Inflation episode to "bad policy" of the Federal Reserve.Clarida, Galí, and Gertler (2000) point toward self-fulfilling expectations due to indeterminacy arising from passive monetary policy as an explanation of the high inflation episode in the U.S. during the 1970s.Lubik andSchorfheide (2003), ,2004) propose a method to quantitatively assess the importance of equilibrium indeterminacy and the propagation of fundamental and sunspot shocks.Following Lubik and Schorfheide (2004) methodology and allowing for nontrivial monetary and fiscal interactions, Bhattarai, Lee, and Park (2016) find that passive monetary and passive fiscal policy regime prevailed in the pre-Volcker period, which resulted in equilibrium indeterminacy, while active monetary and passive fiscal policy prevailed post-Volcker.According to these views, the switch from passive to active monetary policy brought about a stable and determinate environment since the early 1980s.In a related study, Boivin and Giannoni (2006) find that this switch has also been instrumental in reducing observed output and inflation volatility. 1  All these papers share two common features, they: (i) focus on small-scale models and (ii) rely on the standard Representative Agent models.This paper relaxes these two assumptions to investigate the role of (a particular sort of) heterogeneity in the transmission of shocks on U.S. business cycle and in the narrative about the U.S. monetary policy using an empirically relevant medium-scale DSGE model.
Regarding (i), there have been recent progress from a methodological point of view.Bianchi and Nicolò (2019) propose a new method for solving and estimating linear rational expectations (LRE) models under indeterminacy that can handle more complex medium-scale models and can be implemented even when the boundaries of the determinacy region are unknown. 2Building on this, Nicolò (2020) estimates the medium-scale model of Smets and Wouters (2007) for different subsamples while allowing for indeterminacy.
Regarding (ii), a notable exception is Bilbiie and Straub (2013), where the authors estimate a small-scale two-agents New-Keynesian (TANK) model to study the Great Inflation and the Great Moderation periods in the U.S.They put forward an alter-1.Hirose, Kurozumi, and Van Zandweghe (2020), using an estimated NK model with positive trend inflation, show that both systematic monetary policy as well as changes in the level of trend inflation resulted in a switch to determinacy after 1982.
native explanation of the Great Inflation episode arguing that the different monetary policy transmission mechanisms that characterized those periods could be related to a structural change in asset market participation.The main assumption is the presence of the so-called rule-of-thumb (ROT, henceforth) consumers.In line with the seminal papers by Galí, López-Salido, and Vallés (2007) and Bilbiie (2008), ROT consumers are liquidity-constrained households who cannot access financial and capital markets and thus cannot smooth consumption.Bilbiie and Straub (2013) build on Bilbiie (2008)'s finding of an inverted-aggregate-demand-logic (IADL) mechanism, which leads to an upward sloping AD curve for a high enough share of ROT.They find evidence of both a passive monetary policy and limited asset market participation during the pre-Volcker period, thereby implying determinacy in an IADL environment. 3They further show that as the share of agents participating in asset markets had increased, the IS curve's slope flipped and policy became active that results in equilibrium determinacy for the Great Moderation period.The change in the sign of the IS curve's slope in the early 1980s is also documented by Bilbiie and Straub (2012) using single-equation reduced-form GMM estimation.
The ROT assumption enables to move from the standard Representative Agent (RANK) specification while keeping the model tractable from an analytical point of view (see Bilbiie 2020).The presence of ROT consumers proved also to be beneficial for New Keynesian models from an empirical point of view in reproducing empirical dynamics in response to government spending shocks (Galí, López-Salido, andVallés 2007, Bilbiie, Meier, andMüller 2008), investment shocks (Furlanetto, Gisle J. and Martin 2013), and technology shocks (Furlanetto and Seneca 2012).Kaplan, Violante, and Weidner (2014), among others, show that liquidity-constrained agents could be relevant empirically.Moreover, the ROT assumption has been introduced in estimated operational macroeconomic models.Nowadays, important institutions such as the Federal Reserve (Brayton, Laubach, and Reifschneider 2014) and the European Commission (Kollmann et al. 2016) are including this type of agents in their benchmark-estimated models used for forecasting and for the analysis of macroeconomic issues.Coenen, Straub, and Trabandt (2012), Forni, Monteforte, and Sessa (2009) and Albonico, Paccagnini, and Tirelli (2019), among others, estimate mediumscale DSGE models with ROT for the Euro area.For the U.S., the literature focuses more on standard Representative Agent models such as Smets and Wouters (2007).
In this paper, we investigate the relevance of ROT consumers in explaining U.S. business cycle fluctuations, revisiting the findings of Bilbiie and Straub (2013).We introduce the presence of ROT consumers in a medium-scale DSGE model with all the standard bells and whistles similar to Smets and Wouters (2007).We then estimate the model over two different subsamples (the pre-Volcker and the Great Moderation periods), while allowing and testing for (in)determinacy, and compare our results with the standard RANK specification.In this context, indeterminacy can arise due 3. Haque, Groshenny, and Weder (2021) also find support for determinacy in the pre-Volcker period, albeit for different reasons.In the presence of substantial wage rigidity and well-identified commodity price shocks, they show that the Federal Reserve responded aggressively to inflation but negligibly to the output gap in the pre-Volcker period.to different combinations of parameters.For instance, for low values of the degree of ROT, indeterminacy can arise due to passive monetary policy, dubbed the Standard Aggregate Demand Logic (SADL), as in Lubik and Schorfheide (2004).In contrast, for high enough values of the degree of ROT share, IADL might be in place as in Bilbiie (2008), resulting in either indeterminacy due to active monetary policy or determinacy if monetary policy is passive, as found by Bilbiie and Straub (2013).Our paper is also related to Nicolò (2020), who estimates the model of Smets and Wouters (2007) for different subsamples while allowing for indeterminacy and employing the methodology proposed by Bianchi and Nicolò (2019).He shows that monetary policy was passive in the Great Inflation period and active afterward.Similar to Lubik and Schorfheide (2004), he finds that indeterminacy manifested primarily by altering the propagation of structural shocks, while sunspot shocks played only a limited role in explaining macroeconomic volatility.
