## Abstract

We provide evidence on three mechanisms that can reconcile frequent individual price changes with sluggish aggregate price dynamics. To that end, we estimate a semistructural model that can extract information about real rigidities and the distribution of price stickiness from aggregate data. Hence, the model can also speak to the debate about the aggregate implications of sales. Our estimates indicate large real rigidities and substantial heterogeneity in price stickiness. Moreover, the cross-sectional distribution of price stickiness implied by aggregate data is in line with an empirical distribution obtained from microprice data that factors out sales and product substitutions.

## I. Introduction

OUR understanding of the real effects of monetary policy hinges, to a large extent, on the existence of some degree of nominal price rigidity. Since the publication of the seminal Bils and Klenow (2004) paper, the availability of large amounts of microprice data has rekindled interest in this area and allowed us to make progress. Yet estimates of the extent of nominal price stickiness based on microeconomic data versus those based on aggregate data usually produce a conflicting picture.

According to Klenow and Malin's (2011) survey of the empirical literature based on microdata, prices change, on average, at least once a year—somewhat more often than we thought was the case prior to Bils and Klenow (2004). In contrast, making sense of estimates of the response of the aggregate price level to monetary shocks, from dynamic stochastic general equilibrium (DSGE) models or vector autoregressions, usually requires more infrequent price adjustments.1

If nominal price rigidities are to remain the leading explanation for why monetary policy has large and persistent real effects, it is important that we deepen our understanding of mechanisms that can narrow the gap between the evidence of somewhat flexible individual prices and relatively sluggish aggregate prices—mechanisms that can produce a large “contract multiplier,” as Klenow and Malin (2011) put it. If prices change frequently and every price change contributes to fully offset nominal disturbances, then nominal price rigidity cannot be the source of large and persistent monetary nonneutralities. Hence, a large contract multiplier requires that price adjustments somehow fail to perfectly neutralize monetary innovations.

In this paper, we contribute to bridging the gap between micro- and macroevidence on the extent of nominal price rigidity. To that end, we estimate a standard macroeconomic model of price setting and use it to speak to three mechanisms that can boost the contract multiplier.2 First are so-called real rigidities, in the sense of Ball and Romer (1990). Large real rigidities reduce the sensitivity of individual prices to aggregate demand conditions and thus serve as a source of endogenous persistence: for any given frequency of price changes, partial adjustment of individual prices makes for a sluggish response of the aggregate price level to monetary shocks.

The other two mechanisms are motivated by empirical evidence uncovered since Bils and Klenow (2004), and subsequent theoretical literature. Cross-sectional heterogeneity in price rigidity, to the extent documented in the microdata, can lead to much larger monetary nonneutralities than the average frequency of price changes would imply (Carvalho, 2006; Nakamura & Steinsson, 2010). The reason is that while recurrent price changes by firms in more flexible sectors do not contribute as much to offset monetary shocks, they do count for the frequency of price adjustment.3 Heterogeneity can become an even more powerful mechanism when coupled with strong, real rigidities that lead to strategic complementarities in pricing decisions. In those circumstances, firms in stickier sectors become disproportionately important in shaping aggregate dynamics (relative to their sectoral weight) through their influence on pricing decisions of firms that change prices more frequently (Carvalho, 2006).4

The third mechanism is associated with the presence of sales and other temporary price changes. Guimaraes and Sheedy (2011) and Kehoe and Midrigan (2015) show that such price changes may help reconcile frequent microadjustments with a sluggish aggregate price response to nominal disturbances.5 A basic intuition for their results is that temporary price changes fail to offset monetary shocks well, since these shocks tend to induce permanent changes in the level of prices.

Extracting information on the three underlying mechanisms from aggregate data (as opposed to disaggregate data) is a distinct contribution of our paper. To that end, we rely on a relatively standard semistructural model. The price-setting block of the model is a multisector sticky-price economy that allows for heterogeneity in price stickiness and can feature strategic complementarity or substitutability in pricing decisions. In particular, we consider both Taylor and Calvo pricing schemes. The remaining equations specify exogenous stochastic processes that drive firms' frictionless optimal prices. They provide the model with some flexibility to perform in empirical terms and thus allow us to focus on the objects of interest in the price-setting block of the economy.6

We show that, at least in theory, our model is able to separately identify real and nominal rigidities and distinguish economies with homogeneous from those with heterogeneous price stickiness based on aggregate data only. The model can also discriminate between different (nondegenerate) distributions of price rigidity, providing information on which one helps explain aggregate dynamics better. Hence, our analysis can also speak to the debate about whether price changes due to sales and product substitutions are relevant for macrodynamics.

Identification of the distribution of price stickiness based on aggregate data is possible in our framework because sectors that differ in stickiness have different implications for the response of the macroeconomy to shocks at different frequencies. In particular, sectors where prices are stickier are relatively more important in determining the low-frequency response to shocks, and vice versa for more flexible sectors. These differences provide information about the cross-sectional distribution of price stickiness. Separate identification of real and nominal rigidities hinges on the fact that the former induces endogenous persistence in the economy. We show analytically how this introduces a dependence of the aggregate price level on its own lags, which can be exploited to obtain identification of the parameter that governs real rigidities.7 While we consider both Taylor and Calvo pricing, we find that in small samples, the former allows for stronger identification of the objects of interest. Hence, in our baseline estimation, we rely on Taylor pricing and use Calvo pricing to assess the robustness of our findings.8

Much of the recent literature on price setting pays large attention to empirical facts obtained from microprice data—and so do we. However, treating statistics derived from microdata as the true “population moments” that matter for aggregate dynamics can be misleading in our view. First, it is possible that some price adjustments do not convey as much information about changes in macroeconomic conditions as others do. While this possibility is at the core of the debate about whether to exclude sales from price-setting statistics for macropurposes, the argument applies more generally; for example, it also applies to the literature on price setting under information. If so, macrobased estimates should convey useful information about price changes that do matter for aggregate dynamics. Second, and no less important, Eichenbaum et al. (2014) show that the BLS microdata underlying the CPI are plagued with measurement problems when it comes to computing statistics based on individual price changes. While Eichenbaum et al. (2014) focus on pitfalls involved in estimating the distribution of the size of price changes, the problems they document certainly add measurement error to available estimates of the cross-sectional distribution of price stickiness that use those data (e.g., Bils & Klenow, 2004; Nakamura & Steinsson, 2008; Klenow & Kryvtsov, 2008). At the same time, we certainly do not want to ignore all the information that microdata can provide.

To strike a balance between extracting information from aggregate data and exploring information contained in the microdata, we employ a full-information Bayesian approach. We use aggregate (time-series) data on nominal and real GDP as observables, and incorporate the microeconomic information about the cross-sectional distribution of price stickiness through our prior. In the baseline estimation, we use an uninformative (“flat”) prior for the cross-sectional distribution of price stickiness and, hence, let the aggregate data speak. This allows the model to recover the distribution of stickiness that provides the best account of aggregate dynamics. We then supplement the baseline analysis by adopting informative priors based on two empirical distributions of price rigidity: one that takes into account all price changes, including sales and product substitutions (derived from Bils & Klenow, 2004, henceforth “posted prices”) and another that is based on price changes that exclude sales and product substitutions (derived from Nakamura & Steinsson, 2008, henceforth, “regular prices”).9

Our baseline estimation shows that all three mechanisms play an important role in accounting for aggregate dynamics. The estimated model points to heterogeneity in price stickiness and the existence of large, real rigidities, which induce strong strategic complementarities in price setting. Importantly, the macrobased estimate of the cross-sectional distribution of price rigidity accords well with the empirical distribution based on regular price changes. Moreover, formal statistical model comparisons using informative priors based on the two empirical distributions favor specifications based on these price changes, which exclude sales and product substitutions. This suggests that such price changes are relatively more important for aggregate dynamics than those associated with sales and product substitutions.

Finally, we complement the main findings with additional sets of estimations in which we effectively shut down each of the three mechanisms at a time by imposing highly informative (or even dogmatic) priors on the cross-sectional distribution of price stickiness or on the degree of real rigidities. Formal statistical model comparisons reveal that our benchmark model (in which all three mechanisms are at play) outperforms alternative models, such as economies with distribution of price stickiness based on posted prices (as opposed to regular prices), with homogeneous price rigidity, or with strategic neutrality (as opposed to strategic complementarity) in price setting.

### A. Brief Literature Review

Our work is related to the literature that emphasizes the importance of heterogeneity in price rigidity for aggregate dynamics. However, our focus differs from that of other papers. Most of the latter focus on the role of heterogeneity in boosting the contract multiplier in calibrated models (Carvalho, 2006; Carvalho & Schwartzman, 2008; Nakamura & Steinsson, 2010; Carvalho & Nechio, 2011; Dixon & Kara, 2011). These papers do not address the question of whether such heterogeneity does in fact help sticky-price models fit the data better according to formal statistical criteria.

In terms of empirical work on the importance of heterogeneity in price stickiness, Imbs, Jondeau, and Pelgrin (2011) study the aggregation of sectoral Phillips curves and the statistical biases that can arise from using estimation methods that do not account for heterogeneity. They rely on sectoral data for France and find that the results based on estimators that allow for heterogeneity are more in line with the available microeconomic evidence on price rigidity. Lee (2009) and Bouakez, Cardia, and Ruge-Murcia (2009) estimate multisector DSGE models with heterogeneity in price rigidity using aggregate and sectoral data. They also find results that are more in line with the microeconomic evidence than the versions of their models that impose the same degree of price rigidity for all sectors.10 Taylor (1993) provides estimates of the distribution of the duration of wage contracts in various countries inferred solely from aggregate data, while Guerrieri (2006) provides estimates of the distribution of the duration of price spells in the United States based on aggregate data. Both models feature ex-post rather than ex-ante heterogeneity in nominal rigidities, as is the case in our model.11 Coenen et al. (2007) estimate a model with (limited) ex-ante heterogeneity in price contracts using only aggregate data. They focus on the estimate of the Ball-Romer index of real rigidities and the average duration of contracts implied by their estimates, which they emphasize are in line with the results in Bils and Klenow (2004).12

Jadresic (1999) is a precursor to some of the ideas in this paper. He estimates a model with ex-ante heterogeneous price spells using only aggregate data for the U.S. economy to study the joint dynamics of output and inflation. Similar to our findings, his statistical results reject the assumption of identical firms. Moreover, he discusses the intuition behind the source of identification of the cross-sectional distribution of price rigidity from aggregate data in his model, which is the same as in our model. Despite these similarities, our paper differs from Jadresic's in several important dimensions. We use a different estimation method and show the possibility of extracting information about the cross-sectional distribution of price rigidity from aggregate data in a more general context—in particular, in the presence of pricing complementarities. Most important, the focus of our paper goes beyond assessing the empirical support for heterogeneity in price rigidity from aggregate data. We also investigate the similarities between our macro-based estimates and the available microeconomic evidence, and identify price changes that matter for aggregate dynamics.

