empirical comparison of sticky price and sticky information models

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Journal of Macroeconomics 30 (2008) 906–927

www.elsevier.com/locate/jmacro

Empirical comparison of sticky price andsticky information models

Oleg Korenok

VCU School of Business, Department of Economics, 1015 Floyd Ave., Richmond, VA 23248, United States

Received 9 September 2006; accepted 23 April 2007Available online 10 June 2007

Abstract

This paper compares empirically two alternative explanations of the relationship between aggre-gate price and labor share: the sticky price and the sticky information models. To compare models,we derive a similar analytical form for both models and use post WWII US labor share and aggre-gate price series. We use the Bayesian full information likelihood approach for parameter estimation,uncertainty evaluation, and model comparison. Statistical comparison of the two non-nested modelsand estimates of the empirical encompassing model lead to the same result – the sticky informationmodel is dominated by the sticky price model. An unrestricted VAR, however, dominates bothmodels.� 2007 Elsevier Inc. All rights reserved.

JEL classification: E12; E3; C32

Keywords: Sticky price; Sticky information; Model selection; Bayesian model comparison

1. Introduction

The literature on the new-Keynesian Phillips curve has identified several ways in whichthe standard sticky price model does not adequately model the relationship between aggre-gate prices and labor share. Ball (1994) found that the model implies that announced cred-ible disinflation causes booms rather than recessions. Fuhrer and Moore (1995) showedthat the sticky price model falls short of explaining inflation persistence in US. Finally,

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O. Korenok / Journal of Macroeconomics 30 (2008) 906–927 907

Mankiw and Reis (2002) noted that the model has trouble explaining why shocks to mon-etary policy have delayed and gradual effects on inflation.

Mankiw and Reis (2002) propose the sticky information model that addresses the fail-ures of the new-Keynesian Phillips curve. They posit that information about macroeco-nomic conditions spreads slowly because of information acquisition or re-optimizationcosts. Compared to the standard sticky price model, prices in this setup are always read-justed, but decisions about prices are not always based on the latest available information.The model is representative of the wider class of Rational Inattention models developed inPhelps et al. (1970), Lucas (1973), Sims (2003) and Woodford (2003).

To compare these two alternative models, we utilize a theoretical relation betweenaggregate prices and labor share that allows us to leave unspecified household preferences,wage setting and money demand. We introduce then a modeling approach to nest thesticky price and the sticky information models within a single empirical framework. Weuse a single-step estimation method that provides consistent estimates of adjustmentspeeds and reliable confidence bands that enable us to reject flexible price and informationhypotheses.

To estimate information and price stickiness, we follow the approach, advocated byGali and Gertler (1999) and Sbordone (2002), and use the theoretical relation betweenaggregate prices and labor share. For estimation purposes, we transform the relation intoits stationary equivalent: the relation between real marginal cost, measured by unit laborcost/price ratio, and changes in unit labor cost. Our methodology differs in several respectsfrom previous attempts to estimate information stickiness. First, unlike Mankiw and Reis(2002), the relation between labor share and prices is derived from a firm’s optimizationproblem. Second, it does not require taking a stand on household preferences or wage set-ting as in Khan and Zhu (2002) in order to relate inflation and the output gap. Finally, asshown in Sbordone (2002), conclusions based on this approach are robust to alternativespecifications of the production function, which specifies the relation between unobservedmarginal cost and labor share.

We use the method of undetermined coefficients to solve both models. The same solu-tion technique enables us to construct and to estimate an empirical model that encom-passes the sticky price and sticky information models as special cases, even though twotheoretical models are non-nested. This encompassing model facilitates the modelscomparison.

We use a Bayesian version of the full information likelihood approach, which enablesus not only to estimate parameters and the statistical uncertainty around them, but also tocompare non-nested sticky information and sticky price models. Full information likeli-hood approach was advocated by Fuhrer and Moore (1995)1 for empirical evaluationof the sticky price model and hypothesis testing. Our estimation methodology differs froma two-step approach used by Mankiw and Reis (2001), Sbordone (2002), Khan and Zhu(2002), and Kiley (2007). A driving process is estimated in the first step and then, in thesecond step, it is incorporated into a linear rational expectations solution for process ofinterest (unemployment, unit labor cost/price ratio or the output gap, respectively).Parameters, estimated in the first step, are taken as given in the second step. Thus, the

1 See also Linde (2005) on comparison of full likelihood and limited likelihood methods in sticky price modelframework.

908 O. Korenok / Journal of Macroeconomics 30 (2008) 906–927

measure of uncertainty around the parameter of information or price stickiness does notincorporate the uncertainty around parameters of the driving process.

Our empirical comparison favors the sticky price model. This result holds when we esti-mate the models using the two-step approach; it also holds when we estimate the modelsusing 1983:1–2002:1 period. Our estimates of price and information stickiness are closeand imply that firms revise prices or information on average about every 10 months with95% probability interval between 9 and 12 months. Finally, the simple time series modeldominates both theoretical models.

The rest of the paper is organized as follows. The optimization based dynamics of realmarginal cost and the rational expectation solutions are described in Section 2. We explainthe Bayesian estimation of information and price stickiness models and describe the meth-odology for models comparison in Section 3. The data, estimation results and modelscomparison is presented and discussed in Section 4. We test robustness of our results tothe estimation method and sample size in Section 5. Finally, concluding remarks are givenin Section 6.

2. Optimization based dynamics of marginal cost

In this section, we derive the optimization-based dynamics of marginal cost for thesticky price and sticky information models. Also we solve out expectations in both modelsusing the method of undetermined coefficients.

A continuum of monopolistically competitive firms, indexed by i, produce differentiatedgoods. Firms operate with a technology,

Y i;t ¼ H ai;t; ð1Þ

where output Yi,t is firm specific; labor Hi,t is a single factor of production and is uniquefor each agent, and a 2 [0, 1].