We find that introducing ROT consumers in a medium-scale model is irrelevant to explain aggregate business cycle fluctuations in U.S. data.The reason is that the estimated fraction of ROT consumers is so low that it is not affecting the dynamics of the model compared to a standard representative agent model.First, the estimations of both a model with ROT and one without (RANK) point to an indeterminate equilibrium in the pre-Volcker period, due to passive monetary policy, and to a determinate equilibrium in the post-Volcker period with active monetary policy, as in Lubik and Schorfheide (2004) and Nicolò (2020).Second, in the pre-Volcker period, the log-likelihoods of the two models are very close, while in the latter period, the RANK model is preferred by the data.Third, in both subsamples, the RANK and ROT models yield almost the same impulse response functions, variance, and historical decompositions, such that they share the same narrative of U.S. business cycle fluctuations.Therefore, the presence of ROT consumers is not substantive to explain these fluctuations.
Our results show that the estimated fraction of ROT consumers is quite low: 22% and 11% for the two subsamples, respectively.In contrast, Bilbiie and Straub (2013) find the fraction of ROT consumers to be higher: 50% in their pre-Volcker sample and 20% in their post-1984 sample.One might wonder why our estimates turn out to be lower and whether these are in line with the evidence in the literature.Note that while Bilbiie and Straub (2013) estimate a small-scale NK model with ROT consumers, we embed ROT consumers in a model with richer dynamic and stochastic structure along the lines of Smets and Wouters (2007).Then, one possible interpretation is that missing internal propagation and stochastic shocks are misinterpreted as high degree of ROT consumers in estimated small-scale models.In fact, the estimated ROT fraction for the pre-Volcker period in Bilbiie and Straub (2013) turns out to be high enough for the economy to be in the so-called IADL region, whereby monetary contractions turn out to be expansionary and passive monetary policy implies determinacy.On the other hand, we find that the data favor a parameterization corresponding to the SADL region and a corresponding lower fraction of ROT consumers.This finding suggests that the data prefer a model whereby monetary policy has conventional effects-contractionary (expansionary) policy shocks reducing (raising) inflation and economic activity and active (passive) monetary policy implying determinacy (indeterminacy)-when looked at through the lens of an empirically relevant medium-scale model.In addition, we find that the estimated degree of ROT consumers falls in the second subsample, which is in line with Bilbiie and Straub (2013) finding.One reason behind this decline could be changes that took place in financial markets around 1980, which led to financial market liberalization and broader participation in asset markets.Overall, the estimated values for the share of ROT are in line with the aggregate implications of the empirical literature on the marginal propensity to consume (MPC) out of transitory income shocks in microdata (e.g., Johnson, Parker, and Souleles 2006, Parker et al. 2013, Kaplan, Violante, and Weidner 2014).In a recent paper, Bilbiie, Primiceri, and Tambalotti (2022) estimate a tractable Heterogeneous-Agents New Keynesian (HANK) model for the U.S. using data from 1954 to 2019 where they calibrate the fraction of ROT consumers at 0.2.Moreover, using survey data from the U.S. Survey of Consumer Finances, Kaplan, Violante, and Weidner (2014) find the fraction of the so-called poor Hand-to-Mouth consumers to be 14% on average in the U.S. between 1989 and 2010. 4 Our main finding, that including ROT in a RANK model does not change the interpretation of aggregate U.S. fluctuations, does not mean obviously that modeling ROT, or heterogeneous agents more generally, is not important to explain other dimensions of the economy.In recent years, a growing body of literature evolved from simple TANK models to the more complex HANK models, following Kaplan, Violante, and Weidner (2018).However, the ROT assumption is sufficiently simple to allow us to explore the indeterminacy versus determinacy issue in the context of an empirically relevant medium-scale DSGE model, using the Bianchi and Nicolò (2019) methodology.This would not have been feasible with a full HANK model.Moreover, Debortoli and Galí (2017) compare the implications for business cycles fluctuations between an HANK model and a simpler TANK model with ROT consumers.Identifying the three sources of heterogeneity arising in the HANK framework, 5 they show that the most important component of heterogeneity for output fluctuations is the consumption gap between the two types of consumers (constrained and unconstrained).Interestingly, they show that a simple TANK model, with a constant share of constrained households and no heterogeneity within either type, approximates the implications of an HANK model regarding output fluctuations reasonably well, thereby supporting the use of a TANK model in quantitative analysis of U.S. business cycle fluctuations.Notwithstanding, our empirical findings go a step further and suggest that, in fact, estimating a TANK model does not materially change the estimated shocks and frictions relative to a RANK model.As such, our results point toward the irrelevance of ROT consumers and imply that a medium-scale RANK model, like Smets and 4. Poor hand-to-mouth consumers are similar to our ROT consumers.5. Namely, (i) changes in the average consumption gap between constrained and unconstrained households, (ii) variations in consumption dispersion within unconstrained households, and (iii) changes in the share of constrained households.Wouters (2007), does not need to be enlarged by the presence of ROT to study the drivers of U.S. business cycle fluctuations.Along these lines, our results reinforce Bayer, Born, and Luetticke (2020)'s findings, who show that adding data on inequality does not affect aggregate fluctuations in the U.S. 6  The paper is organized as follows.Section 2 briefly presents the model.Section 3 explains the estimation strategy based on Bianchi and Nicolò (2019).Section 4 discusses the main results and Section 5 provides some robustness, while Section 6 concludes.