Finally, our results speak to the ongoing debate on the role of sales in macroeconomic models. That literature started out as a discussion about whether to exclude sales when computing price-setting statistics for macropurposes (Bils & Klenow, 2004; Nakamura & Steinsson, 2008; Klenow & Kryvtsov, 2008). This initial debate was followed by a theoretical literature that developed macroeconomic models with sales and other temporary price changes (Guimaraes & Sheedy, 2011; Kehoe & Midrigan, 2015). More recently, the literature has focused on the cyclicality of sales and consumer behavior, in both theory and the microdata (Coibion et al., 2015; Kryvtsov & Vincent, 2014). We provide statistical evidence on the relative performance of macroeconomic models with different distributions of price rigidity that do and do not exclude sales (and product substitutions).

## II. The Semistructural Model

There is a continuum of monopolistically competitive firms divided into $K$ sectors that differ in the frequency of price changes. Firms are indexed by their sector, $k$$∈1,…,K$, and by $j∈0,1$. The distribution of firms across sectors is summarized by a vector $ω1,…,ωK$ with $ωk>0,$$∑k=1Kωk=1$, where $ωk$ gives the mass of firms in sector $k$. Each firm produces a unique variety of a consumption good and faces a demand that depends negatively on its relative price.

In any given period, profits of firm $j$ from sector $k$ (henceforth referred to as firm $kj$) are given by
$Πtk,j=Ptk,jYtk,j-CYtk,j,Yt,ξt,$
where $Ptk,j$ is the price charged by the firm, $Ytk,j$ is the quantity that it sells at the posted price (determined by demand), and $CYtk,j,Yt,ξt$ is the total cost of producing such quantity, which may also depend on aggregate output $Yt$ and is subject to shocks ($ξt$). We assume that the demand faced by the firm depends on its relative price $Ptk,jPt$, where $Pt$ is the aggregate price level in the economy, and on aggregate output. Thus, we write firm $kj$'s profit as
$Πtk,j=ΠPtk,j,Pt,Yt,ξt$
and make the usual assumption that $Π$ is homogeneous of degree 1 in the first two arguments and single-peaked at a strictly positive level of $Ptk,j$ for any level of the other arguments.13
The aggregate price index combines sectoral price indices, $Ptk$'s, according to the sectoral weights, $ωk$'s:
$Pt=ΓPtk,ωkk=1,…,K,$
where $Γ$ is an aggregator that is homogeneous of degree 1 in the $Ptk$'s. In turn, the sectoral price indices are obtained by applying a symmetric, homogeneous-of-degree-1 aggregator $Λ$ to prices charged by all firms in each sector:
$Ptk=ΛPtk,jj∈0,1.$

We assume the specification of staggered price setting inspired by Taylor (1979, 1980). Firms set prices that remain in place for a fixed number of periods. The latter is sector specific, and we save on notation by assuming that firms in sector $k$ set prices for $k$ periods. Thus, $ω=ω1,…,ωK$ fully characterizes the cross-sectional distribution of price stickiness that we are interested in. Finally, across all sectors, adjustments are staggered uniformly over time.

Before we continue, a brief digression about the Taylor pricing model is in order. As will become clear, this model allows us to distinguish real rigidities from nominal rigidities and to infer the cross-sectional distribution of price stickiness implied by aggregate data. Hence, it serves our purposes well. However, that model is at odds with the microeconomic evidence on the duration of price spells. Klenow and Kryvtsov (2008), for example, provide evidence that the duration of individual price spells varies at the quote line level. However, this evidence does not invalidate the use of the Taylor model for our purposes. In particular, in the appendix, we provide an alternative model in which the duration of price spells varies at the firm level, and yet the aggregate behavior of the model is identical to the one presented here. The alternative model can match additional microfacts documented in the literature. Hence, it provides a cautionary note on attempts to test specific models of price setting by confronting them with descriptive microprice statistics. For ease of exposition, we proceed with the standard Taylor pricing specification. But readers should keep in mind that the aggregate implications in which we are interested do survive in models that can match the microeconomic evidence in many dimensions.

When setting its price $Xtk,j$ at time $t$, given that it sets prices for $k$ periods, firm $kj$ solves
$maxEt∑i=0k-1Qt,t+iΠXtk,j,Pt+i,Yt+i,ξt+i,$
where $Qt,t+i$ is a (possibly stochastic) nominal discount factor. The first-order condition for the firm's problem can be written as
$Et∑i=0k-1Qt,t+i∂ΠXtk,j,Pt+i,Yt+i,ξt+i∂Xtk,j=0.$
(1)
Note that all firms from sector $k$ that adjust prices at the same time choose a common price, which we denote $Xtk$.14 Thus, for a firm $kj$ that adjusts at time $t$ and sets $Xtk$, the prices charged from $t$ to $t+k-1$ satisfy
$Pt+k-1k,j=Pt+k-2k,j=⋯=Ptk,j=Xtk.$
Given the assumptions on price setting and uniform staggering of price adjustments, with an abuse of notation, sectoral prices can be expressed as
$Ptk=ΛXt-iki=0,…,k-1.$

Instead of postulating a fully specified model to obtain the remaining equations to be used in the estimation, we assume exogenous stochastic processes for nominal output ($Mt≡PtYt$) and for the unobservable $ξt$ process; hence, we refer to our model as semistructural. Given our focus on estimation of parameters that characterize price-setting behavior in the economy in the presence of heterogeneity, our goal in specifying such exogenous time series processes is to close the model with a set of equations that can provide it with flexibility relative to a fully structural model. Such flexibility is useful because it allows us to draw conclusions about price setting that do not depend on details of structural models that are not the focus of our analysis.15

### A. A Log-Linear Approximation

We assume that the economy has a deterministic zero-inflation steady state characterized by $Mt=M¯,ξt=ξ¯,Yt=Y¯,Qt,t+i=βi(0<β<1)$, and for all $k,j,$$Xtk,j=Pt=P¯$, and log-linearize equation 1 around it to obtain16
$xtk=1-β1-βkEt∑i=0k-1βipt+i+ζyt+i-yt+in,$
(2)
where lowercase variables denote log deviations of the respective uppercase variables from the steady state. The parameter $ζ>0$ is the Ball and Romer (1990) index of real rigidities. The new variable $Ytn$ is defined implicitly as a function of $ξt$ by
$∂ΠXtk,j,Pt,Ytn,ξt∂Xtk,jXtk,j=Pt=0.$
In the log-linear approximation, $ytn$ moves proportionately to $logξt/ξ¯$. Strictly speaking, it is the level of output that would prevail in a flexible-price economy. In a fully specified model, this would tie it down to preference and productivity shocks. Here, we do not pursue a structural interpretation of the exogenous processes driving the economy.17 Nevertheless, for ease of presentation, we follow the literature and label $ytn$ the “natural level of output.” The definition of nominal output yields
$mt=pt+yt.$
(3)
Effectively, the two exogenous processes, $mt$ and $ytn$, serve respectively as aggregate demand and aggregate supply shocks in the model: a positive innovation to $mt$ ($ytn$) moves output and the price level in the same (opposite) directions. Finally, we postulate that the aggregators that define the overall level of prices $Pt$ and the sectoral price indices give rise to the following log-linear approximations:18
$pt=∑k=1Kωkptk,$
(4)
$ptk=∫01ptk,jdj=1k∑j=0k-1xt-jk.$
(5)

Large, real rigidities (small $ζ$ in equation [2]) reduce the sensitivity of prices to aggregate demand conditions and thus magnify the nonneutralities generated by nominal price rigidity. In fully specified models, the extent of real rigidities depends on primitive parameters such as the intertemporal elasticity of substitution, the elasticity of substitution between varieties of the consumption good, and the labor supply elasticity. It also depends on whether, for example, the economy features economy-wide or segmented factor markets or whether there is an explicit input-output structure.19

In the context of our model, $ζ$ is itself a primitive parameter. Following standard practice in the literature, we refer to economies with $ζ<1$ as displaying strategic complementarities in price setting. To clarify the meaning of this expression, replace equation (3) in equation (2) to obtain
$xtk=1-β1-βkEt∑i=0k-1βiζmt+i-yt+in+1-ζpt+i.$
(6)

That is, new prices are set as a discounted weighted average of current and expected future driving variables $mt+i-yt+in$ and prices $pt+i$. $ζ<1$ implies that firms choose to set higher prices if the overall level of current and expected future prices is higher, all else equal. $ζ>1$ means that prices are strategic substitutes in that under those same circumstances, adjusting firms choose relatively lower prices.