The demand curve for product i can be represented as

Y i;t ¼P i;t

P t

� ��h

Y t; ð2Þ

where Pi,t is the optimization based price choice of the firm i, Pt is the index of prices, Yt is

the aggregator function defined as Y t ¼ ðR 1

0Y

h�1h

i;t diÞh

1�h, and h is the Dixit–Stiglitz elasticityof substitution among differentiated goods.

A supplier of good i does not believe that his price decisions can affect either the aggre-gate demand Yt or the aggregate price Pt because good i is a very small part of households’consumption. Thus the supplier chooses the price Pi,t, taking aggregate demand and aggre-gate price as given. Optimization then involves setting a price,

P i;t

P t¼ lSi;t; ð3Þ

where Si,t is real marginal cost of supplier and l ¼ hh�1

is a markup. Taking into accountthe firm technology (1), the real marginal cost of the individual firm can be written as

Si;t ¼1

aW tH i;t

P tY i;t; ð4Þ


where Wt is a labor compensation on a common, perfectly competitive labor market.Assuming a specific aggregator function for labor input and using the demand curve,we can rewrite (4) as

Si;t ¼ StP i;t

P t

� ��h !1�a

a

; ð5Þ

where St ¼ 1a

W tHtP tY t

is the average level of real marginal costs in the economy. This expressionshows that the real marginal cost of the firm i does not need to be the same as the average.In the model the fact that firms charge different prices determines their different levels ofsales, and hence firm’s different level of marginal costs.

Combining the profit maximizing rule (3) with the marginal cost (5) yields a relationbetween the average level of marginal cost in economy, individual firm price choicesand the aggregate price level,

P i;t

P t¼ lSt

P i;t

P t

� ��h !1�a

a

; ð6Þ

we rewrite (6) as

pi;t � pt ¼ st �hð1� aÞ

aðpi;t � ptÞ; ð7Þ

where we use lower case variables to denote the log transformation. Here and below, weignore constants because we focus on deviations from the mean in empirical applications.(7) can be solved for the individual price in terms of the aggregate price and marginal cost:

pi;t ¼ ast þ pt; ð8Þ

where a ¼ aaþhð1�aÞ, a < 1 when h > 0. We interpret (8) as a positive relation between desired

relative price of the individual firm and real marginal cost. The relation is a direct conse-quence of profit maximizing rule for the individual firm.

In the above discussion, all firms choose the price of good i each period. The pricechoice is independent of prices that were charged in the past. The choice is based onthe full information about current demand and cost. These conditions characterize theperfectly flexible price model. Now we want to consider two deviations from the perfectlyflexible model: one, where the fraction of firms have to keep the last period price, the stickyprice model, and the other, where the fraction of firms have to use past or outdated infor-mation, the sticky information model.

2.1. Sticky price model

Following Calvo (1983), we assume that in every quarter a fraction of firms kp can set anew price independently of the past history of price changes. Other firms have to keep thelast period price. The expected time between price updates is therefore 1

kpquarters. The

model nests the monopolistically competitive economy with perfectly flexible prices inthe case kp = 1.

Because the opportunity to change price is random, a firm wants to set it’s price equalto an average of desired prices until next adjustment:


xi;t ¼ kp

X1j¼0

ð1� kpÞjEtpi;tþj; ð9Þ

where xi,t is the optimal price choice. Note that index for each individual firm can bedropped since all firms according to (8) make similar choices. Also note that we set thestochastic discount factor to one.2

The overall price level is the weighed average of optimal and previous period prices

pt ¼ kpxt þ ð1� kpÞpt�1: ð10Þ

The system of three equations (the firm’s desired and actual price (8) and (9), and the over-all price level (10)) can be solved for the relation between inflation and real marginal costs

pt ¼1

nst þ Etptþ1; ð11Þ

where n ¼ ð1�kpÞak2

pis a nonlinear function of structural parameters. Eq. (11) implies that cur-

rent inflation can rise because increase in cost or agent’s expectation of high inflation in thefuture.

From the definition of St and the definition of average unit labor cost it follows that realmarginal cost is proportional to unit labor cost/price ratio,

st ¼ ulct � pt: ð12ÞUsing (12), we follow Sbordone (2002) and derive the relation between the real marginalcost and changes in unit labor cost,

st ¼ z1st�1 þ Dulct � ð1� z1ÞX1j¼0

z�j2 EtDulctþj; ð13Þ

where z1 and z2 are the real roots of the characteristic polynomial of the difference equa-tion in pt, pðzÞ ¼ z2 � 2þ 1

n

� �zþ 1 ¼ 0, with 0 < z1 < 1 < z2. Eq. (13) describes the process

for marginal cost in the sticky price model as a weighted average of the past marginalcosts, current and expected future changes in unit labor cost.

2.2. Sticky information model

As in the flexible price model, Mankiw and Reis (2002) assume that an individual firmmakes a price adjustment each period. The novelty of the approach is the assumption thatthe information used for decision-making is not necessarily the current one. Only a frac-tion of firms kinf uses current information in pricing decisions, while the remaining fraction1 � kinf uses past or outdated information. The expected time between price updates istherefore 1

kinfquarters. The model nests the monopolistically competitive economy with

perfectly flexible prices when kinf = 1.One can think about the sticky information model as a variant of the sticky price model,

where a fraction of firms that are unable to set prices optimally instead of keeping the lastperiod price use a more complex updating scheme. Firms, when they have an opportunityto set prices optimally, solve not only for optimal current prices but also for the infinite path

2 Gali and Gertler (1999) and Sbordone (2002) demonstrated that setting stochastic discount factor to one havelittle impact on estimates of price stickiness and other structural parameters.


of future prices based on available information. When they are not able to set prices opti-mally, firms set the price to the appropriate value from this solution set.

Each period, a representative firm sets the price to

xji;t ¼ Et�jpi;t; ð14Þ

where j represents the latest period when the firm updated its information set. As in thesticky price model index for each individual firm can be dropped since all firms accordingto (8) make similar choices. The overall price in this case is an average among the firmswho update information sets in different periods,

pt ¼ kinf

X1j¼0

ð1� kinfÞjxjt : ð15Þ

Combining (8) and (14) with (15) yields,

pt ¼ kinf

X1j¼0

ð1� kinfÞjEt�jðast þ ptÞ: ð16Þ

Note that the structure of expectations here is different from the usual backward iterationof expectations: expectations are formed for the current t value of the variable at the dif-ferent t � j periods in the past (for example Et�jpt).