MODEL
We develop a dynamic stochastic general equilibrium (DSGE) model following Smets and Wouters (2007) in particular.Smets and Wouters (2007) model has become the workhorse model for the empirical analysis of the U.S. economy.It includes all the standard features and frictions of New-Keynesian models, while still remaining tractable.We depart from their model only in few aspects.First, we introduce the presence of ROT consumers, on the footsteps of Galí, López-Salido, and Vallés (2007) and Bilbiie (2008).There is a fraction θ of households who do not have access to financial and capital markets and consume all their disposable labor income in each period.Second, we consider a separable utility function in consumption and hours, to stay close to Bilbiie andStraub (2012, 2013).Wage decisions are made by unions that optimally reset the nominal wage according to a Calvo (1983) scheme.The supply side is composed of final producers operating under perfect competition and intermediate monopolistically competitive firms.Prices are sticky following a Calvo (1983) mechanism.Intermediate goods are packed by final firms with a Kimball (1995) aggregator.
The model includes the usual frictions considered in New-Keynesian mediumscale models: external habits in consumption, variable capital utilization, investment adjustment costs, sticky wages and prices, indexation on past, and trend inflation.
Given that the model is rather standard, we leave a more detailed description of the model equations in the Appendix.

Households
There is a continuum of households indexed by i ∈ [0, 1].A share 1 − θ of households are Ricardian (i = o), such that they can access financial markets, hold government bonds, accumulate physical capital, and rent capital services to firms.The remaining θ households are ROT consumers (i = rt), as specified above.
6. Nevertheless, Bayer, Born, and Luetticke (2020) show that the estimated shocks from their HANK model have significantly contributed to the evolution of U.S. wealth and income inequality.
Households maximize the following utility function: where individual and aggregate consumption (c i t , c t ) are adjusted by the deterministic growth trend g z , h i t stands for individual hours worked, 0 < β < 1 is the subjective discount factor, σ measures the inverse of the intertemporal elasticity of substitution, and φ l is the inverse of Frisch elasticity.The parameter 0 < b < 1 measures the degree of external habits in consumption.
Ricardian households budget constraint is standard: The capital accumulation equation is: with the investment adjustment costs function defined as: where δ is the capital depreciation rate and γ I is a parameter measuring the degree of investment adjustment costs.ε i t is a shock to the marginal efficiency of investment (see Justiniano, Giorgio E. and Andrea 2010).
ROT households maximize (1) subject to the following budget constraint: (5) A generic aggregate variable is expressed as

Labor Market
Each household supplies the bundle of labor services t that firms demand.For each labor type j, the wage setting decision is allocated to a specific labor union.At the given nominal wage W j t , households supply the amount of labor that firms demand.Following Colciago (2011), demand for labor type j is split uniformly across the households, so that households supply identical amount of labor services, h t = h i t .λ w t represents an exogenous shock to the net wage markup.
Wage setting.Nominal wages are sticky à la Calvo (1983).In each period, union j can optimally reset the nominal wage with probability (1 − ξ w ).Those unions that cannot reoptimize the wage adjust the wage according to the scheme , where π is the steady-state (or trend) inflation rate.Non-reset wages are partially indexed to past inflation and trend inflation, with χ w ∈ [0, 1] allowing for any degree of combination of indexation between the two components.The aggregate wage is thus: where Wt is the optimal reset wage.Following Colciago (2011), we assume that the representative union's objective function is a weighted average (1 − θ , θ ) of the two household types' utility functions, subject to the labor demand (2) and ( 5).The resulting first-order condition is:

Production
Final good firms.The final good Y t is produced under perfect competition.A continuum of intermediate inputs Y z t is combined as in Kimball (1995).The final good producers maximize profits: with G strictly concave and increasing and G(1) = 1 and λ p t is the net price markup, which is assumed to be exogenous.
Intermediate good firms.Intermediate firms z are monopolistically competitive and use as inputs capital and labor services, u z t K z t and h z t , respectively.The production technology is a Cobb-Douglas function where are fixed production costs.ε a t is a temporary total factor productivity shock.The term g z is a deterministic growth trend.
Price setting.Intermediate goods prices are sticky à la Calvo (1983).A firm z can optimally reset its price with probability (1 − ξ p ). Firms that cannot reoptimize adjust the price according to the scheme , where χ p ∈ [0, 1] allows for any degree of combination of indexation to past or trend inflation.
The aggregate price index is: where The representative firm chooses the optimal price Pz t that maximizes expected profits subject to the demand schedule.The resulting first-order condition is: where

Government
The government budget constraint is: We assume that it is balanced every period.Government spending evolves exogenously.
The monetary authority sets the nominal interest rate according to the same Taylor rule as in Smets and Wouters (2007): where Y f lex t is the level of output prevailing in a flexible prices and wages environment and ε r t is an exogenous interest rate shock.

Data
To estimate the model, we use Bayesian techniques and the measurement equations that relate the macroeconomic data to the endogenous variables of the model are defined as: where dl denotes the percentage change measured as log difference, l denotes the log, and hatted variables denote log deviations from steady state.The observables are the seven quarterly U.S. macroeconomic time series used in Smets and Wouters (2007), and they match the number of fundamental shocks that affect the economy.
The series considered are: the growth rate in real GDP, consumption, investment and wages, log of hours worked, inflation rate measured by the GDP deflator, and the federal funds rate.Similar to Smets and Wouters (2007), γ denotes a deterministic growth trend common to the real variables GDP, consumption, investment, and wages (γ = 100(g z − 1)), h is the (log) steady-state hours worked (normalized to zero), π is the quarterly steady-state net inflation rate, and R is the quarterly steady-state net nominal interest rate.
We include seven fundamental shock processes in the estimation (the same as in Smets and Wouters 2007): a technology shock, a risk premium shock, an investment shock, a monetary policy shock, a government spending shock, a price markup shock, and a wage markup shock.All shocks have an autoregressive component of order 1.The first four shocks are AR(1) processes with i.i.d.normally distributed innovations.The government spending shock is also correlated with the technology shock.The two markup shocks also have an MA(1) component.