### B. Nominal (⁠$mt$⁠) and Natural (⁠$ytn$⁠) Output

We postulate an $AR(p1)$ process for nominal output, $mt$,
$mt=ρ0+ρ1mt-1+⋯+ρp1mt-p1+ɛtm,$
(7)
and an $AR(p2)$ process for the natural output level, $ytn$,
$ytn=δ0+δ1yt-1n+⋯+δp2yt-p2n+ɛtn,$
(8)
where $ɛt=ɛtm,ɛtn$ is i.i.d. $N01×2,Ω2$, with $Ω2=σm200σn2.$

### C. State-Space Representation and Likelihood Function

We solve the semistructural model, equations 3 to 8, with Gensys (Sims, 2002), to obtain
$Zt=Cθ+G1θZt-1+Bθɛt,$
(9)
where $Zt$ is a vector collecting all variables and additional dummy variables created to account for leads and lags and $ɛt$ is as already defined. The vector $θ$ collects the primitive parameters of the model:
$θ=(K,p1,p2,β,ζ,σm,σn,ω1,…,ωK,ρ0,…,ρp1,δ0,…,δp2).$
In all estimations that follow, we make use of the likelihood function $Lθ|Z*$, where $Z*$ is the vector of observed time series (i.e., a subset of $Z$). Given that our state vector $Zt$ includes many unobserved variables, such as the natural output level and sectoral prices, the likelihood function is constructed through application of the Kalman filter to the solved log-linear model, equation (9). Letting $H$ denote the matrix that singles out the observed subspace $Zt*$ of the state vector $Zt$ (i.e., $Zt*=HZt$), our distributional assumptions can be summarized as
$Zt|Zt-1∼NCθ+G1θZt-1,BθΩBθ',Zt*|Zτ*τ=1t-1∼NMt|t-1θ,Vt|t-1θ,$
where $Mt|t-1θ≡HCθ+HG1θZ^t|t-1,Vt|t-1θ≡HBθΣ^t|t-1Bθ'H',$$Z^t|t-1$ denotes the expected value of $Zt$ given $Zτ*τ=1t-1$, and $Σ^t|t-1$ is the associated forecast-error covariance matrix.

### D. Identification of the Cross-Sectional Distribution from Aggregate Data

In estimating our multisector model, we use only time series data on aggregate nominal and real output as observables. It is thus natural to ask whether the structure of the model is such that these aggregate data reveal information about the cross-sectional distribution of price stickiness $ω=ω1,…,ωK$. As in Jadresic (1999), we start by looking at a simple case where it is easy to show that $ω$ can be inferred from observations of those two aggregate time series. This helps develop the intuition for a more general case for which we also show identification. We then assess the small-sample properties of estimates of $ω$ inferred from aggregate data through a Monte Carlo exercise. As in our estimation, we assume throughout that the discount factor, $β$, is known.

The key simplifying assumption to show identification in the first case is the absence of pricing interactions: $ζ=1$. In that case, from equation (6), new prices $xtk$ are set on the basis of current and expected future values of the two exogenous processes $mt$ and $ytn$. For simplicity and without loss of generality, assume further that the latter follow the AR(1) processes:
$mt=ρ1mt-1+ɛtm,and$
(10)
$ytn=δ1yt-1n+ɛtn.$
(11)
New prices are then set according to
$xtk=Fβ,ρ1,kmt-Fβ,δ1,kytn,$
where
$Fβ,a,k≡1+1-β1-βkβa-βak1-βa.$
Replacing this expression for newly set prices in the sectoral price equation (5) and aggregating according to equation (4) produces the following expression for the aggregate price level:
$pt=∑j=0K-1∑k=j+1KFβ,ρ1,kωkkmt-j-∑j=0K-1∑k=j+1KFβ,δ1,kωkkyt-jn.$
(12)

If we observe $mt$ and $yt$ (and thus $pt$), estimates of the coefficients on $mt-j$ in equation (12) allow us to infer the sectoral weights $ω$. The reason is that $Fβ,ρ1,k$ is “known,” since $ρ1$ can be estimated directly from equation (10). Thus, knowledge of the coefficient on the longest lag of $mt-j$ (attained when $j=K-1$) allows us to uncover $ωK$. The coefficient on the next longest lag ($mt-K-2$) depends on $ωK-1$ and $ωK$, which reveals $ωK-1$. We can thus recursively infer the sectoral weights from the coefficients $Fβ,ρ1,kωkk$. Moreover, identification obtains with any estimation method that produces consistent estimates of these coefficients.20

Checking for identification of $ω$ in the presence of pricing interactions ($ζ≠1$) is slightly more involved. To gain intuition on why this is so, fix the case of pricing complementarities ($ζ<1$). Then, because of the delayed response of sticky-price firms to shocks, firms with flexible prices will only react partially to innovations to $mt$ and $ytn$ on impact. They will eventually react fully to the shocks, but also with a delay.

It turns out that the “recursive identification” that applies when $ζ=1$ also works in this case. The reason is that in equilibrium, pricing interactions manifest themselves through the dependence of the aggregate price level on its own lags. This is how they serve as a propagation mechanism. Specifically, the expression for the equilibrium price level becomes
$pt=∑j=1K-1ajpt-j+∑j=0K-1bjmt-j-∑j=0K-1bjyt-jn,$
(13)
where $a1,…,aK-1$, $b0,…,bK-1$ are functions of the model parameters. Knowledge of the coefficients on the lags of the aggregate price level and on lagged nominal output again allows us to solve for the sectoral weights—and for $ζ$.21

The intuition behind the identification result in the absence of pricing interactions is clear: the impact of older developments of the exogenous processes on the current price level depends on prices that are sticky enough to have been set when the shocks hit. This provides information on the share of the sector with that given duration of price spells (and sectors with longer durations). More generally, in the presence of pricing interactions, fully forward-looking pricing decisions may also reflect past developments of the exogenous processes. This dependence manifests itself through lags of the aggregate price level. The intuition behind the mechanism that allows for identification extends in a natural way: sectors where prices are stickier are relatively more important in determining the impact of older shocks to the exogenous processes on the current price level, and vice versa for sectors where prices are more flexible. Moreover, the relative sizes of the coefficients on past prices and past nominal output in equation (13) pin down the index of real rigidities $ζ$.

These results on identification are of little practical use if they do not work well in finite samples. To analyze this issue, we rely on a Monte Carlo exercise. We generate artificial data on aggregate nominal and real output using parameter values that roughly resemble what we find when we estimate the model with actual data. Then we estimate the parameters of the model by maximum likelihood. We conduct both a large- and a small-sample exercise. Details and results are reported in the online appendix.

The bottom line is that for large samples, the estimates are quite close to the true parameter values and fall within a relatively narrow range. For samples of the same size as our actual sample, we also find the aggregate data to be informative of the distribution of sectoral weights. However, in this case, the estimates are less precise, and some of them are relatively biased. This finding motivates our supplementary estimation exercise in section VB, where we incorporate prior information from the microeconomic evidence on price setting.

## III. Empirical Methodology and Data

With the challenges involved in bridging the gap between price-setting statistics based on microdata and time series of aggregate nominal and real output, the choice of empirical methodology is critical. We employ a Bayesian approach, which allows us to integrate those two sources of information and compare different models in a formal way.

With some abuse of notation, the Bayesian principle can be briefly stated as
$fθ|Z*=fZ*|θfθ/fZ*∝Lθ|Z*fθ,$
where $f$ denotes density functions, $Z*$ is the vector of observed time series, $θ$ is the vector of primitive parameters, and $Lθ|Z*$ is the likelihood function.

As observables, we use time series of aggregate nominal and real output. For constructing our prior distribution over the vector of sectoral weights, $fω1,…,ωK$, we derive empirical distributions from Bils and Klenow (2004) and Nakamura and Steinsson (2008), as discussed in detail in section IIIA. We then detail our prior distributions, sources of data, and estimation approach.

### A. Prior over $ω$

We specify priors over the set of sectoral weights $ω=ω1,…,ωK$, which are then combined with the priors on the remaining parameters to produce the joint prior distribution for the set of all parameters of interest. We impose the combined restrictions of nonnegativity and summation to unity of the $ω$'s through a Dirichlet distribution, which is a multivariate generalization of the beta distribution. Notationally, $ω∼Dα1,…,αK$ with density function
$fωω|α1,…,αK∝∏k=1Kωkαk-1,∀αk>0,∀ωk≥0,∑k=1Kωk=1.$
The Dirichlet distribution is well known in Bayesian econometrics as the conjugate prior for the multinomial distribution, and the hyperparameters $α1,…,αK$ are most easily understood in this context, where they can be interpreted as the “number of occurrences” for each of the $K$ possible outcomes that the econometrician assigns to the prior information.22 Thus, for given $α1,…,αK$, the parameter $α0≡∑kαk$ captures, in some sense, the overall level of information in the prior distribution. The information about the cross-sectional distribution of price stickiness comes from the relative sizes of the $αk$'s. The latter also determine the marginal distributions for the $ωk$'s. For example, the expected value of $ωk$ is simply $αk/α0$, whereas its mode equals $α0-K-1αk-1$ (provided that $αi>1$ for all $i$).

Whenever we want to estimate a cross-sectional distribution of price rigidity based solely on aggregate data, we can impose an uninformative (“flat”) prior, in which all $ω$ vectors in the $K$-dimensional unit simplex are assigned equal prior density. This corresponds to $αk=1$ for all $k$—and thus $α0=K$. This allows us to extract the information that the aggregate data contain about the cross-sectional distribution of price stickiness.

To incorporate microeconomic information in the estimation, we relate the relative sizes of the hyperparameters ($α1,…,αK$) to the empirical sectoral weights derived from the microdata and choose the value $α0>K$ to determine the tightness of the prior distribution around the empirical distribution. Specifically, let $ω^$ denote the set of sectoral weights from a given empirical distribution. We specify the relative sizes of the hyperparameters ($α1,…,αK$) so that the mode of the prior distribution for $ω$ coincides with the empirical sectoral weights $ω^$. This requires setting $αk=1+ω^kα0-K$. The case of flat priors analyzed previously obtains when $α0=K$. Henceforth, we refer to $α0/K$ as the degree of prior informativeness.