With some algebra (see the Appendix for details), Eq. (16) can be transformed into anexpression that describes the dynamics of real marginal cost,

st ¼ gDulct þ gst�1 þ gkinf

X1j¼0

ð1� kinfÞjEt�j�1ðð1� aÞst � ð1� aÞst�1 � DulctÞ; ð17Þ

where g ¼ 1�kinf

1�kinf ð1�aÞ < 1.

Dynamics of (17) is comparable to the dynamics of real marginal cost in the sticky pricemodel. In both models, it depends on a weighted average of past realizations of real mar-ginal cost, current changes in unit labor cost, and expectations. This similarity enables usto compare the sticky price and sticky information models on the basis of observed seriesfor the unit labor cost/price ratio and the first difference in unit labor cost.

The main difference between (13) and (17) is the expectation term. In the sticky pricemodel, firms reset prices on rare occasions. The expectations term, which is the weightedaverage of current expectations about future changes in unit labor cost, is the way firmstake into account periods when costs change, but prices do not. In the sticky informationmodel, on the other hand, firms reset prices every period. But they are not able to use cur-rent information about cost to make the price choice. The expectations term, which is theweighted average of past expectations about current costs, is the way firms take intoaccount outdated information.

2.3. Rational expectations solution

We solve out for unobserved expectations by assuming agents are rational and using themethod of undetermined coefficients.3

3 For careful and extensive discussion of the method of undetermined coefficients see Taylor (1986).


Assume that Dulct follows a general linear process with the MA representation,

Dulct ¼X1k¼0

wk�t�k; ð18Þ

where wk, k = 0,1,2 . . ., is a sequence of parameters and �t are serially uncorrelated ran-dom variables with zero mean.

For the sticky price model, a solution for the stochastic process st satisfies (13) and (18).One can guess that the solution has the following general form:

st ¼X1k¼0

ck�t�k: ð19Þ

Finding the solution for st is equivalent to finding ck that satisfy (13) and (18). To solve forck, we substitute st and st�1 from (19) and Dulct from (18) in (13). By moving (18) forwardand taking expectations with respect to information in period t, we substitute expectationsof future differences of unit labor cost as well. The substitutions result in

X1k¼0

ck�t�k ¼ z1

X1k¼0

ck�t�1�k þX1k¼0

wk�t�k � ð1� z1ÞX1k¼0

X1j¼0

z�j2 wkþj�t�k: ð20Þ

By equating coefficients of �t, �t�1, �t�2, . . . on both sides of (20) we solve for ck in terms ofz1, z2 and wk,

c0 ¼ w0 � ð1� z1ÞX1j¼0

z�j2 wj;

ck ¼ z1ck�1 þ wk � ð1� z1ÞX1j¼k

zk�j2 wj;

k ¼ 1; 2; . . .

ð21Þ

The forward looking nature of prices and therefore unit labor cost/price ratio results in thesolution for ck which depends on future wk.

For the sticky information model, a solution for the stochastic process st has to satisfy(17) and (18). We make a similar guess that (19) is the solution, then we substitute st andst�1 from (19) and Dulct from (18) in (17). Finally, we substitute expectations with respectto information in period t � j � 1 in (17) by taking appropriate expectations of (18) and(19). The substitutions result in:

X1k¼0

ck�t�k ¼ gX1k¼0

wk�t�k þ gX1k¼0

ck�t�k�1 þ gkinf

X1j¼0

ð1� kinfÞj ð1� aÞX1

k¼jþ1

ck�t�k

� ð1� aÞX1k¼j

ck�t�k�1 �X1

k¼jþ1

wk�t�k

!: ð22Þ

By equating coefficients of �t, �t�1, �t�2, . . . on both sides of equality we solve for ck in termsof g, ki and wk:


c0 ¼ gw0;

ck ¼gð1� kinfÞk

1� gð1� aÞð1� ð1� kinfÞkÞwk þ

gð1� ð1� aÞð1� ð1� kinfÞkÞÞ1� gð1� aÞð1� ð1� kinfÞkÞ

ck�1;

k ¼ 1; 2; . . .

ð23Þ

The difference in RE solutions for the two models can be traced to the difference in profitmaximization problems. In the sticky price model solution (21) ck depends not only onpast and current values of wk, but also on all future values. Future values appear becauseagents have to take into account future costs when they are not able to set the price. In thesticky information model RE solution (23), ck depends only on past and current values ofwk because agents can maximize profit in each period using only current and pastinformation.

The method of undetermined coefficients enables us to describe the dynamics of realmarginal cost and changes in unit labor cost in a similar analytical form. The similar ana-lytical form, in turn, allows us to construct the empirical encompassing model that neststhe sticky information and sticky price models, even though two theoretical models arenon-nested.

3. Estimation

In this section, we describe a Bayesian full information likelihood approach for estima-tion and models comparison. The posterior distribution, the main object of interest inBayesian analysis, is the product of a likelihood and a prior:

pðhjdataÞ ¼ pðhÞLðh; dataÞ;

where h is a vector of parameters. We use the general representation for (19), together withexogenous process Dulct, (18), to form the likelihood function. Then we formulate priorsfor structural parameters in the model. Finally, we discuss Bayesian comparison of non-nested sticky price and sticky information models and describe construction of the empir-ical encompassing model that nests two models.

3.1. Likelihood function

Before formulating the likelihood, we modify the solution for the real marginal cost,Eq. (19), to make the estimation of structural parameters possible. First, we truncatethe infinite MA representation of st at kmax. Second, following the Sargent and Hansen(1980) approach, we add the error term vt. Without the second error term, only one ran-dom shock �t determines behavior of both unit labor cost and marginal cost, which makesthe system of Eqs. (18) and (19) stochastically singular. We interpret the second error termas compensating for the fact that our versions of the sticky price and the sticky informa-tion models are potentially incomplete. One can think of other shocks to inflation, such asthe oil shock, which are unrelated to unit labor cost. Alternatively, vt can be interpreted asmeasurement errors.