Calibration and Priors
We calibrate a number of parameters.In particular, the discount factor β is fixed at 0.9975, corresponding to a 2.6% annual real interest rate at the prior mean.The steady-state depreciation rate δ is 0.025, corresponding to a 10% depreciation rate per year.The elasticity of the demand for goods is set at 6, which implies a 20% net price markup in steady state.We set the government spending-to-GDP ratio at 20%, in line with its sample average.Table 1 reports the prior distributions for the structural parameters of the model and the exogenous processes that drive the dynamics of the economy, which are set in accordance with Smets and Wouters (2007).The only differences relate to the Taylor rule coefficient associated with the response of the monetary authority to changes in the inflation rate (φ π ) and the fraction of ROT consumers (θ ), which is absent in the RANK model of Smets and Wouters (2007).For φ π , Smets and Wouters (2007) specify a normal distribution truncated at 1, centered at 1.50 and with standard deviation 0.25 and impose determinacy.Instead, here, we want to deal with the possibility of indeterminacy. Figure 1 shows the determinacy/indeterminacy regions as φ π and θ vary.For low values of the fraction of ROT agents, the model behaves like a standard NK model, so that it admits a unique stable rational expectations equilibrium when the Taylor principle is satisfied, that is, φ π > 1.However, as it is well known from the literature, when θ is sufficiently high the result flips, so that the model needs a passive monetary policy, that is, φ π < 1, for determinacy to arise.Bilbiie (2008) call this possibility the IADL.The threshold value for θ that makes the model enter the IADL region of the parameter space depends on the properties of the model and on parameter calibration.While Bilbiie (2008) show that in standard three-equation NK model with ROT agents this threshold value for θ can be relatively low, Colciago (2011) shows that nominal wage rigidity increases the threshold value substantially (see also Ascari, Colciago, and Rossi 2017). 7In our medium-scale model, with parameters calibrated at their prior means, this threshold value in Figure 1 is around 0.6.Moreover, other possibilities arise in a medium-scale model, because some parameter combinations yield instability and some other a degree of indeterminacy greater than one.The next section explains how we deal with the determinacy/indeterminacy issue in the estimation, following Bianchi and Nicolò (2019).Regarding priors, we consider a prior that assigns roughly equal probability of observing indeterminacy as 7. Few papers analyze determinacy region in a medium-scale model with ROT.Motta andTirelli (2012, 2014) highlight the role of the interaction between the fraction of ROT and the degree of habits in consumption.Neither paper includes capital and the related frictions.Albonico, Paccagnini, and Tirelli (2019) show the results for the determinacy regions of a medium-scale model with respect to both the degree of habits and its specification.
well as a unique solution.In particular, for φ π , we set a flatter normal prior distribution centered at 1 and with standard deviation 0.35 following Nicolò (2020).The fraction of ROT θ is assumed to follow a Beta distribution with mean 0.3 and standard deviation 0.1, in line with Bilbiie and Straub (2013).

Methodology
Bianchi and Nicolò (2019) develop a new method to solve and estimate LRE models that accommodates both determinacy and indeterminacy.Their characterization of indeterminate equilibria is equivalent to Lubik andSchorfheide (2003, 2004) and Farmer, Khramov, and Nicolò (2015).We closely follow Bianchi and Nicolò (2019) and in the following briefly sketch their methodology while referring the readers to their paper for detailed exposition.The LRE model can be compactly written in the canonical form as: where s t is the vector of endogenous variables, is the vector of model parameters, ε t is the vector of fundamental shocks, and η t are one-step ahead forecast errors for the expectational variables.Bianchi and Nicolò (2019) propose to augment the original model by appending an independent process, which could be either stable or unstable.First, for our medium-scale ROT model with priors set as above, the occurrence of indeterminacy of degree two (or higher) is a-priori very low and so in what follows we focus on one degree of indeterminacy.Second, the priors are such that there is roughly a 50-50 prior probability of determinacy and one degree of indeterminacy.Following Bianchi and Nicolò (2019), we append the following autoregressive process to the original LRE model: where ν t is the sunspot shock and η f ,t can be any element of the forecast error vector η t .As proven by Bianchi and Nicolò (2019), it is without loss of generality that we include the forecast error associated with the inflation rate η π,t = π t − E t−1 (π t ) as η f ,t in the augmented representation.The key insight consists of choosing this auxiliary process in a way to deliver the "correct" solution.When the original model is determinate, the auxiliary process must be stationary so that the augmented representation also satisfies the Blanchard-Kahn condition.Accordingly, we set ϕ * such that its absolute value is inside the unit circle.Then the autoregressive process for ω t does not affect the solution for the endogenous variables s t .On the other hand, under indeterminacy, the additional process should be explosive so that the Blanchard-Kahn condition is satisfied for the augmented system, though it is not for the original model.Hence, the absolute value of ϕ * is set outside the unit circle.Under indeterminacy, we estimate the standard deviation of the sunspot shock, σ ν , and so, we specify a uniform distribution over the interval [0, 1] following Nicolò (2020).In addition, the newly defined sunspot shock, ν t , is potentially related to the structural shocks of the model.Nicolò (2020) finds that the correlation between this newly defined sunspot shock and the price markup shock is the only one statistically different from zero, implying that the price markup shock has a contemporaneous effect on inflation through this channel.Hence, in what follows, we report estimation results corresponding to the correlations with the remaining shocks set to zero. 8For the correlation between the sunspot shock and the price markup shock, we set a uniform prior distribution over the interval [−1, 1] as in Nicolò (2020).
We use Bayesian techniques to estimate the model parameters and to test for (in)determinacy using posterior model probabilities.First, we find the mode of the posterior distribution by maximizing the log posterior function, which combines the prior information on the parameters with the likelihood of the data.In a second step, the Metropolis-Hastings algorithm is used to simulate the posterior distribution and to evaluate the marginal likelihood of the model. 9