### B. Priors on Remaining Parameters

The remaining parameters of the model fall into three categories that we deal with in turn. Our goal in specifying their prior distributions is to avoid imposing any meaningful penalties on most parameter values—except for those that really seem extreme on an a priori basis. The first set comprises the $ρ$'s and $δ$'s, parameterizing the exogenous AR processes for nominal and natural output, respectively. These are assigned loose gaussian priors with mean zero, $N0,52$. We choose to fix the lag length at 2 for both processes: $p1=p2=2$.23 The second set of parameters consists of the standard deviations of the shocks to nominal $σm$ and natural output $σn$. These are strictly positive parameters to which we assign loose gamma priors with mode at 0.025 and a 5th to 95th percentile interval equal to (0.009, 0.195). The last parameter is the Ball-Romer index of real rigidity, $ζ$, which should also be nonnegative. This is captured with a very loose gamma prior distribution, with mode at unity and a 5th to 95th percentile interval equal to $0.47,16.9$. Hence, any meaningful degree of pricing complementarity or substitutability should be a result of the estimation rather than of our prior assumptions.24

### C. Macroeconomic Time Series

We estimate the model using quarterly data on nominal and real output for the U.S. economy. These are measured as seasonally adjusted GDP at, respectively, current and constant prices, from the Bureau of Economic Analysis. We take natural logarithms and remove a linear trend from the data. Whereas the assumptions underlying the model include one of an unchanged economic environment, the U.S. economy has undergone profound changes in recent decades, including the so called Great Moderation and the Volcker disinflation. As a consequence, we choose not to confront the model with the full sample of postwar data. We use the period from 1979 to 1982 as a presample and evaluate the model according to its ability to match business cycle developments in nominal and real output in the period 1983 to 2007.25

### D. Empirical Distributions of Price Stickiness

We work with the statistics on the frequency of price changes for the 350 categories of goods and services (“entry-level items”) reported by Bils and Klenow (2004, henceforth BK), and with the 272 entry-level items covered by Nakamura and Steinsson (2008, henceforth NS). In the latter case, we use the statistics for regular prices (those excluding sales and product substitutions). We refer to the corresponding empirical distributions of price rigidity as distributions with (BK) and without (NS) sales.

Our goal is to map those statistics into an empirical distribution of sectoral weights over spells of price rigidity with different durations. We work at a quarterly frequency and, for computational reasons, in our baseline specification, we consider economies with at most eight quarters of price stickiness. Sectors correspond to price spells, which are multiples of one quarter. We denote an empirical cross-sectional distribution of price rigidity by $ω^kk=18$, where $ω^1$ denotes the fraction of firms that change prices every quarter, $ω^2$ the fraction with an expected duration of price spells between one (exclusive) and two quarters (inclusive), and so on. The sectoral weights are aggregated accordingly by adding up the corresponding CPI expenditure weights. We proceed in this fashion up to the sector with seven-quarter price spells. Finally, we aggregate all the remaining categories, which have mean durations of price rigidity of eight quarters and beyond, into a sector with two-year price spells. This gives rise to the empirical cross-sectional distributions of price stickiness that we use in our estimation, which are summarized in table 1. We denote the sectoral weight for sector $k$ obtained from this procedure by $ω^k$. For each of the BK and NS distributions, we also compute the average duration of price spells, $k¯^=∑k=18ω^kk$, and the cross-sectional standard deviation of the underlying distribution, $σ^k=∑k=18ω^kk-k¯^2$.

Table 1.
Empirical Cross-Sectional Distributions of Price Stickiness
ParameterWith Sales (BK)Without Sales (NS)
$ω^1$ 0.395 0.273
$ω^2$ 0.240 0.071
$ω^3$ 0.116 0.098
$ω^4$ 0.118 0.110
$ω^5$ 0.037 0.060
$ω^6$ 0.033 0.129
$ω^7$ 0.030 0.061
$ω^8$ 0.032 0.198
$k¯^a$ 2.54 4.23
$σ^ka$ 1.86 2.66
ParameterWith Sales (BK)Without Sales (NS)
$ω^1$ 0.395 0.273
$ω^2$ 0.240 0.071
$ω^3$ 0.116 0.098
$ω^4$ 0.118 0.110
$ω^5$ 0.037 0.060
$ω^6$ 0.033 0.129
$ω^7$ 0.030 0.061
$ω^8$ 0.032 0.198
$k¯^a$ 2.54 4.23
$σ^ka$ 1.86 2.66

$a$In quarters. $∑ω^k$ might differ from unity due to rounding.

### E. Simulating the Posterior Distribution

The joint posterior distribution of the model parameters is obtained through application of a Markov-chain Monte Carlo (MCMC) Metropolis algorithm. The algorithm produces a simulation sample of the parameters that converges to the joint posterior distribution under certain conditions.26 We provide details of our specific estimation process in the appendix. The outcome is a sample of 1 million draws from the joint posterior distribution of the parameters of interest, based on which we draw the conclusions that we start to report in the next section.

Having obtained a sample of the posterior distribution of parameters from any given model, we can estimate the marginal likelihood (ML) of the data given the model as
$mlj=fZ*|Mj=∫Lθ|Z*,Mjfθ|Mjdθ$
(14)
and use it for model comparison purposes. In equation (14), $Mj$ refers to a specific configuration of the model and prior distribution, and $fθ|Mj$ denotes the corresponding joint prior distribution. Specifically, we approximate log(ML$j$) for each model using Geweke's (1999) modified harmonic mean. We use these estimates to evaluate the empirical fit of the models relative to one another. The ML ratio of two model configurations yields the Bayes factor, which, when neither configuration is a priori considered more likely, constitutes the posterior odds. It indicates how likely the two models are relative to one another given the observed data $Z*$.

## IV. Results Based on Macrodata Only (Flat Prior over $ω$⁠)

This section reports our main findings. We first show that the aggregate data point to the existence of strong strategic complementarities in price setting and heterogeneity in price stickiness. Importantly, the estimated distribution of price stickiness resembles the empirical distribution based on regular price changes—that is, excluding sales and product substitutions. We then develop intuition for the role of each mechanism through simple counterfactual exercises in which, departing from the estimated model, we “shut down” each mechanism at a time.

### A. Estimates

Table 2 reports the results for the case of uninformative priors in terms of marginal distributions of the parameters.27 The empirical distributions of price rigidity from table 1 are reproduced in the last columns, for ease of comparison. In what follows, we use posterior means as point estimates for the sectoral weights, reported in the third column of the table.28

Table 2.
Parameter Estimates under a Flat Prior
Empirical Distributions
$αk=1$ for all $k(α0=8)$With SalesWithout Sales
$ζ$ 4.440 0.042 0.050 — —
(0.466; 16.863) (0.015; 0.111)
$ω1$ 0.094 0.264 0.276 0.395 0.273
(0.007; 0.348) (0.099; 0.493)
$ω2$ 0.094 0.072 0.086 0.240 0.071
(0.007; 0.348) (0.007; 0.212)
$ω3$ 0.094 0.020 0.027 0.116 0.098
(0.007; 0.348) (0.002; 0.078)
$ω4$ 0.094 0.027 0.037 0.118 0.110
(0.007; 0.348) (0.002; 0.107)
$ω5$ 0.094 0.144 0.156 0.037 0.059
(0.007; 0.348) (0.017; 0.337)
$ω6$ 0.094 0.123 0.144 0.033 0.129
(0.007; 0.348) (0.011; 0.345)
$ω7$ 0.094 0.120 0.143 0.030 0.061
(0.007; 0.348) (0.010; 0.353)
$ω8$ 0.094 0.112 0.132 0.032 0.198
(0.007; 0.348) (0.010; 0.323)
$k¯$ 4.501 4.394 4.37 2.54 4.25
(3.245; 5.760) (3.214; 5.462)
$σk$ 2.139 2.523 2.62 1.86 2.66
(1.584; 2.678) (2.112; 2.893)
$ρ0$ 0.000 0.000 0.000 — —
($-$8.224; 8.224) ($-$0.001; 0.001)
$ρ1$ 0.000 1.426 1.426 — —
($-$8.224; 8.224) (1.273; 1.576)
$ρ2$ 0.000 −0.446 −0.446 — —
($-$8.224; 8.224) ($-$0.593; −0.296)
$σm$ 0.059 0.005 0.005 — —
(0.009; 0.195) (0.005; 0.006)
$δ0$ 0.000 0.002 0.003 — —
($-$8.224; 8.224) ($-$0.002; 0.007)
$δ1$ 0.000 0.541 0.532 — —
($-$8.224; 8.224) (0.270; 0.763)
$δ2$ 0.000 0.146 0.149 — —
($-$8.224; 8.224) ($-$0.027; 0.331)
$σn$ 0.059 0.069 0.081 — —
(0.009; 0.195) (0.030; 0.172)
Empirical Distributions
$αk=1$ for all $k(α0=8)$With SalesWithout Sales
$ζ$ 4.440 0.042 0.050 — —
(0.466; 16.863) (0.015; 0.111)
$ω1$ 0.094 0.264 0.276 0.395 0.273
(0.007; 0.348) (0.099; 0.493)
$ω2$ 0.094 0.072 0.086 0.240 0.071
(0.007; 0.348) (0.007; 0.212)
$ω3$ 0.094 0.020 0.027 0.116 0.098
(0.007; 0.348) (0.002; 0.078)
$ω4$ 0.094 0.027 0.037 0.118 0.110
(0.007; 0.348) (0.002; 0.107)
$ω5$ 0.094 0.144 0.156 0.037 0.059
(0.007; 0.348) (0.017; 0.337)
$ω6$ 0.094 0.123 0.144 0.033 0.129
(0.007; 0.348) (0.011; 0.345)
$ω7$ 0.094 0.120 0.143 0.030 0.061
(0.007; 0.348) (0.010; 0.353)
$ω8$ 0.094 0.112 0.132 0.032 0.198
(0.007; 0.348) (0.010; 0.323)
$k¯$ 4.501 4.394 4.37 2.54 4.25
(3.245; 5.760) (3.214; 5.462)
$σk$ 2.139 2.523 2.62 1.86 2.66
(1.584; 2.678) (2.112; 2.893)
$ρ0$ 0.000 0.000 0.000 — —
($-$8.224; 8.224) ($-$0.001; 0.001)
$ρ1$ 0.000 1.426 1.426 — —
($-$8.224; 8.224) (1.273; 1.576)
$ρ2$ 0.000 −0.446 −0.446 — —
($-$8.224; 8.224) ($-$0.593; −0.296)
$σm$ 0.059 0.005 0.005 — —
(0.009; 0.195) (0.005; 0.006)
$δ0$ 0.000 0.002 0.003 — —
($-$8.224; 8.224) ($-$0.002; 0.007)
$δ1$ 0.000 0.541 0.532 — —
($-$8.224; 8.224) (0.270; 0.763)
$δ2$ 0.000 0.146 0.149 — —
($-$8.224; 8.224) ($-$0.027; 0.331)
$σn$ 0.059 0.069 0.081 — —
(0.009; 0.195) (0.030; 0.172)

The first two columns report the medians of, respectively, the marginal prior and posterior distributions; the third column gives the mean of the marginal posterior distribution; numbers in parentheses correspond to the 5th and 95th percentiles; the last columns reproduce the empirical distributions from table 1.