Assuming stationary AR(q) for the driving process, Dulct, the processes for the changein unit labor cost and marginal cost together can be written as


st ¼Xkmax

k¼0

ck�t�k þ vt;

Dulct ¼Xq

j¼1

qjDulct�j þ �t;

ð24Þ

where kmax is our truncation point. From the second expression it follows that

�t�k ¼ Dulct�k �Xq

j¼1

qjDulct�k�j;

which, when substituted into the first expression, results in

st ¼Xkmax

k¼0

ck Dulct�k �Xq

j¼1

qjDulct�k�j

! !þ vt:

Let K be a vector of structural parameters of interest. From the rational expectations solu-tion (21) and (23), it follows that the coefficients of marginal cost process, ci, are functionsof structural parameters, K, and coefficients of the driving process, w = [w0,w1, . . .], wherewi are taken from (18). It is easy to solve for wi as a function of q, q = [q1,q2, . . .], i.e. in theAR(1) case, wi ¼ qi

1.Taking into account the above transformations, (24) can be written as

st ¼Xkmax

k¼0

ckðK;qÞ Dulct�k �Xq

j¼1

qjDulct�k�j

! !þ vt;

Dulct ¼Xq

j¼1

qjDulct�j þ �t:

ð25Þ

The coefficients ck(K,q) are different for two models. The analytical form for ck(K,q) is ageneralization of (21) and (23).

Error terms (�t,vt) are independent by construction. We assume that they are drawsfrom a multivariate normal distribution with zero mean and variance covariance matrixR. We collect all parameters of the model to be estimated in a single vector h = (R,q,K) 0.The likelihood function then can be written (see the Appendix for details), apart from aconstant, as

Lðh; dataÞ / jRjð�1=2T Þ expð�1=2trR�1QÞ; ð26Þwhere

Q ¼v0v v0�

�0v �0�

� �:

We denote the vector of vt by v and the vector of �t by �. Both vectors can be solved from (25).

3.2. Estimated structural parameters and priors selection

Not all structural parameters in our models are identified, as a results we have to chooseset of estimated parameters and calibrate the rest.


In the sticky price model, the identified parameter n is a nonlinear combination of laborshare a, the Dixit–Stiglitz elasticity of substitution among differentiated goods h, and theprice stickiness kp. The parameter of price stickiness kp is our object of interest. FollowingSbordone (2002), we calibrate the labor share a = 0.75 and the Dixit–Stiglitz elasticity ofsubstitution among differentiated goods h = 6 (which implies the average value of markupof 1.2).4 This calibration ensures identification for the parameter of price stickiness. Also itfixes the value of a at one-third.

In the sticky information model, both a and kinf are identified. However, to treat stickyprice and sticky information models symmetrically, we fix the value of a at one-third as inthe sticky price model.5 Two estimated parameters in the sticky information model com-pared to one parameter in the sticky price model could unfairly favor sticky price modelbecause marginal likelihood penalizes additional parameter that fails to improve the mod-els explanatory power.

The information about prior distributions for all parameters is summarized in Table 1.For the price stickiness parameter kp and for the information stickiness parameter kinf wechoose the same non-informative Uniform prior distribution on the range from zero toone. This range covers all possible values for price and information stickiness includingperfectly flexible prices, kp = 1, or perfectly flexible information kinf = 1. A prior for q

was chosen to be Normal with a large variance, centered at zero, and with supportrestricted to the region where roots of qðLÞ ¼ 1�

Pqj¼1qjL

j ¼ 0 lie outside the complexunit circle. A bivariate inverted Wishart distribution is chosen as a prior for joint distribu-tion of parameters rv, rv,� and r�.

3.3. Models comparison

We compare two parametric models for the data, defined by a probability density func-tion L(datajhsp) and L(datajhsi), where hsp = (kp,R,q) are parameters in the sticky pricemodel and hsi = (kinf,R,q) are parameters in the sticky information model. Then the fullparameter family space is Hsp [ Hsi. Its prior can be constructed as probability p thatspace is Hsp and 1 � p that space is Hsi. The posterior probability that the sticky pricemodel is true has the following form:

psp ¼pR

hsp2HspLðdatajhspÞpðhspÞdhsp

pR

hsp2HspLðdatajhspÞpðhspÞdhsp þ ð1� pÞ

Rhsi2Hsi

LðdatajhsiÞpðhsiÞdhsi

;

and the posterior probability that the sticky information model is true is equal topsi = 1 � psp.

The main problem in evaluating the posterior probability is the estimation of margin-alized likelihood. Marginalized likelihood is

Rh2H LðdatajhÞpðhÞdh ¼ mðdataÞ and one can

use the basic likelihood identity to write

mðdataÞ ¼ LðdatajhÞpðhÞpðhjdataÞ ;

4 We also estimated n directly and received similar results.5 We estimated a and kinf jointly and received similar results.

Table 1Prior distributions

p(q) N(0[2,1], I[2,2])I(q(L))

p(kp) U([0,1])p(kinf) U([0,1])p(R) IW2(0.01I[2,2], 4)p(x) U([0,1])

Notes: Here q is a vector of parameters for driving process, kp is parameter of price stickiness, kinf is parameter ofinformation stickiness, R is variance–covariance matrix, and x is a weighting parameter of the empiricalencompassing model. N stands for the Normal distribution with support limited by I(q(L)) the indicator functionused to denote the region where roots of q(L) = 0 lie outside the complex unit circle, U – Uniform distribution,IW – Inverted Wishart distribution.


where the numerator is a product of the likelihood and the prior, and the denominator is theposterior density of h. To calculate the marginal likelihood, we use the Gelfand and Dey(1994) procedure modified and implemented by Geweke (1999).6 Note that, to calculatethe posterior probability that one of the models is true, the methodology does not requirethe models to be nested. In addition, following Kiley (2007), for competing models we com-pute R2 defined as one minus sum of squared forecast errors vt of the marginal cost equationdivided by the sum of squared deviations of real marginal cost st from its sample mean.