RESULTS
We estimate both our baseline model and a model without ROT (where θ = 0) for the pre-Volcker (55:Q4-79:Q2) and the Great Moderation (84:Q1-07:Q3) periods separately. 10Table 2 shows the log-data densities of the four possibilities (determinacy versus indeterminacy, ROT versus RANK) for both subsamples.Comparing the log-likelihoods, both models (ROT and RANK) point definitely toward 8.We also confirm that this is actually favored by the data.9.All estimations are done using Dynare (https://www.dynare.org/wp-repo/dynarewp001.pdf).The posterior distributions are based on 500,000 draws, with the first 100,000 draws being discarded as burnin draws.The average acceptance rate is around 25-30%.10.We exclude the years of the Volcker disinflation and the end of the second subsample is marked by the onset of the Great Recession.indeterminacy in the first subsample and determinacy in the second subsample.The probability of indeterminacy and determinacy in the two subsamples, respectively, are calculated as in Lubik and Schorfheide (2004) and are equal to one in both cases.
Then, let us focus on the first subsample under indeterminacy.The ROT model is marginally preferred to the RANK model.Comparing the two alternative models, the Bayes factor is 1.7, which according to the classification in Kass and Raftery (1995) is "not worth more than a bare mention" as evidence against the RANK model. 11Indeed, the two models are very close, so that our estimates deliver two main results.
First, consistent with most of the results in the literature (e.g., Lubik and Schorfheide 2004, or more recently Nicolò 2020), the RANK model in the first subsample yields indeterminacy, because of a passive monetary policy rule (the estimated posterior mean for φ π is 0.798, see Table 1).However, contrary to the evidence in Bilbiie and Straub (2013), this is also the case for the ROT model.The estimated posterior mean for the fraction of ROT, θ , is low, equal to 0.219, far below the threshold value for the IADL region in our model (recall the discussion in Section 3.2 and Figure 1).Figure 2 shows that data are informative for the posterior distribution for θ .Bilbiie and Straub (2013) found that the data preferred determinacy when estimating 11.We report the Bayes Factor as suggested in Kass and Raftery (1995), calculated as 2(log-data density H1-log-data density H0), where the null hypothesis (H0) is always the less preferred model (while the alternative hypothesis, H1, is the preferred one).Hence, we weight evidence against the null hypothesis.a small-scale ROT model for the pre-Volcker period, as a result of passive monetary policy and a high fraction of ROT (their posterior mean for θ is 0.5), that is, the model is in the IADL region of the parameter space.According to our medium-scale model instead, the ROT model delivers indeterminacy, exactly for the same reason as the RANK model: a passive monetary policy (the estimated posterior mean for φ π is 0.796, see Table 1).The estimated ROT fraction is too low to put the model in the IADL region.Second, the estimated ROT fraction is actually so low that the two models are extremely similar, delivering almost identical estimated posterior means of all the parameters, variance, and historical decompositions, and impulse response functions to shocks.Table 1 shows the posterior means for all the parameters; there are barely any differences across the two models and the estimates are consistent with the standard value in the RANK-DSGE literature.Table 3 presents the variance decompositions for the pre-Volcker period.For both models, output and consumption volatility is mainly determined by the technology and the risk-premium shocks (the later being relatively more important for consumption).Government spending shock is also important for output fluctuations.In both models, inflation volatility is mainly driven by the wage markup, the technology, and the price markup shocks, but also by the sunspot shock.So, inflation dynamics was driven by self-fulfilling expectations both for the RANK and the ROT model.This is confirmed by the historical decomposition of inflation and the output gap, as shown in Figures 3-6.The narrative about the main drivers of U.S. business cycle fluctuations that comes out from the estimated DSGE model is the same in both models, and corroborates the results in Nicolò (2020).In the Great Inflation period of the '70s, the dynamics of the output gap is mainly driven by       risk-premium shocks, which generate "stagflation" dynamics under indeterminacy.A positive risk-premium shock has a contractionary effect on the economy, but because of passive monetary policy agents form self-fulfilling inflationary expectations (see the impulse responses in Figure B.4 in the Online Appendix).In the same period, high inflation is caused by technology shocks, demand shocks, and the sunspot shock.Passive monetary policy alters the dynamics of inflation in response to shocks, particularly to technology, risk premium, and monetary policy shocks.The presence of ROT consumers does not alter this interpretation of U.S. business cycle fluctuations during this subsample, because their fraction is too low.The impulse response functions to the different shocks almost overlap for the two models (indicated as ROT-IND and RANK-IND in the figures) with two expected exceptions: the responses of aggregate consumption to the government spending shock and to the investment shock. 12 Figure 7 shows that the positive reaction of output to a government spending shock induces higher consumption of the ROT consumers that only partially compensates the decrease in consumption of optimizing consumers, who adhere to standard Ricardian equivalence dynamics.As a result, aggregate consumption decreases much less in the ROT-IND model than in the RANK-IND one.Similarly, Figure 8 shows that in response to the investment shock, the increase in income pushes up the consumption of ROT consumers, while optimizing consumers decrease their consumption to finance 12. Hence, in the main text, we just include the impulse response functions to these two shocks, while the others are confined to the Online Appendix.the increase in investment.As a result, aggregate consumption decreases slightly on impact, but then, it increases faster in the ROT-IND model with respect to the RANK-IND one.However, these differences are quantitatively negligible regarding the narrative of U.S. business cycle fluctuations according to the two models.The historical decomposition figures demonstrate that these two shocks are not quantitatively important drivers of consumption fluctuations.The variance decompositions in Table 3 are also unaffected. 13 #jmcb13057-fig- 0005.figTo sum up, the estimations of the two empirically rich models in the pre-Volcker subsample yield two main results that contrast with the ones in Bilbiie and Straub (2013), who estimate a small-scale model.First, a model with ROT consumers delivers indeterminacy due to passive monetary policy, just like a standard RANK model.Second, the estimate of the fraction of ROT consumers is so low that the RANK and the ROT models deliver almost exactly the same dynamics and interpretation of aggregate U.S. business cycle fluctuations.
Therefore, the presence of ROT consumers is not substantive to explain these fluctuations.Indeed, the difference in the log-data densities between the RANK-IND and the ROT-IND models is negligible.The next section presents further robustness checks on the two main results of our paper.
13.If anything, somewhat surprising, the fraction of the (forecast error) variance of consumption explained by these two shocks is higher in the RANK-IND model than in the ROT-IND one.While substantially so in percentage terms, the numbers are still minuscule.The results for the second subsample are less surprising and in line with the existing literature.Both the RANK and the ROT models point toward determinacy and active monetary policy (see Table 2).The posterior mean for θ , as seen in Table 4, is very low (0.1), such that the two models are even more similar.Again, the estimated posterior means of all the other parameters of the model (see Table 4), the variance (see Table 5) and historical decompositions, and the impulse response functions are very similar across the two specifications, and they are in accordance with the results in Nicolò (2020).The Bayes factor (equal to 11) favors the RANK model "very strongly," according to Kass and Raftery's (2015) classification.In accordance with the literature (Stock andWatson 2003, Primiceri 2005, Sims and   Justiniano and Primiceri 2008), the standard deviations of the fundamental shocks are substantially lower in this Great Moderation subsample, pointing to a change in both the shock volatilities and the conduct of monetary policy as the explanation for the conquest of American inflation.