The cross-sectional distribution that we infer from aggregate data conforms quite well with the empirical distribution based on regular price changes. The macro-based estimates imply that approximately 28% of firms change prices every quarter, 43% change prices at least once a year, and 13% change prices once every two years. The average duration of price spells is thirteen months, and the standard deviation of the duration of price spells is approximately eight months. These numbers are quite close to the empirical distribution based on regular price changes (last column of the table). The correlation between our macro-based estimates and those empirical weights is 0.63. The correlation of the estimates with the empirical distribution based on posted prices is somewhat lower, at 0.43. This is a first, informal indication that the distribution that excludes sales and product substitutions helps the model fit aggregate dynamics better. We will investigate this possibility by performing formal model comparisons using a standard measure of fit.

The index of real rigidities implies strong pricing complementarities. The posterior mean of $ζ$ is 0.05, and the 95th percentile equals 0.11, which falls within the 0.10 to 0.15 range that Woodford (2003) argues can be made consistent with fully specified models. As Carvalho (2006), highlighted, such complementarities interact with heterogeneity in price stickiness to amplify the aggregate effects of nominal rigidities in this type of sticky-price model.

### B. The Role of the Three Mechanisms

Our baseline estimation reveals that all three mechanisms are useful to account for the dynamics of the aggregate price level and output. To further highlight this result, we contrast the estimated benchmark model with counterfactual economies in which we shut down each of the mechanisms at a time. In addition, we compare model-implied dynamics for inflation and output to those of a restricted bivariate VAR, including nominal and real output.

In estimating the VAR, we impose the same assumption used in the models: that nominal output is exogenous and follows an AR(2) process. We allow real output to depend on four lags of both itself and nominal output and to be contemporaneously affected by innovations to nominal output. Estimation is by ordinary least squares.

Figure 1 shows the (mean) impulse response functions of inflation ($πt$, first row) and output ($yt$, second row) to positive innovations $ɛtm$ (left column) and $ɛtn$ (right column) of 1 standard deviation in size.29 In addition, we report the responses of the price level ($pt$, third row), constructed from those of inflation.

Figure 1.

Impulse Response Functions of Models and Bivariate VAR

Figure 1.

Impulse Response Functions of Models and Bivariate VAR

The impulse responses of the benchmark model are fairly close to those of the VAR. As expected, the counterfactual models do a worse job at approximating those impulse response functions. This happens at both short and medium horizons and in response to both shocks. The impulse response functions of inflation imply more volatility when compared to the benchmark or the VAR. This implies swifter adjustments of the aggregate price level (shown in the third row) in these counterfactual economies, relative to the benchmark or the VAR: it responds more (especially at short horizons) and peaks earlier, which in turn causes the output gap $(yt-ytn)$ to respond less and peak earlier (shown in the second row).

We first consider the counterfactual homogeneous-firm economy, where firms change prices every four quarters (i.e., roughly the mean duration of price spells in the benchmark economy) and is otherwise identical to the benchmark. The response of inflation features a kink after four quarters. The aggregate price level and output peak earlier, compared to the benchmark model, in which low-frequency firms still play an important role after the fourth quarter. Furthermore, in the benchmark, pricing interactions between low-frequency and high-frequency firms generate more persistent dynamics than what is implied by the mean frequency of price adjustments, which helps bring the model closer to the data. Introducing multiple sectors that differ in price rigidity smoothes out kinks, thereby producing empirically more plausible responses.

Turning to the counterfactual economy with the distribution of price stickiness based on posted prices, the overall degree of nominal rigidities turns out to be too low to fit the data well. According to that distribution (see the fourth column of table 2—the BK distribution), most price adjustments happen within two quarters after a shock. In addition, while pricing interactions are still operative, their impact is muted because the mass of firms with a large degree of price rigidity is small. These properties cause the aggregate price level to respond more and peak earlier.

Finally, real rigidities are especially important to generate persistent aggregate dynamics. In particular, the right column of figure 1 shows that the counterfactual model with strategic neutrality in price setting ($ζ=1$) fails to generate a hump-shaped response of the price level and output when shocks are relatively transient ($ytn$). In the absence of pricing interactions, optimizing firms are not held back by nonadjusting firms and thus change their prices by a full amount. This property induces fast adjustment of the aggregate price level, which is at odds with the data.

Overall, we find that the three features—real rigidities, heterogeneity in price stickiness, and exclusion of temporary price changes—are all quantitatively important. In particular, our exercise suggests that real rigidities that generate pricing interactions across heterogeneous firms play a key role in generating persistent aggregate dynamics.

## V. Results Based on Informative Priors

Results in the previous section show the importance of the three mechanisms under consideration in our estimated model, but they do not speak to whether models estimated without these features can still provide a good account of the aggregate data. For instance, a homogeneous-firm economy with a greater degree of nominal rigidities, a heterogeneous-firm economy with the distribution of price stickiness based on posted prices yet with stronger complementarities in price setting, and an economy with no real rigidities yet with a greater number of sticky-price firms all have the ability to generate more muted and persistent responses of the aggregate price level. In other words, while the previous counterfactual analysis allows us to understand the role of each mechanism in the estimated model, it does not provide a formal way to assess whether those features are crucial.

We therefore complement our main findings with three additional sets of estimation exercises that feature tighter (or even dogmatic) priors on the cross-sectional distribution of price stickiness, $ω=ω1,…,ωK$, or on the extent of real rigidities, $ζ$. The main goal is to see whether and how the aggregate data allow us to distinguish between different configurations that have the potential to produce sluggish aggregate dynamics. Overall, we find that the insights from the counterfactual analysis continue to apply in our estimation exercises.

### A. Homogeneous Firms

We first ask how sharply the data allow us to discriminate between multisector models with heterogeneity in price stickiness and one-sector models with homogeneous firms. To that end, we estimate one-sector models with price spells ranging from two to eight quarters. We keep the same prior distributions for all parameters besides the sectoral weights. A one-sector model with price spells of length $k$, say, can be seen as a restriction of the multisector model, with a degenerate distribution of sectoral weights ($ωk=1$, $ωk'=0$ for all $k'≠k$).

We pick the best-fitting one-sector model according to the marginal likelihood of the data given the models. Results are reported in table 3. The best-fitting model is the one in which all price spells last for seven quarters. This seems unreasonable in light of the microeconomic evidence. Given the extent of nominal rigidity, not surprisingly the degree of pricing complementarity is smaller. The posterior distributions for the parameters of the nominal output process are quite similar to the ones obtained in the multisector models. Perhaps this should be expected given that this variable is one of the observables used in the estimation. In contrast, the distributions of the parameters of the unobserved driving process are different under the restriction of homogeneous firms. We defer a discussion of what might drive this result to the end of this section.

Table 3.
Best-Fitting Homogeneous Economy
Prior$K=7$, $ω7=1$
$ζ$ 4.440 0.362 0.419
(0.466; 16.863) (0.193; 0.830)
$ρ0$ 0.000 0.000 0.000
($-$8.224; 8.224) ($-$0.001; 0.001)
$ρ1$ 0.000 1.430 1.428
($-$8.224; 8.224) (1.284; 1.568)
$ρ2$ 0.000 $-$0.454 $-$0.452
($-$8.224; 8.224) ($-$0.590; $-$0.310)
$σm$ 0.059 0.005 0.005
(0.009; 0.195) (0.005; 0.006)
$δ0$ 0.000 0.003 0.004
($-$8.224; 8.224) ($-$0.003; 0.011)
$δ1$ 0.000 0.064 0.071
($-$8.224; 8.224) ($-$0.154; 0.319)
$δ2$ 0.000 0.135 0.141
($-$8.224; 8.224) ($-$0.027; 0.327)
$σn$ 0.059 0.216 0.230
(0.009; 0.195) (0.087; 0.421)
Prior$K=7$, $ω7=1$
$ζ$ 4.440 0.362 0.419
(0.466; 16.863) (0.193; 0.830)
$ρ0$ 0.000 0.000 0.000
($-$8.224; 8.224) ($-$0.001; 0.001)
$ρ1$ 0.000 1.430 1.428
($-$8.224; 8.224) (1.284; 1.568)
$ρ2$ 0.000 $-$0.454 $-$0.452
($-$8.224; 8.224) ($-$0.590; $-$0.310)
$σm$ 0.059 0.005 0.005
(0.009; 0.195) (0.005; 0.006)
$δ0$ 0.000 0.003 0.004
($-$8.224; 8.224) ($-$0.003; 0.011)
$δ1$ 0.000 0.064 0.071
($-$8.224; 8.224) ($-$0.154; 0.319)
$δ2$ 0.000 0.135 0.141
($-$8.224; 8.224) ($-$0.027; 0.327)
$σn$ 0.059 0.216 0.230
(0.009; 0.195) (0.087; 0.421)

The first two columns report the medians of, respectively, the marginal prior and posterior distributions; the third column gives the mean of the marginal posterior distribution; numbers in parentheses correspond to the 5th and 95th percentiles.