There is an alternative to the direct comparison of non-nested models. This alternativecomparison is possible because rational expectations solutions for both models have sim-ilar analytical form. We construct the empirical encompassing model that nests the stickyprice and sticky information models by introducing a weighting parameter x and defininga new set of parameters c�k as

c�k ¼ xckðkinf ; qÞ þ ð1� xÞckðkp; qÞ; ð27Þ

and replace with them ck in (25). The resulting model encompasses the sticky price modelas a special case when x = 0. The model encompasses also the sticky information modelwhen x = 1.

To estimate the nested model, we extend the vector of parameters of the likelihood func-tion by combining structural parameters for both models, K* = (kp,kinf)

0 and adding theweighting parameter x, h = (R,q,K*,x) 0. A uniform prior is chosen for the weightingparameter over the interval [0, 1]. If the estimate of weighting parameter x is not significantlydifferent from zero, we interpret the result as empirical support to the sticky price model. Ifthe parameter is not significantly different from one, the result can be interpreted as an empir-ical support to the sticky information model. Otherwise the result is inconclusive.

4. Data and estimation results

The unit labor cost/price ratio is calculated from the historical quarterly observationsof the aggregate price level (GDP deflator, FRED database) and unit labor cost series(US Department of Labor) for the sample from 1960:1 to 2002:1.

Based on the Schwartz information criteria, we choose an AR(2) representation for thedifference in unit labor cost. By inverting the estimated coefficients of the AR(2), q, onecan solve for the MA coefficients, wi. We truncate the MA representation of unit labor

6 The program mlike is used which can be found at www.econ.umn.edu/bacc.

http://www.econ.umn.edu/bacc


cost/price ratio at kmax = 20. The choice of kmax is motivated by two reasons: by the accu-racy of solution, and by the fact that with a finite sample we want to keep the number atreasonable level. At k = 20 the estimates of coefficients for the unit labor cost/price ratio,ci, are close to zero.

A detailed description of the Markov Chain Monte Carlo (MCMC) algorithm is givenin Appendix A.3. The length of MCMC chain is 100,000 draws with 20,000 burn. To mon-itor convergence we use standard techniques, such as the Kolmogorov–Smirnov test, theZG test, and convergence of chains from different starting points.

The discussion of empirical results in a Bayesian analysis is based on comparing theprior and posterior distributions of the structural parameters. If the data is informative,it should change our prior beliefs about the parameters. Table 2 summarizes these compar-isons.7 We compare the mean of the prior distribution in column 2 with the mean of theposterior distribution in column 4. Prior and posterior standard deviations can be found incolumns 3 and 5. In addition, we report the 95% highest posterior density interval (HPDI)in columns 6 and 7 that are used for hypothesis testing.

4.1. Sticky price model

Panel 1.A of Table 2 presents our estimates of the system of equations (25) for thesticky price model. Our main parameter of interest is the price stickiness kp, row 3 ofthe panel 1.A. The data is informative both about location and about dispersion of theprice stickiness. The mean of the posterior, 0.29, is lower than the prior mean, 0.5. Theposterior standard deviation, 0.02, is ten times lower than the prior standard deviation,0.29. The 95% HPDI for the price stickiness does not include zero, with the lower bound0.25 and upper bound 0.34. Fig. 1 demonstrates the shape of the posterior distribution.

Based on Fig. 1 and the estimates we reject the hypothesis of perfectly flexible prices.Also, our estimates imply that the average time between price changes is equal to 10months with 95% interval around average time between price changes between 9 and 12months. Our estimate of average time between price adjustments is somewhat higher thanSbordone’s estimate, 9 months, though not significantly, as the HPDI includes 9 months.

4.2. Sticky information model

Panel 2.A of Table 2 presents the summary of the prior and posterior distributions ofthe sticky information model parameters. The posterior mean of the parameter of infor-mation stickiness, kinf, row 3 of the panel 2.A is equal to 0.30. Our posterior estimateof standard deviation is 0.02, which is ten times lower than standard deviation of ourprior. Fig. 2 reports the shape of the posterior distribution. The mean is higher than thevalue assumed by Mankiw and Reis (2002), 0.25, or estimated by Mankiw and Reis(2001) and Khan and Zhu (2002). Our estimate implies more frequent information updat-ing, once in every ten month, compared to once in a year, implied by kinf = 0.25.

Based on our estimates we reject the hypothesis of perfectly flexible information. The upperbound of the 95% HPDI 0.34 is far from 1. The HPDI estimates imply that with 95% prob-ability agents update their information set from once in three quarters to once a year.

7 Replication programs for all results in the paper are available upon request.

Table 2Parameter estimates: full sample: Panel 1. Sticky price model, Panel 2. Sticky information model and Panel 3.Nested model

Prior distribution Posterior distribution

Mean s.d. Mean s.d. HPDI

2.5% 97.5%

Panel 1A. Joint estimates

q1 0.0000 1.0000 0.3476 0.0713 0.2114 0.4854q2 0.0000 1.0000 0.2632 0.0701 0.1307 0.3857kp 0.5000 0.2887 0.2915 0.0226 0.2489 0.3357rv 0.0100 0.0400 0.0003 3.6e�5 0.0002 0.0004rv,� 0.0000 0.0260 �5.2e�6 2.4e�5 �5.1e�5 4.1e�5r� 0.0100 0.0400 0.0002 1.7e�5 0.0001 0.0002