ROBUSTNESS
Our results point to the irrelevance of ROT consumers for aggregate business cycle fluctuations in the U.S., that is, the dynamics of the model with and without ROT consumers are very similar such that both models provide a similar interpretation of U.S. business cycles.In what follows, we first check if our results survive if we calibrate the share of ROT consumers, θ , to a higher value as found in some works in the literature.Then, we check whether our results hinge on the assumption of sticky wages that could dampen the role of ROT consumers as potential amplifier of shocks.Next, for similar reason, we look at a more realistic specification of the fiscal side of the model, relaxing the assumption of a balance budget.Finally, we check the robustness of the indeterminacy result in the pre-Volcker sample.

Alternative Calibration for θ
In our estimates, the estimated fraction of ROT consumers turns out to be quite low: 22% and 11% for the two subsamples, respectively.Bilbiie and Straub (2013) estimate a small-scale TANK model and find the fraction of ROT consumers to be higher: 50% in their pre-Volcker sample and 20% in their post-1984 sample.In fact,  the estimated ROT fraction for the pre-Volcker period in Bilbiie and Straub (2013) turns out to be high enough for the economy to be in the so-called inverse aggregate demand logic (IADL) region whereby a passive monetary policy implies determinacy.In contrast, our results show that estimating a similar TANK model with richer dynamic and stochastic structure implies a smaller role for ROT consumers in both sub-samples.One interpretation could be that missing propagation mechanisms and structural shocks are misinterpreted as high degree of ROT consumers in estimated small-scale models.These estimates are consistent with the empirical consensus about the MPC out of transitory income shocks (see, e.g., Johnson, Parker, and Souleles 2006, Parker et al. 2013, Kaplan, Violante, and Weidner 2014).Based on this, Bilbiie, Primiceri, and Tambalotti (2022) estimate a tractable HANK model for the U.S. from 1954 to 2019 setting the fraction of ROT at 0.2.Kaplan, Violante, and Weidner (2014) find a fraction of poor hand-to-mouth consumers of 14% on average in the U.S. between 1989 and 2010.Moreover, we can interpret the decline of the share of ROT in the second subsample due to increased financial market liberalization starting in the '80s, which led to a broader participation in asset markets. 14 Nevertheless, alternative direct estimates of the MPC from Jappelli and Pistaferri (2020) and Fagereng, Holm, and Natvik (2021) find the MPC to be around 0.4 at an annual level.Hence, to check the robustness of our results, we calibrate the fraction of ROT consumers to 0.4 for both subsamples and reestimate the model.Table 6 shows the log-data densities and estimated posterior model probabilities.For ease of comparison, in Table 6, we report our baseline results too.First, we find that the pre-Volcker period continues to be characterized by indeterminacy due to passive monetary policy.However, and in contrast to our baseline results, the post-84 period is now also characterized by indeterminacy and passive monetary policy as the posterior puts more than 80% weight in the indeterminacy region.Nonetheless, our baseline estimations, whereby we estimate the fraction of ROT consumer, fit significantly 14.Bilbiie and Straub (2013) also find a smaller fraction of ROT during the Great Moderation period.better in both subsamples, suggesting that a low fraction of ROT consumers is preferred by the data through the lens of the full-system Bayesian estimation.

The Cyclicality of Inequality and the Degree of Wage Stickiness
Bilbiie (2020) characterizes the conditions for the presence of ROT consumers to lead to an amplification or a dampening of monetary and fiscal policy shocks.Bilbiie (2020) shows that the key object is the constrained agent's (i.e., ROT consumer's) income elasticity to aggregate income.When this elasticity is larger (smaller) than one, the model dynamics amplifies (dampens) the effects of monetary and fiscal policies relative to RANK models.Bilbiie (2020) labels this finding as the "cyclical inequality" channel: when the constrained agent's income overreacts (underreacts) to aggregate income, inequality between unconstrained and constrained is countercyclical (procyclical), and the model delivers amplification (dampening) relative to a RANK model.Bilbiie (2020) suggests that the cyclical behavior of inequality between the two types of agents, however, could depend on the degree of wage stickiness.According to Bilbiie (2020), the "cyclical inequality" channel relies crucially on flexible wages, as a TANK model with sticky wages, along the lines of Colciago (2011) and Ascari, Colciago, andRossi (2016, 2017), would imply smaller monetary and fiscal multipliers.This section investigates to what extent the assumption of sticky wages in our model affects the cyclicality of inequality, and in so doing, it affects the magnitude and features of business cycles.We simulate the estimated ROT model for the two subsamples for different degrees of wage stickiness-keeping the other parameters at their posterior mean-and compute the following statistics: (i) cyclicality of Ricardian consumer's income, ρ( y o t , y t ); (ii) cyclicality of ROT consumer's income, ρ( y rt t , y t ); (iii) cyclicality of inequality, ρ(ineq t , y t ), where inequality is the defined as the difference between the log-deviations of Ricardian and ROT income, that is, ineq t = y o t − y rt t ; and (iv) volatility of output fluctuations measured as the standard deviation of aggregate income, std( y t ).
Table 7 shows the results.Both the incomes of the Ricardian and of the ROT consumers are strongly procyclical with the former more procyclical than the latter.An increase in wage stickiness increases the procyclicality of both constrained and unconstrained agents' income.However, note that what matters for amplification/dampening according to Bilbiie (2020) is not the cyclicality of different agents' income, but rather the cyclicality of inequality, which relies on the income elasticity of constrained and unconstrained agents to changes in aggregate income.Table 7 shows that inequality is countercyclical in both subsamples when wages are relatively more flexible.As wage stickiness increases, inequality turns and becomes more and more procyclical.
Does this imply that business cycles in our model become dampened as wage stickiness increases?To see this, we look at the volatility of aggregate output.Our results suggest a non-monotonic relationship between the degree of wage stickiness and the volatility of output.When stickiness is very low, an initial increase in wage stickiness dampens output fluctuations.In contrast, when stickiness is moderate, further  increase in wage stickiness amplifies output fluctuations.These results imply that it is possible in principle to have amplification in our medium-scale model-in the sense of higher output volatility-even when wages become stickier and inequality becomes more procyclical.One might find this counterintuitive given Bilbiie (2020) results.However, note that Bilbiie (2020) findings regarding amplification/dampening pertains to the real effects of demand-type shocks, that is, the monetary and fiscal policy multipliers.On the other hand, our model features a combination of both demand and supply shocks, as is common in estimated medium-scale models. 15