The multisector model with $K=8$ nests the best fitting homogeneous-firms model. Thus, under measures of fit that do not “correct” for the number of parameters, the former model will necessarily perform at least as well as the latter model. To circumvent that problem, we base our comparison on the marginal likelihood, which already accounts for the fact that the multisector model has more parameters than the homogeneous-firms model.30 We find the fit of the multisector model with the flat prior for $ω$ is much better than that of the best-fitting one-sector model: the posterior odds in favor of the former model are of the order of $1011:1$.31

Our model comparison criterion has the disadvantage that it does not provide information on what drives the improved empirical fit of the multisector model. To shed some light on this question, we compare model-implied dynamics for inflation and output to those of the bivariate VAR, as in the previous section. The impulse response functions from the (benchmark) multisector model and those from the VAR in figure 2 are the same as those in figure 1. We include the impulse responses implied by the best-fitting homogeneous firms model. Since the impulse response functions are conditional on specific parameter values (the posterior means), the comparison does not correct for the larger number of parameters in the multisector model. Thus, it is only meant to provide some indication of the sources of the large differences in the posterior odds of the models.32

Figure 2.

Impulse Response Functions of Models and Bivariate VAR: The Role of Heterogeneity

Figure 2.

Impulse Response Functions of Models and Bivariate VAR: The Role of Heterogeneity

Figure 2 indicates that relative to the one-sector model, the estimated multisector model does a better job at approximating the impulse response functions produced by the VAR at both short and medium horizons, in response to both shocks. The overwhelming statistical support for heterogeneity does not seem to depend on any single feature of the dynamic response of macroeconomic variables to the shocks. However, one noticeable feature of the one-sector model is, once again, the kink (or abrupt change) in the response of inflation (and thus the price level) around the time when all prices have responded to the shock, which now corresponds to the seventh quarter after a shock. This feature of the model, which is now more pronounced because the exogenous process $ytn$ is more transient, is clearly at odds with the data. Compared to the results for the homogeneous firms model in the previous section, the overall size of the responses of inflation and the price level is now closer to the data (and the benchmark model) thanks to the greater degree of nominal rigidities (prices are now fixed for seven quarters instead of four quarters).

These results suggest one explanation for why the estimated parameters associated with the unobserved driving process ($ytn$) are different in the one-sector economy. While the multisector model can rely on the distribution of sectoral weights to balance the response of the economy to shocks at different horizons, the one-sector model lacks this mechanism. Given the facts that nominal output is observed and that its parameter estimates imply quite persistent dynamics in both economies, perhaps the one-sector economy needs to rely on the unobserved process as a more transient and volatile component that can help it do a better job at matching higher-frequency features of the data.

### B. Sales and Product Substitutions

We now turn to estimations that incorporate information from price-setting statistics derived from microdata. This exercise serves two purposes. First, it produces a formal statistical comparison of two model economies, with distributions of price stickiness based on posted or regular prices. Second, while identification is possible solely based on aggregate data, estimates may be less precise in small samples, as suggested by our Monte Carlo exercises. Incorporating information from the microprice data through priors may help the model better identify the role of real rigidities ($ζ$) separately from that of nominal rigidities (implied by $ω=ω1,…,ωK$). This is useful per se and also provides some information about the validity of our baseline estimation in section IVA.

Table 4 presents3 the results for two sets of informative priors ($α0/K=2,$ 5) for each empirical distribution. The bottom row of table 4 reports the log of the marginal likelihood of the various models. For the less informative set of priors ($α0/K=2$), the two empirical distributions that inform the prior lead to models that perform similarly in terms of fit and close to the model estimated under a flat prior. The aggregate data appear to be informative of the parameters of interest. The estimates of $ω$, even under the prior distribution that includes sales, move toward the empirical distribution of price rigidity obtained when discarding sales. For the more informative set of priors ($α0/K=5$), the model with prior based on the empirical distribution without sales fits the data better according to the marginal likelihood criterion; the difference of 4.4 log points implies a posterior odds ratio of roughly $80:1$ in favor of that model.

Table 4.
Parameter estimates with informative priors
Informative Prior, $α0/K=2$Informative Prior, $α0/K=5$Flat PriorEmpirical Distributions
With SalesWithout SalesWith SalesWithout SalesBenchmarkWith SalesWithout Sales
$ζ$ 0.032 0.042 0.018 0.041 0.042 — —
(0.01; 0.08) (0.02; 0.11) (0.01; 0.05) (0.02; 0.10) (0.015; 0.111)
$ω1$ 0.324 0.277 0.425 0.309 0.264 0.395 0.273
(0.17; 0.51) (0.14; 0.45) (0.31; 0.55) (0.21; 0.43) (0.099; 0.493)
$ω2$ 0.123 0.069 0.190 0.059 0.072 0.240 0.071
(0.04; 0.24) (0.01; 0.18) (0.11; 0.29) (0.02; 0.13) (0.007; 0.212)
$ω3$ 0.035 0.033 0.059 0.051 0.020 0.116 0.098
(0.01; 0.09) (0.01; 0.09) (0.02; 0.11) (0.02; 0.10) (0.002; 0.078)
$ω4$ 0.049 0.047 0.081 0.072 0.027 0.118 0.110
(0.01; 0.13) (0.01; 0.12) (0.03; 0.15) (0.03; 0.14) (0.002; 0.107)
$ω5$ 0.106 0.109 0.056 0.066 0.144 0.037 0.059
(0.02; 0.26) (0.02; 0.26) (0.01; 0.15) (0.02; 0.15) (0.017; 0.337)
$ω6$ 0.100 0.142 0.052 0.144 0.123 0.033 0.129
(0.01; 0.27) (0.04; 0.31) (0.01; 0.15) (0.07; 0.25) (0.011; 0.345)
$ω7$ 0.090 0.086 0.042 0.058 0.120 0.030 0.061
(0.01; 0.25) (0.01; 0.24) (0.00; 0.13) (0.02; 0.14) (0.010; 0.353)
$ω8$ 0.088 0.160 0.044 0.200 0.112 0.032 0.198
(0.01; 0.24) (0.05; 0.32) (0.01; 0.13) (0.11; 0.31) (0.010; 0.323)
$k¯$ 3.776 4.367 2.811 4.262 4.394 2.54 4.25
(2.91; 4.69) (3.45; 5.25) (2.31; 3.40) (3.60; 4.91) (3.214; 5.462)
$σk$ 2.515 2.612 2.184 2.725 2.523 1.86 2.66
(2.17; 2.85) (2.28; 2.91) (1.81; 2.56) (2.50; 2.93) (2.112; 2.893)
$ρ0$ 0.000 0.000 0.000 0.000 0.000 — —
(−0.00; 0.00) ($-$0.00; 0.00) ($-$0.00; 0.00) ($-$0.00; 0.00) ($-$0.00; 0.00)
$ρ1$ 1.425 1.427 1.424 1.429 1.426 — —
(1.27; 1.57) (1.27; 1.58) (1.27; 1.57) (1.28; 1.58) (1.27; 1.58)
$ρ2$ $-$0.445 $-$0.447 $-$0.444 $-$0.449 $-$0.446 — —
($-$0.59; $-$0.30) ($-$0.59; $-$0.30) ($-$0.59; $-$0.29) ($-$0.60; $-$0.30) ($-$0.59; $-$0.30)
$σm$ 0.005 0.005 0.005 0.005 0.005 — —
(0.00; 0.01) (0.00; 0.01) (0.00; 0.01) (0.00; 0.01) (0.00; 0.01)
$δ0$ 0.002 0.002 0.002 0.002 0.002 — —
($-$0.00; 0.01) ($-$0.00; 0.01) ($-$0.00; 0.01) ($-$0.00; 0.01) ($-$0.00; 0.01)
$δ1$ 0.514 0.545 0.465 0.563 0.541 — —
(0.30; 0.72) (0.32; 0.75) (0.28; 0.65) (0.38; 0.75) (0.27; 0.76)
$δ2$ 0.176 0.151 0.201 0.146 0.146 — —
(0.01; 0.34) ($-$0.01; 0.32) (0.06; 0.34) ($-$0.01; 0.30) ($-$0.03; 0.33)
$σn$ 0.068 0.066 0.072 0.062 0.069 — —
(0.03; 0.16) (0.03; 0.16) (0.03; 0.17) (0.03; 0.15) (0.03; 0.17)
Log ML 807.56 808.27 803.77 808.16 808.03
Informative Prior, $α0/K=2$Informative Prior, $α0/K=5$Flat PriorEmpirical Distributions
With SalesWithout SalesWith SalesWithout SalesBenchmarkWith SalesWithout Sales
$ζ$ 0.032 0.042 0.018 0.041 0.042 — —
(0.01; 0.08) (0.02; 0.11) (0.01; 0.05) (0.02; 0.10) (0.015; 0.111)
$ω1$ 0.324 0.277 0.425 0.309 0.264 0.395 0.273
(0.17; 0.51) (0.14; 0.45) (0.31; 0.55) (0.21; 0.43) (0.099; 0.493)
$ω2$ 0.123 0.069 0.190 0.059 0.072 0.240 0.071
(0.04; 0.24) (0.01; 0.18) (0.11; 0.29) (0.02; 0.13) (0.007; 0.212)
$ω3$ 0.035 0.033 0.059 0.051 0.020 0.116 0.098
(0.01; 0.09) (0.01; 0.09) (0.02; 0.11) (0.02; 0.10) (0.002; 0.078)
$ω4$ 0.049 0.047 0.081 0.072 0.027 0.118 0.110
(0.01; 0.13) (0.01; 0.12) (0.03; 0.15) (0.03; 0.14) (0.002; 0.107)
$ω5$ 0.106 0.109 0.056 0.066 0.144 0.037 0.059
(0.02; 0.26) (0.02; 0.26) (0.01; 0.15) (0.02; 0.15) (0.017; 0.337)
$ω6$ 0.100 0.142 0.052 0.144 0.123 0.033 0.129
(0.01; 0.27) (0.04; 0.31) (0.01; 0.15) (0.07; 0.25) (0.011; 0.345)
$ω7$ 0.090 0.086 0.042 0.058 0.120 0.030 0.061
(0.01; 0.25) (0.01; 0.24) (0.00; 0.13) (0.02; 0.14) (0.010; 0.353)
$ω8$ 0.088 0.160 0.044 0.200 0.112 0.032 0.198
(0.01; 0.24) (0.05; 0.32) (0.01; 0.13) (0.11; 0.31) (0.010; 0.323)
$k¯$ 3.776 4.367 2.811 4.262 4.394 2.54 4.25
(2.91; 4.69) (3.45; 5.25) (2.31; 3.40) (3.60; 4.91) (3.214; 5.462)
$σk$ 2.515 2.612 2.184 2.725 2.523 1.86 2.66
(2.17; 2.85) (2.28; 2.91) (1.81; 2.56) (2.50; 2.93) (2.112; 2.893)
$ρ0$ 0.000 0.000 0.000 0.000 0.000 — —
(−0.00; 0.00) ($-$0.00; 0.00) ($-$0.00; 0.00) ($-$0.00; 0.00) ($-$0.00; 0.00)
$ρ1$ 1.425 1.427 1.424 1.429 1.426 — —
(1.27; 1.57) (1.27; 1.58) (1.27; 1.57) (1.28; 1.58) (1.27; 1.58)
$ρ2$ $-$0.445 $-$0.447 $-$0.444 $-$0.449 $-$0.446 — —
($-$0.59; $-$0.30) ($-$0.59; $-$0.30) ($-$0.59; $-$0.29) ($-$0.60; $-$0.30) ($-$0.59; $-$0.30)
$σm$ 0.005 0.005 0.005 0.005 0.005 — —
(0.00; 0.01) (0.00; 0.01) (0.00; 0.01) (0.00; 0.01) (0.00; 0.01)
$δ0$ 0.002 0.002 0.002 0.002 0.002 — —
($-$0.00; 0.01) ($-$0.00; 0.01) ($-$0.00; 0.01) ($-$0.00; 0.01) ($-$0.00; 0.01)
$δ1$ 0.514 0.545 0.465 0.563 0.541 — —
(0.30; 0.72) (0.32; 0.75) (0.28; 0.65) (0.38; 0.75) (0.27; 0.76)
$δ2$ 0.176 0.151 0.201 0.146 0.146 — —
(0.01; 0.34) ($-$0.01; 0.32) (0.06; 0.34) ($-$0.01; 0.30) ($-$0.03; 0.33)
$σn$ 0.068 0.066 0.072 0.062 0.069 — —
(0.03; 0.16) (0.03; 0.16) (0.03; 0.17) (0.03; 0.15) (0.03; 0.17)
Log ML 807.56 808.27 803.77 808.16 808.03