B. Second stage estimates

kp 0.5000 0.2887 0.2921 0.0153 0.2629 0.3229rv 0.0100 0.0400 0.0003 3.4e�5 0.0002 0.0004


q1 0.0000 1.0000 0.3300 0.0701 0.1956 0.4605q2 0.0000 1.0000 0.1953 0.0678 0.0562 0.3237kinf 0.5000 0.2887 0.3042 0.0211 0.2623 0.3464rv 0.0100 0.0400 0.0003 3.1e�5 0.0003 0.0004rv,� 0.0000 0.0260 �2.2e�5 1.9e�5 �5.7e�5 1.4e�5r� 0.0100 0.0400 0.0002 1.2e�5 0.0001 0.0002


kinf 0.5000 0.2887 0.2849 0.0153 0.2573 0.3152rv 0.0100 0.0400 0.0003 4.0e�5 0.0003 0.0004

Panel 3q1 0.0000 1.0000 0.3359 0.0691 0.2022 0.4657q2 0.0000 1.0000 0.2520 0.0692 0.1155 0.3827kp 0.5000 0.2887 0.3295 0.1935 0.0710 0.7577kinf 0.5000 0.2887 0.4854 0.2743 0.0245 0.9348x 0.5000 0.2887 0.3383 0.1738 0.0115 0.6424rv 0.0100 0.0400 0.0003 2.6e�5 0.0003 0.0004rv,� 0.0000 0.0260 �7.2e�6 1.8e�5 �4.2e�5 2.6e�5r� 0.0100 0.0400 0.0002 1.2e�5 0.0001 0.0002

Notes: Here kp is the price stickiness parameter in the sticky price model; kinf is the information stickinessparameter in the sticky information model; q1 and q2 are reduced form parameters of unit labor cost growth rate;and rv, rv,� and r� are parameters of variance–covariance matrix of errors; x is a weighting parameter of theempirical encompassing model.


4.3. Comparing sticky price and sticky information models

Panel 1.A of Table 4 summarizes the comparison of the sticky price and sticky informa-tion models. The sticky price model dominates the sticky information model empirically.The log of the marginal likelihood of the sticky price model, 819, is significantly higherthan the log of marginal likelihood of the sticky information model, 813. Also the R2 ofthe sticky price model 0.66 is higher than the R2 of the sticky information model 0.60.

0.2 0.25 0.3 0.35 0.40

2

4

6

8

10

12

14

16

18

20

Fig. 2. Posterior distribution for the parameter of information stickiness, kinf.

0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.380

2

4

6

8

10

12

14

16

18

Fig. 1. Posterior distribution for the parameter of price stickiness, kp.


Our estimates from the empirical encompassing model that nests the sticky price andsticky information models as special cases also show that the sticky price model dominatesthe sticky information model. Panel 3 of Table 2 reports our estimates for the nested model.In particular, by comparing the prior and posterior moments for the weighting parameter xin row 6 of the table we observe that the mean decreases from 0.50 to 0.27 and the standarddeviation falls by more than 40% from 0.29 to 0.17. Our estimate of the mean implies that ineconomy, populated by both sticky price and sticky information firms, sticky price firms

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

0.5

1

1.5

2

2.5

Fig. 3. Posterior distribution of weighting parameter, x.


constitute 70% of population, while sticky information firms only 30%. The weightingparameter is not significantly different from zero: the left boundary of the 95% HPDI is equalto 0.01. The same conclusion follows from Fig. 3, which reports the full posterior distributionof the weighting parameter. One can see that a lot of posterior density is concentrated nearzero.

Finally, panel 1.A in Table 4 also reports the value of marginal likelihood for the unre-stricted VAR(2) model.8 The simple time series alternative dominates both sticky price andsticky information models. Both the log of marginal likelihood, 886, and the R2, 0.91, ofthe VAR(2) model are much higher than respective values of our theoretical models.

5. Robustness check

To estimate the sticky price or sticky information model, previous empirical studiesused different estimation methods and sample sizes. In order to check the robustness ofour results and facilitate comparison, we reproduce our results with alternative estimationmethods and sample sizes.

5.1. Two-step vs. one-step estimation

To estimate the parameters of information or price stickiness, Sbordone (2002), Man-kiw and Reis (2001), Khan and Zhu (2002) and Kiley (2007) use a two-step procedure. In

8 Similarly to the theoretical models, marginal likelihood for the unrestricted VAR(2) model is computed viaGeweke (1999) algorithm. We use the natural conjugate Normal–Wishart prior distribution for the parameters ofthe VAR(2) model

bjR � Nð0½8;1�;R� 0:05I ½4;4�mÞ;R � IW 2ð0:01I ½2;2�; 4Þ;

where m ¼ ð1=rols1;1; 1=r

ols2;2; 2=r

ols1;1; 2=r

ols2;2Þ0.


the first step, the authors estimated a statistical model for a driving process. It is equivalentto estimating the second expression in (24). In the second step, they estimated structuralparameters assuming that the estimates from the first step are given. The estimate of uncer-tainty around structural parameters then is biased downward because by taking the firststep estimates as given one ignores the uncertainty associated with them.9

To evaluate the size of bias for our data we estimate structural parameters taking valuesof OLS estimates of qi as given. Panels 1.B and 2.B of Table 2 demonstrate that in the sec-ond stage standard errors and highest posterior density intervals of the structural param-eters are smaller compared to simultaneous estimates, panels 1.A and 2.A, respectively.The standard deviation of kp falls by 45% and the standard deviation of kinf falls by29%. Having unbiased estimate of standard deviation, we are more confident rejectingthe perfectly flexible price and perfectly flexible information hypotheses.

The two-step estimation does not change the ordering between sticky price and stickyinformation models. The sticky price model still has higher marginal likelihood and higherR2 than the sticky information model, panel 1.B of Table 4. We conclude that the stickyprice dominance is robust to alternative estimation methods.

5.2. Full sample vs. stable monetary policy sample

Table 3 and panel 2 of Table 2 present estimation results and models comparison overthe 1983:1–2002:1 period. This sample is potentially more relevant for the evaluation ofthe structural models since a number of studies point to structural changes in monetarypolicy in early 1980s. Also, to estimate parameters of price and information stickiness,Laforte (2007) focuses on this sample, while Kiley (2007) compares estimates from bothsamples.