Fiscal Policy Rules
In this section, we relax the assumption that the government budget is balanced in every period and introduce a richer fiscal structure.In our two-agents environment, this may be potentially relevant.In fact, it is well known from the literature that ROT consumers break the Ricardian equivalence.Thus, for example, while in a representative agent model government lump-sum transfers/taxes have no effects, they do have effects, however, when a fraction of agents are non-Ricardian. 16This may be important for our estimates, as ROT and Ricardian agents have different reactions to changes in fiscal variables and, in line with Bilbiie (2020)'s arguments, this could also alter the cyclicality of inequality. 17 We introduce fiscal feedback rules for distortionary taxes on consumption, labor income, and capital together with a rule for lump-sum transfer/taxes, closely following Leeper, Traum and Walker (2017) and Zubairy (2014), and then re-estimate the 15.A detailed analysis of how the cyclical inequality channel affects amplification/dampening for supply shocks is beyond the scope of this paper and we leave it for future research.
16. See, for example, Giambattista and Pennings (2017).17.We thank an anonymous referee for pointing this out.model.In the linear version of the model, all fiscal instruments respond to government debt. 18We find that our baseline results remain essentially unchanged.The estimated fraction of ROT along with the estimates for the other structural and shock parameters is very similar with respect to our baseline estimates, thereby also delivering similar results in terms of log data densities (see Table 8), impulse responses functions, and historical and variance decompositions.In addition, our conclusions regarding the cyclicality of inequality remain unchanged in the model with taxes.

Pre-Volcker Sample
Our main result concerns the irrelevance of ROT consumers for aggregate business cycle fluctuations in U.S. data.Given previous results in the literature, this is surprising for the pre-Volcker sample in particular.In this section, we check the robustness of this result for the pre-Volcker sample with respect to changes to: (i) the prior for the fraction of ROT consumers, θ ; (ii) the specification of the Taylor rule; and (iii) the subsample splits.
Prior for θ .Our baseline prior for θ is in line with Bilbiie and Straub (2013).To give a fair chance to higher values of θ , we re-estimate the model for the pre-Volcker period with a uniform prior (0,1) for θ .In this case, results are sensitive to the initial values, that is, they depend on the region of the parameter space the estimations are launched in (as shown in Table A.1 in the Online Appendix). 19Starting from a parameter configuration from the usual determinacy region (SADL, in Bilbiie's (2008) terminology), we find the same results as above, and the data strongly favor an indeterminate model.However, when we initialize the estimation algorithm in the IADL region, we do find results consistent with Bilbiie and Straub (2013).That is, we find determinacy due to a passive monetary policy (posterior mean of φ π = 0.50) and a high value of ROT consumers (posterior mean of θ = 0.65), and hence, the parameter estimates put the model in the IADL region.The log-data density, however, notably drops to (−702.59), while it is equal to (−609.66) for the indeterminate model estimated when the algorithm is initialized in the SADL region.The Bayes factor comparing these two log-data densities is as large as 185.9 signaling a very strong evidence against the determinate model with a high value of θ .
Forward-looking Taylor rule.We run a robustness check assuming a forward-looking Taylor rule where the interest rate reacts to expected inflation as opposed to contemporaneous inflation as in our baseline model.Bilbiie (2008) show that the "inverted Taylor principle" holds in the IADL case in his small-scale NK model for a smaller fraction of ROT consumers with a forward-looking Taylor rule compared to a contemporaneous Taylor rule.In addition, Bilbiie and Straub (2013) use a forward-looking Taylor rule whereby the monetary authority responds to expected inflation.First, we find that the determinacy-indeterminacy boundary with a forward-looking rule in our medium-scale model is the same as in Figure 1.Second, Table A.2 in the Online Appendix shows that the estimation results are very similar to our baseline results with contemporaneous inflation in the Taylor rule.20 Subsamples.Table 9 displays the results of different experiments with four different subsamples for the Great Inflation years.The first two correspond to the two subsamples in Nicolò (2020), who argues that it is important to split the original sample in pre-and post-1970, because the '70s are characterized by slower productivity growth, resulting in a distinct balanced growth path.Not surprisingly, our results are in line with Nicolò (2020) and the data favor the indeterminate model in both subsamples.Moreover, comparing the log-data densities, we show that there is "positive" evidence against the ROT model compared to the RANK one.Hence, considering this split of our original pre-Volcker sample would reinforce our argument of rejecting the usefulness of a model with ROT consumers to fit the U.S. business cycle.
The third subsample (60:Q1-79:Q2) is the sample used by Lubik and Schorfheide (2004) and also by Bilbiie and Straub (2013).In this case, we find results similar to our baseline, so that the data favor the indeterminate model with basically no difference in terms of fit between the ROT and the RANK models.Hence, the fact that our results differ from the ones in Bilbiie and Straub (2013) is not due to us employing a different sample for the pre-Volcker period.
Finally, we experiment also with 66:Q1-79:Q2 that is the sample used in their seminal paper by Smets and Wouters (2007).To our surprise, here, the results differ and it is worth spending few words on this result, because it might have been overlooked by the literature.Our results are consistent with Smets and Wouters (2007) because the data favor a determinate model for this particular subsample.Determinacy follows from the estimate of an active monetary policy and a small fraction of ROT consumers.In Kass and Raftery (1995) terminology, there is positive evidence against indeterminacy.This is true for both the ROT and the RANK models, again signaling that the two models are empirically indistinguishable, despite the log-data density being marginally larger for the ROT model.Hence, whether or not the estimation finds indeterminacy in the pre-Volcker sample seems to be sensitive to the choice of the dates.We conjecture that the reason why the 66:Q1-79:Q2 sample yields determinacy is because of the increase in the real interest rate in the last years of '60s that pushes the estimation toward an active monetary policy. 21The determinacy result seems to be confined to this particular sample period, so this could be just a minor point.However, given that papers in the literature might choose this sample period to compare their results with Smets and Wouters (2007), we think it is important to point out that choosing this particular sample has an impact on the long standing debate about bad versus good monetary policy in the pre-Volcker period.