The first four columns report the posterior medians under informative priors, and the fifth column reproduces the posterior medians under a flat prior from table 2. Numbers in parentheses correspond to the 5th and 95th percentiles. The last two columns reproduce the empirical distributions from table 1.

To further assess the relative merits of the two empirical distributions (with and without sales) for the purpose of helping the model explain aggregate dynamics, we estimate a series of additional models, progressively increasing the degree of prior informativeness (i.e., increasing $α0/K$). Specifically, we estimate additional models with $α0/K=10,20,100$, and 1,000. The estimation results reveal that the difference between the fit of estimated models increase as the priors become more informative. The difference approaches six log points, which translates into a posterior odds ratio of roughly $400:1$ in favor of the model with prior distribution that excludes sales and product substitutions.33

For intuition, we finally report the impulse response functions as in the previous subsections. Figure 3 shows that in contrast to the counterfactual analysis, the estimated model with prior based on the empirical distribution derived from posted prices (under $α0/K=1,000$) now generates significantly sluggish responses of the aggregate price level, which brings the model closer to the data and the benchmark. However, the improved performance requires an unusually small value (0.018) of $ζ$. Most important, disproportionately larger weights on flexible sectors in the model with sales ($ω1+ω2=63.5%$) continue generating early switches in the sign of the inflation response and, hence, early peaks in the responses of the aggregate price level. This feature is at odds with the data and hampers the fit of the model.

Figure 3.

Impulse Response Functions of Models and Bivariate VAR: The Role of Sales

Figure 3.

Impulse Response Functions of Models and Bivariate VAR: The Role of Sales

### C. No Real Rigidities

Finally, we estimate the model imposing an essentially dogmatic prior on the degree of real rigidities, thereby ruling out meaningful pricing interactions among firms within and across sectors. The dogmatic prior for $ζ$ is uniform on [0.99, 1.01]; those for the other parameters are the same as in the baseline estimation.

Table 5 presents the results. Compared to the benchmark, (a) the model without pricing interactions fits the data worse; (b) the weights are redistributed from less sticky to more sticky sectors, which leads to an average duration of price spells ($k¯$) of 6.2 quarters—significantly larger than those implied by microdata with or without sales (2.5 and 4.3 quarters, respectively); and (c) the standard deviation of shocks to natural output ($σn$) is smaller.

Table 5.
Parameter estimates without real rigidities
Empirical Distributions
No Pricing InteractionsBenchmarkWith SalesWithout Sales
$ζ$ 0.999 0.042 — —
(0.99; 1.01) (0.015; 0.111)
$ω1$ 0.026 0.264 0.395 0.273
(0.00; 0.07) (0.099; 0.493)
$ω2$ 0.044 0.072 0.240 0.071
(0.01; 0.12) (0.007; 0.212)
$ω3$ 0.013 0.020 0.116 0.098
(0.00; 0.05) (0.002; 0.078)
$ω4$ 0.021 0.027 0.118 0.110
(0.00; 0.08) (0.002; 0.107)
$ω5$ 0.162 0.144 0.037 0.059
(0.02; 0.38) (0.017; 0.337)
$ω6$ 0.157 0.123 0.033 0.129
(0.02; 0.35) (0.011; 0.345)
$ω7$ 0.234 0.120 0.030 0.061
(0.03; 0.52) (0.010; 0.353)
$ω8$ 0.277 0.112 0.032 0.198
(0.06; 0.52) (0.010; 0.323)
$k¯$ 6.192 4.394 2.54 4.25
(5.62; 6.66) (3.214; 5.462)
$σk$ 1.790 2.523 1.86 2.66
(1.40; 2.18) (2.112; 2.893)
$ρ0$ 0.000 0.000 — —
(−0.00; 0.00) ($-$0.00; 0.00)
$ρ1$ 1.307 1.426 — —
(1.17; 1.43) (1.27; 1.58)
$ρ2$ $-$0.343 $-$0.446 — —
($-$0.47; $-$0.21) ($-$0.59; $-$0.30)
$σm$ 0.005 0.005 — —
(0.00; 0.01) (0.00; 0.01)
$δ0$ 0.001 0.002 — —
($-$0.00; 0.00) ($-$0.00; 0.01)
$δ1$ 0.367 0.541 — —
(0.04; 0.70) (0.27; 0.76)
$δ2$ 0.342 0.146 — —
(0.07; 0.59) ($-$0.03; 0.33)
$σn$ 0.020 0.069 — —
(0.01; 0.03) (0.03; 0.17)
Log ML 798.55 808.03
Empirical Distributions
No Pricing InteractionsBenchmarkWith SalesWithout Sales
$ζ$ 0.999 0.042 — —
(0.99; 1.01) (0.015; 0.111)
$ω1$ 0.026 0.264 0.395 0.273
(0.00; 0.07) (0.099; 0.493)
$ω2$ 0.044 0.072 0.240 0.071
(0.01; 0.12) (0.007; 0.212)
$ω3$ 0.013 0.020 0.116 0.098
(0.00; 0.05) (0.002; 0.078)
$ω4$ 0.021 0.027 0.118 0.110
(0.00; 0.08) (0.002; 0.107)
$ω5$ 0.162 0.144 0.037 0.059
(0.02; 0.38) (0.017; 0.337)
$ω6$ 0.157 0.123 0.033 0.129
(0.02; 0.35) (0.011; 0.345)
$ω7$ 0.234 0.120 0.030 0.061
(0.03; 0.52) (0.010; 0.353)
$ω8$ 0.277 0.112 0.032 0.198
(0.06; 0.52) (0.010; 0.323)
$k¯$ 6.192 4.394 2.54 4.25
(5.62; 6.66) (3.214; 5.462)
$σk$ 1.790 2.523 1.86 2.66
(1.40; 2.18) (2.112; 2.893)
$ρ0$ 0.000 0.000 — —
(−0.00; 0.00) ($-$0.00; 0.00)
$ρ1$ 1.307 1.426 — —
(1.17; 1.43) (1.27; 1.58)
$ρ2$ $-$0.343 $-$0.446 — —
($-$0.47; $-$0.21) ($-$0.59; $-$0.30)
$σm$ 0.005 0.005 — —
(0.00; 0.01) (0.00; 0.01)
$δ0$ 0.001 0.002 — —
($-$0.00; 0.00) ($-$0.00; 0.01)
$δ1$ 0.367 0.541 — —
(0.04; 0.70) (0.27; 0.76)
$δ2$ 0.342 0.146 — —
(0.07; 0.59) ($-$0.03; 0.33)
$σn$ 0.020 0.069 — —
(0.01; 0.03) (0.03; 0.17)
Log ML 798.55 808.03

The first column reports the posterior medians under the tight prior on $ζ$, and the second column reproduces the posterior medians under a flat prior from table 2. Numbers in parentheses correspond to the 5th and 95th percentiles. The last two columns reproduce the empirical distributions from table 1.

We report the impulse response functions in figure 4. The absence of real rigidities is partially compensated by the (perhaps unrealistically) larger degree of nominal rigidities. However, the lack of pricing interaction continues to lead the price level to respond more and to peak earlier, as in the previous counterfactual analyses. Such discrepancy between the model and the data is more pronounced when nominal output shocks hit the economy (the left column in the figure). The reason is that the estimated processes for $mt$ are similar regardless of real rigidities, as nominal output is observed.

Figure 4.

Impulse Response Functions of Models and Bivariate VAR: The Role of Real Rigidities

Figure 4.

Impulse Response Functions of Models and Bivariate VAR: The Role of Real Rigidities

## VI. Robustness

Our findings are robust to different assumptions about exogenous processes, the number of sectors, and other models of price setting. We in particular consider two alternative price-setting models: the Calvo (1983) model and what we term the “random Taylor” model. The latter produces the exact same results as our conventional Taylor model, yet can speak to a much larger set of empirical facts about price setting derived from microdata. The online appendix provides details of our robustness analysis and the implications of the alternative models of price setting.