In the stable monetary policy sample the estimated price and information stickiness aresomewhat lower than in the full sample, but the decrease is not significant. ComparingTables 2 and 3, we observe that the estimate of price stickiness falls from 0.29 to 0.27and the estimate of information stickiness falls from 0.30 to 0.28. The 95% HPDIs forthe price or information stickiness parameters in Table 2 include estimates of mean fromTable 3 and vice versa.

The estimates based on 1983:1–2002:1 period do not change the ordering between stickyprice and sticky information models. The sticky price model still has higher marginal like-lihood, 569, and higher R2, 0.78, than the sticky information model, 560 and 0.71 respec-tively. See panel 2 of Table 4. The only important difference is that in the encompassingmodel, panel 3 of Table 3, the lower bound of the HPDI of the weighting parameter xis somewhat further from zero 0.09, still the sticky price model is preferred model sincemost of the posterior probability mass is closer to zero than to one. Finally, theVAR(2) model, still dominates both theoretical models with higher marginal log-likeli-hood and R2, 606 and 0.93 respectively, the last line in panel 2 of Table 4.

To conclude, our main result that the sticky price model empirically dominates thesticky information model appears to be robust to the discussed estimation methods. Also,our estimates based on 1983:1–2002:1 sub-sample do not change the main result.

9 See Pagan (1984) for the formal asymptotic argument in linear setup. The author shows that the estimate ofvariance is inconsistent and biased downward.

Table 3Parameter estimates: stable monetary policy sample: Panel 1. Sticky price model, Panel 2. Sticky informationmodel and Panel 3. Nested model

Prior distribution Posterior distribution

Mean s.d. Mean s.d. HPDI

2.5% 97.5%

Panel 1q1 0.0000 1.0000 0.3927 0.0985 0.1894 0.5883q2 0.0000 1.0000 0.2311 0.0875 0.0658 0.4013kp 0.5000 0.2887 0.2673 0.0208 0.2280 0.3066rv 0.0100 0.0400 0.0003 4.2e�5 0.0002 0.0004rv,� 0.0000 0.0260 �1.7e�5 3.1e�5 �7.8e�5 4.4e�5r� 0.0100 0.0400 0.0002 2.4e�5 0.0001 0.0002

Panel 2q1 0.0000 1.0000 0.3737 0.1011 0.1823 0.5711q2 0.0000 1.0000 0.1511 0.0944 -0.0321 0.3490kinf 0.5000 0.2887 0.2787 0.0211 0.2388 0.3205rv 0.0100 0.0400 0.0003 3.3e�5 0.0003 0.0004rv,� 0.0000 0.0260 �3.2e�5 2.3e�5 �7.7e�5 1.3e�5r� 0.0100 0.0400 0.0002 1.6e�5 0.0001 0.0002

Panel 3q1 0.0000 1.0000 0.3766 0.0944 0.1931 0.5587q2 0.0000 1.0000 0.1907 0.0869 0.0293 0.3583kp 0.5000 0.2887 0.4158 0.2834 0.0496 0.9376kinf 0.5000 0.2887 0.4698 0.3440 0.0005 0.9670x 0.5000 0.2887 0.3781 0.1432 0.0894 0.6457rv 0.0100 0.0400 0.0003 2.6e�5 0.0002 0.0003rv,� 0.0000 0.0260 �1.3e�5 2.2e�5 �5.7e�5 2.8e�5r� 0.0100 0.0400 0.0002 1.7e�5 0.0001 0.0002

Notes: See notes to Table 2.

Table 4Comparison of the sticky price and the sticky information models: Panel 1. Full sample and Panel 2. Stablemonetary policy sample

Model R2 lnm(data)


Sticky price model 0.66 819.4Sticky information model 0.60 810.8VAR(2) 0.91 885.8


Sticky price model 0.66 384.9Sticky information model 0.59 370.7

Panel 2Sticky price model 0.78 569.2Sticky information model 0.71 560.4VAR(2) 0.93 606.3

Notes: R2 for each model is computed as one minus sum of squared forecast errors vt of the marginal costequation divided by the sum of squared deviations of real marginal cost st from its sample mean; lnm(data) is alog-marginal likelihood of the model.



6. Conclusions

Our formal statistical comparison of the sticky price and the sticky information modelsand estimates of the empirical encompassing model lead to the same result – the stickyprice model dominates empirically the sticky information model. Unfortunately, theoret-ical benefits of the sticky information model do not translate into improvement in empir-ical performance. Our result is based on a simple equilibrium model and is mostly datadriven. More sophisticated empirical comparisons (Kiley, 2007; Laforte, 2007; Korenokand Swanson, 2005, forthcoming; Korenok et al., 2006) show that our result extends tomore complex General Equilibrium models, to models with alternative measures of mar-ginal costs, and to alternative estimation techniques.

Acknowledgments

This paper is a revised version of Chapter 1 of my Ph.D. dissertation at Rutgers Uni-versity. I am indebted to my supervisors Bruce Mizrach and Argia Sbordone for guidanceon an earlier drafts. I am also grateful to Roberto Chang, Kevin Grier, Oleksiy Kryvtsov,Carol S. Lehr, Florian Pelgrin, Laura Razzolini, Ricardo Reis, Norman R. Swanson, GregTkacz, Hiroki Tsurumi, Christian Zimmermann and seminar participants at Bank ofCanada, Bowdoin College, Rutgers University, Virginia Commonwealth University, Uni-versity of Connecticut and University of Oklahoma for helpful suggestions and comments.All errors are mine.