CONCLUSION
We estimate a medium-scale model with ROT consumers over two different subsamples (the pre-Volcker and the Great Moderation periods), while allowing and testing for (in)determinacy, and compare our results with the standard RANK 21.Real interest rates were mostly rising in the late 1960s, which suggests Fed's strong responsiveness to inflation during that time.Indeed, Coibion and Gorodnichenko (2011) find a strong response to inflation and an associated high probability of determinacy in the late 1960s.This suggests that the increase in the real rate in the mid-to-late 1960s more than compensates for the loose policy during the 70s such that overall, we find the posterior mass lying mostly in the determinacy region in the 1966Q1-1979Q2 subsample.specification.Our main finding is that including ROT in a RANK model is irrelevant to explain U.S. aggregate business cycle fluctuations.The reason being that the ROT model preferred by the data has a very low fraction of ROT consumers, which only marginally affects the dynamics of the model relative to a RANK specification.The two models are empirically equivalent.In both subsamples, the RANK and ROT models yield almost the same impulse response functions, variance, and historical decompositions, such that they share the same narrative of U.S. business cycle fluctuations In line with Lubik and Schorfheide (2004) and Nicolò (2020), we find that passive monetary policy and self-fulfilling fluctuations characterize the pre-Volcker period for both the ROT and the RANK model.This contrasts with previous findings in the literature by Bilbiie and Straub (2013), who employ a small-scale model.In the pre-Volcker period, the log-likelihoods of the ROT and the RANK models are very close, while in the second period, the RANK model is preferred by the data.
Our main finding, which including ROT in a RANK model does not change the interpretation of aggregate U.S. business cycle fluctuations, does not mean that modeling ROT, or heterogeneous agents more generally, is not important to explain other dimensions of the data.However, in line with some others in the HANK literature (e.g., Bayer, Born, and Luetticke 2020), it suggests that adding heterogeneity may not be substantive to explain aggregate fluctuations, at least for U.S. data.

APPENDIX A: SYSTEM OF NONLINEAR EQUATIONS
After deriving the first conditions of the model, we adjust variables to guarantee that the model has a balanced growth path.Lower case letters stand for detrended vari- Given that the model is then log-linearized, we omit price and wage dispersion variables.We add exogenous shock processes for the following variables: ε a t , ε b t , ε i t , ε r t , λ p t , λ w t , g t .ROT lump-sum taxes are also modeled as exogenous shocks, which we are not estimating, thus they remain constant at their steady state.Given that the government budget constraint is balanced every period, we can omit this equation.We introduce distortionary taxes on consumption, labor income and capital, and lump-sum transfers/taxes for both consumers.This alters the problem of households and following are the modified equations for the model with taxes:  .We also assumed that transfers are symmetric to both types of individuals.
We calibrate the fiscal parameters as follows.The steady-state tax rates are based on Leeper, Traum, and Walker (2017); thus, τ c = 0.023, τ l = 0.186, τ k = 0.218.The response of transfers to debt is based on their estimates, and thus, set at 0.03.The feedback parameters of taxes are borrowed from Zubairy (2014), who estimates φ τ l b = 0.02 and φ τ k b = 0.017.We set φ τ c b similarly to 0.02.

Fig. 1 .
Fig. 1.Determinacy Region for φ π against θ; the Remaining Structural Parameters of the Model Are Set at the Prior Mean.

TABLE 1 Parameter
Estimates for the Sample 55-79 15384616, 0, Downloaded from https://onlinelibrary.wiley.com/doi/10.1111/jmcb.13057 by Universita Milano Bicocca, Wiley Online Library on [25/05/2023].See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions)on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Note: The prior probability of determinacy is 0.51.ROT and RANK stand for Rule of Thumb and Representative Agent New Keynesian, respectively.Log marginal data densities are approximated by Geweke's (1999) harmonic mean estimator.The posterior probabilities are calculated based on the output of the Metropolis algorithm.KR stands for Kass and Raftery.

TABLE 7 Simulation
Results on Cyclicality of Inequality and Volatility of Output 15384616, 0, Downloaded from https://onlinelibrary.wiley.com/doi/10.1111/jmcb.13057 by Universita Milano Bicocca, Wiley Online Library on [25/05/2023].See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions)on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License

TABLE 9
Note: The prior probability of determinacy is 0.51.ROT and RANK stand for Rule of Thumb and Representative Agent New Keynesian, respectively.Log marginal data densities are approximated by Geweke's (1999) harmonic mean estimator.The posterior probabilities are calculated based on the output of the Metropolis algorithm.KR stands for Kass and Raftery.