## VII. Conclusion

If prices change frequently and every price change contributes wholly to offset nominal disturbances, then nominal price rigidity cannot be the source of large and persistent monetary nonneutralities. Hence, bridging the micro-macro gap on the extent of price rigidity requires that price adjustments somehow fail to perfectly neutralize monetary innovations.

This paper provides evidence on three features—real rigidities, heterogeneity in price stickiness, and sales and product substitutions—that can reconcile frequent individual price changes with sluggish adjustments of the aggregate price level. To that end, we estimate a semistructural model that allows us to extract information about real rigidities and cross-sectional heterogeneity in price stickiness and to discriminate between different distributions of price stickiness based on aggregate data.

Our estimation results point to the presence of large, real rigidities and a significant degree of heterogeneity in price stickiness. Moreover, the cross-sectional distribution of price stickiness implied by aggregate data is in line with a micro-based empirical distribution that factors out sales and product substitutions. All three mechanisms play an important role in accounting for aggregate dynamics. Real rigidities that induce strong pricing interactions across heterogeneous firms are particularly important. This finding warrants additional research on the nature and sources of real rigidities, which are unspecified in our semistructural model.

As a by-product, we develop a price-setting model that produces the same aggregate dynamics as our multisector model with Taylor pricing and yet can match various empirical facts on price setting, including the evidence of variation in the duration of price spells at the quote-line level. Hence, the model provides a cautionary note on attempts to test specific models of price setting by confronting them with descriptive price-setting statistics (Klenow & Kryvtsov, 2008).

Finally, our findings reinforce the general point that heterogeneity can matter for aggregate dynamics. Sectors in our model, however, are assumed to differ only in the degree of price stickiness. Other sources of sectoral heterogeneity, which may be correlated systematically with sector-specific price stickiness, will certainly have interesting implications for aggregate dynamics.34 Heterogeneity along other dimensions is likely to generate the need for additional observables, such as sectoral price and quantity data, in order to identify the underlying cross-sectional distribution. We leave this potentially interesting endeavor for future research.

## Notes

1

See, for example, the survey by Maćkowiak and Smets (2008).

2

Information frictions can also lead to large contract multipliers. Not surprisingly, that literature picked up steam after the empirical literature based on microprice data flourished. Classic contributions include Caballero (1989), Reis (2006), and Maćkowiak and Wiederholt (2009), who obtain large monetary nonneutralities in models with information frictions in which prices change continuously. More recently, Bonomo et al. (2014) obtain a large contract multiplier in an estimated model with menu costs and partially costly information.

3

Carvalho and Schwartzman (2015) show how this intuition can be formalized in terms of a selection effect relative to the timing of price changes, which arises in the class of time-dependent pricing models.

4

Nakamura and Steinsson (2010) conclude that this interaction is not important in their calibrated menu-cost model.

5

Coibion et al. (2015) provide evidence that sales are essentially acyclical, which is consistent with the models in Guimaraes and Sheedy (2011) and Kehoe and Midrigan (2015). Kryvtsov and Vincent (2014), on the other hand, argue that sales do not help reconcile micro- and macroevidence on price rigidity. They document a large degree of cyclicality of sales in the U.K. microdata and develop a model that can explain their findings. In bad times, consumers intensify their search for bargain prices, and firms increase the frequency of sales. This “complementarity” between search effort and sales frequency breaks down the strategic substitutability of sales that would otherwise arise (as in Guimaraes and Sheedy, 2011), and leads to cyclical sales.

6

Several earlier papers in the literature combine structural equations with empirical specifications for other parts of the model (e.g., Sbordone, 2002; Guerrieri, 2006; Coenen, Levin, & Christoffel, 2007).

7

Real and nominal rigidities are not separately identified in the version of our model with Calvo pricing and homogeneous price stickiness. This, however, is the only exception to the class of models that we entertain.

8

The original Taylor model, in which all prices are fixed for the same number of periods, has been criticized for being at odds with both micro- and macrofacts. First, that model tends to produce impulse response functions with kinks, which are uncommon in impulse response functions produced by estimated VARs. Second, in that model, all price spells have the same duration, which is at odds with the microeconomic evidence (Klenow & Kryvtsov, 2008). The Calvo model is usually more successful along both dimensions. However, as will become clear, our estimated multisector models with Taylor pricing produce smoother dynamics that resemble those of a VAR, especially when pricing decisions are strategic complements. In addition, we show in the online appendix that our version of the Taylor model can be recast into a model that yields identical aggregate dynamics and yet can be made consistent with a large number of microprice facts. Hence, although it still undeperforms the Calvo model when it comes to fitting aggregate data, our multisector Taylor model can account for both aggregate dynamics and microprice facts significantly better than the conventional one-sector Taylor model.

9

To be clear, the microinformation also plays a role in the baseline case, as we choose the support of the price stickiness distribution, which is fixed preestimation, to be roughly consistent with the empirical distributions.

10

Bouakez, Cardia, and Ruge-Murcia (2014) find similar results in an extension of their earlier paper to a larger number of sectors.

11

Their frameworks are thus closer to the generalized time-dependent model of Dotsey, King, and Wolman (1997) than to our model with ex-ante heterogeneity.

12

Their estimated model features indexation to an average of past inflation and a (nonzero) constant inflation objective. Thus, their finding is that the average time between “contract reoptimizations” is comparable to the average duration of price spells documented by Bils and Klenow (2004).

13

This is analogous to assumption 3.1 in Woodford (2003).

14

In the appendix, we discuss how the model can be enriched with idiosyncratic shocks that can help it match some microfacts about the size of price changes without affecting any of its aggregate implications.

15

The results are conditional on the particular model of price setting that we adopt. In the appendix, we discuss the extent to which our conclusions generalize to alternative price-setting specifications.

16

We write all such approximations as equalities, ignoring higher-order terms.

17

We think such an interpretation is unreasonable because we take nominal output to be exogenous. In that context, an interpretation of $ytn$ as being driven by preference and technology shocks would imply that these shocks have no effect on nominal output (i.e., that they have exactly offsetting effects on aggregate output and prices).

18

This is what comes out of a fully specified multisector model with the usual assumption of Dixit-Stiglitz preferences.

19

See, for example, Woodford (2003), Carvalho, Lee, and Park (2018), and Carvalho and Nechio (2016).

20

Jadresic (1999) discusses identification in a similar context. The main differences are that he considers a regression based on a first-differenced version of the analogous equation in his model and assumes $ρ1=1$ and that the term corresponding to $∑j=0K-1∑k=j+1KFβ,δ1,kωkkΔyt-jn$ is an i.i.d. disturbance.

21

In the appendix, we illustrate how the process works in a two-sector model.

22

Gelman et al. (2003) offer a good introduction to the use of Dirichlet distribution as a prior distribution for the multinomial model.

23

In principle, we could specify priors over $p1,$$p2$ and estimate their posterior distributions as well. However, the computational cost of estimating all the models in the paper is already high, and we restrict ourselves to this specification with a fixed number of lags. Our conclusions are robust to alternative assumptions about the number of lags (see section VI).

24

We do not include $β$ in the estimation and set $β=0.99$.

25

We make use of the presample, 1979 to 1982, by initializing the Kalman filter in the estimation stage with the estimate of $Zt$ and corresponding covariance matrix obtained from running a Kalman filter in the presample. We use the parameter values in each draw. For the initial condition for the presample, we use the unconditional mean and a large variance-covariance matrix.

26

These conditions are discussed in Gelman et al. (2003, part III).

27

Figures that show prior and posterior distributions from the estimation exercises in sections IVA to VC are provided in the appendix. We use a gaussian kernel density estimator to graph the marginal posterior density for each parameter. The priors on $k¯$ and $σk$ are based on 100,000 draws from the prior Dirichlet distribution.

28

The results are almost insensitive to using alternative point estimates, such as the values at the joint posterior mode, or taking medians or modes from the marginal distributions and renormalizing so that the weights sum to unity.

29

Following the notation of the semistructural model, in the VAR, $ɛtm$ denotes innovations to nominal output and $ɛtn$ denotes the other (orthogonal) innovations.

30

The reason is that the vector of parameters is “integrated out” in equation (14).

31

Log ML are 808.03 and 781.33, respectively.

32

In addition, while looking at the impulse response functions is clearly instructive, the usual caveat applies because our likelihood-based estimation method fits the model to the entire autocovariance function of the data, not any specific moments.

33

As we tighten the priors on the sectoral weights, the fit of models estimated under priors with sales (BK) deteriorates. In contrast, the fit of models estimated under priors based on regular prices (NS) remains essentially unchanged. This is consistent with our previous finding that sectoral weights estimated under flat priors are somewhat similar to the empirical distribution without sales and product substitutions.We refer interested readers to the online appendix for further details on this exercise.

34

For example, we refer readers to Barsky, House, and Kimball (2007), who show aggregate implications of flexible-price durable sectors; Nakajima, Sudo, and Tsuruga (2010), who analyze a model with cross-sectional heterogeneity in real rigidities; and Eusepi, Hobijn, and Tambalotti (2011), who introduce sector-specific labor shares.

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## Author notes

This paper circulated also under the title “Real Rigidities and the Cross-Sectional Distribution of Price Stickiness: Evidence from Micro and Macro Data Combined.” For helpful comments and suggestions, we thank Luis Álvarez, Marco Del Negro, Stefano Eusepi, Henrik Jensen, Ed Knotek, Oleksiy Kryvtsov, John Leahy, Frank Schorfheide, two anonymous referees, and, especially, Giorgio Primiceri. We also thank seminar participants at the ESWC 2010, AEA 2010, Banque de France conference “Understanding Price Dynamics: Recent Advances,” ESEM 2009, SED 2009, NAMES 2009, Bank of Canada, ECB-WDN Workshop, Rutgers University, LAMES 2008, Danmarks Nationalbank, University of Copenhagen, Riksbank, and NY Fed.

A supplemental appendix is available online at http://www.mitpressjournals.org/doi/suppl/10.1162/rest_a_00811.