Appendix A

A.1. Equation for inflation

We start with (16). We substitute (12) into (16) to get

pt ¼ ki

X1j¼0

ð1� kinfÞjEt�jðaulct þ ð1� aÞptÞ; ðA:1Þ

the expression that describes the aggregate price level as a weighted average of past expec-tations of current value of unit labor cost and current price. Taking out the first term andredefining the summation index,

pt ¼ kinfðð1� aÞpt þ aulctÞ þ kinf

X1j¼0

ð1� kinfÞðjþ1ÞEt�1�jðaulct þ ð1� aÞptÞ; ðA:2Þ

which can be rearranged in the following form,

1

ð1� kinfÞðpt � kinfðð1� aÞpt þ aulctÞÞ ¼ kinf

X1j¼0

Et�1�jðaulct þ ð1� aÞptÞð1� kinfÞj:

ðA:3ÞFor the period t � 1 price level can be written as,

pt�1 ¼ kinf

X1j¼0

ð1� kinfÞjEt�1�jðð1� aÞpt�1 þ aulct�1Þ; ðA:4Þ


Subtracting (A.4) from (A.2) we obtain,

pt ¼ kinfðð1� aÞpt þ ulctÞ þ kinf

X1j¼0

ð1� kinfÞðjþ1ÞEt�1�jðaulct þ ð1� aÞptÞ

� kinf

X1j¼0

ð1� kinfÞjEt�1�jðð1� aÞpt�1 þ aulct�1Þ !

: ðA:5Þ

We rearrange terms in (A.5) and define Dulct = ulct � ulct�1:

pt ¼ kinfðaulct þ ð1� aÞptÞ þX1j¼0

kinfð1� kinfÞjEt�1�jðð1� aÞpt þ aDulctÞ

� k2inf

X1j¼0

Et�1�jðaulct þ ð1� aÞptÞð1� kinfÞj: ðA:6Þ

Substituting into the last term of (A.6) from (A.3),

pt ¼kinfa

1� kinf

ðulct � ptÞ þX1j¼0

kinfð1� kinfÞjEt�1�jðaDulct þ ð1� aÞptÞ: ðA:7Þ

Using Eq. (12) we substitute pt out

Dulct � st þ st�1 ¼kinfa

1� kinf

st þX1j¼0

kinfð1� kinfÞjEt�1�jðDulct � ð1� aÞðst � st�1ÞÞ;

ðA:8Þand solve for real marginal cost st

1� kinfð1� aÞ1� kinf

st ¼ Dulct þ st�1 þX1j¼0

kinfð1� kinfÞjEt�1�jðð1� aÞðst � st�1Þ � DulctÞ;

ðA:9Þto get Eq. (17) in the text.

A.2. Likelihood function

In the text, we assume that (�t,vt)0 = yt are independent draws from a bi-variate normal

distribution with zero mean, variance R, and probability density function

f ðytjRÞ ¼ ð2pÞ�1jRj�12 exp � 1

2y0tR

�1yt

� �: ðA:10Þ

Then the likelihood function for the sample y = (y1,y2, . . . ,yT) is given by

LðRjyÞ ¼YT

t¼1

f ðytjRÞ / jRj�T

2 exp � 1

2

XT

t¼1

y0tR�1yt

!¼ jRj�

T2 exp � 1

2try0yR�1

� �:

Change variables from (�t,vt)0 to (st,Dulct)

0 and use of definition of Q gives us the likeli-hood function

LðR; q;Kjs;DulcÞ / jRjð�1=2T Þ expð�1=2trR�1QÞ; ðA:11Þthat is equivalent to Eq. (26) in the text.


A.3. The Gibbs-sampling algorithm

The Gibbs-sampling algorithm consists of three steps:

1. Conditional on the values of (R,q) 0, generate K using the random walk Metropolis Has-tings algorithm:(a) Set initial values of K0. Initial values can be set arbitrarily, but we set K0 as means ofprior distributions for structural parameters of the model.(b) Draw parameters Ki+1 from the following generating function:

gðKiþ1jKiÞ � NðKi;RKÞ;

where RK is a variance–covariance matrix. We set RK = 0.05I to minimize autocorrela-tion of draws while keeping acceptance rate above 5%. Accept the new draw Ki+1 withthe acceptance probability a(Ki,Ki+1) which is defined as

aðKi;Kiþ1Þ ¼ pðKiþ1; q;RjdataÞpðKi; q;RjdataÞ

:

Note that there is no ratio of generating functions since we use the random walkMetropolis–Hasting algorithm.

2. Conditional on the values of (R, K) 0 generate q using the Metropolis–Hastings algo-rithm. It is convenient to rewrite the likelihood (26) using the fact that the inversecovariance matrix can be written as

R�1 ¼1 0

� rv;�

r2�

1

" #r2

v �r2

v;�

r�2

� ��1

0

0 1r2�

24

35 1 � rv;�

r2�

0 1

" #:

Then, we restate the likelihood,

Lðrv; rv;�; r�; q;KÞ ¼ r2v �

r2v;�

r2�

!�1=2T

� exp �1=2tr r2v �

r2v;�

r2�

!�1

v� rv;�

r2�

�

� �0v� rv;�

r2�

�

� �0@

1A

� ðr2� Þ�1=2 exp �1=2tr

�0�

r2�

� �: ðA:12Þ

We use second part of the likelihood to generate a vector of parameters q,

pðqjK;R; dataÞ ¼ pðqÞðr2� Þ�1=2 exp �1=2tr�0

�

r2�

� �:

This choice of random walk generating density improves the convergence properties ofthe algorithm significantly. If any of the roots of q(L) = 0 lies inside the complex unitcircle we discard the draw and generate the new draw. We repeat until all roots lie in-side the complex unit circle. The rest of step 2 is equivalent to step 1.


3. Conditional on the values of (q,K) 0, we generate R from,

pðR; jq;K; dataÞ ¼ jRj�1=2ðTþd0þqþ1Þ expð�1=2trR�1ðQþ S0ÞÞ;

which is the kernel of an Inverted Wishart distribution with parameters d = d0 + T,S = Q + S0. This form follows from the likelihood function (26) and the prior specifiedin Table 1.

At the end of iteration, we have draws of the structural and covariance parameters. Werepeat iterations 100,000 times. To avoid dependence on the starting value of the algo-rithm, we delete the first 20,000 draws. We use next 80,000 draws to compute the statisticalsummaries of marginal posterior densities for the structural parameters (i.e. mean, stan-dard deviation and HPDIs).

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empirical comparison of sticky price and sticky information models

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