Estimating Strategic Complementarity in a State-DependentPricing Model�

Marco BonomoEPGE, Getulio Vargas Foundation

marco.bonomo@fgv.br

Arnildo da Silva CorreaCentral Bank of Brazil

arnildo.correa@bcb.gov.br

Marcelo Cunha MedeirosDepartment of Economics, Ponti�cal Catholic

University of Rio de Janeiro�Puc-Riomcm@econ.puc-rio.br

January 2013

AbstractThe macroeconomic e¤ects of shocks in models of nominal rigidity depend crucially on the degree

of strategic complementarity among price setters. However, the empirical evidence on its magnitude isindirect and ambiguous: the one based on macroeconomic data suggest strong strategic complementaritiesin price-setting, which seems to be contradicted by some recent studies based on micro data. In this paperwe estimate directly the degree of strategic complementarity based on individual price data underlying theCPI-FGV from Brazil for the 1996-2006 period, bene�ting from large amount of macroeconomic variationin Brazilian sample during this period. Our identi�cation strategy is to infer the degree of strategiccomplementarity from the relation between the frictionless optimal price and macroeconomic variablesthat results from a microfounded model. However, since we observe price changes, and not the changein the frictionless optimal price, in order to estimate the model we assume that �rms follow a two-sidedasymmetric Ss rule. The resulting econometric model for price-adjustments is a non-standard orderedprobit model. By explicitly assuming the Ss pricing rules, our methodology is able to disentangle thee¤ect of strategic complementarity from selection e¤ect. The results, which are based on individual pricechanges and not on macro e¤ects, indicate a substantial degree of strategic complementarity, contributingto reconcile micro and macro based evidence.

1 Introduction

In recent years there has been an increasing interest in the study of price rigidity. It became well understoodthat not only the extent of price rigidity matters for monetary policy e¤ects, but also the type of dynamic

�This paper is based on the �rst chapter of the second author�s Ph.D. thesis submitted to the Department of Economicsof the Ponti�cal Catholic University of Rio de Janeiro (Puc-Rio). We would like to thank Ariel Burstein, Carlos Carvalho,Marc Giannoni, John Leahy, Virgiliu Midrigan, Axel Weber and participants at the 2011 SED Meeting, the 2011 BrazilianEconometric Meeting, and the XII Annual Seminar of In�ation Targeting of the Central Bank of Brazil for helpful commentsand discussions. We are grateful to IBRE-FGV (Brazilian Economics Institute of Getulio Vargas Foundation) for providingthe data set and to Solange Gouvea for helping us obtain access to the data. We also thank to Pedro Guinsburg for excellentresearch assistance. The views herein are those of the authors and do not necessarily re�ect those of the Central Bank of Brazil.

1

price setting policy. In particular, monetary policy shocks tend to have much less e¤ect in models where pricesetters use state-dependent rules than in time-dependent models.1 The reason is that in the former models�rms that adjust their prices tend to be further away from their pro�t maximizing price than �rms that keeptheir price invariant (the so called selection e¤ect), leading to higher average price changes in response toshocks.2 It has been known, since Ball and Romer (1990), that the e¤ects of monetary policy are ampli�edwhen there are strategic complementarities in prices, that is, when individual desired prices increase withother prices for a given level of nominal aggregate demand. More recently, Gertler and Leahy (2008) haveshown that strong enough strategic complementarity in prices may make aggregate price-level stickiness ina state-dependent model comparable to those in time-dependent models. Thus, the macroeconomic e¤ect ofshocks depend not only on the details of the price-rigidity mechanism of the model but also on the structureof its �exible price counterpart.Additionally, the recent access to vast amounts of micro price data enabled the direct measurement of

some those micro features, as the frequency and size of price adjustments, while stimulating research strategiesto unveil features that are not directly observable. Price adjustments were found to be more frequent thanpreviously thought (e.g. Bils and Klenow 2004, Klenow and Krivtsov 2008 and Nakamura and Steinson 2008).Although it seems di¢ cult to �nd a pricing model that �ts perfectly the micro data, the evidence seems to beconsistent with state-dependency, as frequency of price adjustments seem to move with the macroeconomicenvironment (e.g. Dias, Marques, and Silva 2007, Klenow and Krivtsov 2008, Midrigan 2008, Gagnon 2009,Barros et al. 2010). Finally, some empirical work based on model simulation found indirect evidence ofweak or no strategic complementarities in price-setting (Klenow and Willis 2006, Burstein and Hellwig 2007,Kryvtsov and Midrigan (2012), and Bils, Klenow and Malin 2009, 2012). As a result, this recent researchseems to point out to a large gap between the features consistent with micro data� relatively high frequencyof adjustments, state-dependency, and low degree of strategic complementarity� and those necessary toexplain macroeconomic e¤ects.In this paper we revisit the issue of what degree of strategic complementarity is consistent with micro

data. We use individual price data from Brazilian CPI of Getulio Vargas Foundation (CPI-FGV), followingGouvea (2007) and Barros et al (2010). We bene�t from the large amount of macroeconomic variation inBrazilian sample. Studying the determinants of individual price variation (e.g. Barros et al. 2010) does notprovide us the adequate framework, since price changes potentially re�ect not only frictionless optimal pricechanges - that would enable us to assess the degree of price complementarities -, but also the frequency ofprice adjustments. We circumvent this issue by postulating that �rms follow a two-sided Ss pricing rule.According to those pricing rules, adjustments are trigered whenever the price discrepancy �the di¤erencebetween the price and its frictionless optimal level � attains a certain threshold. Thus, the pricing ruleallows us to relate the price discrepancy � and the conditional probability of adjusment � to the changein the frictionless optimal price since the last adjustment date. As a consequence, assumptions for non-observable shocks enable the derivation of an expression that relates the probability of adjustments to theconditional mean of changes in the frictionless optimal price since the last adjustment. This expression allowsus to directly estimate the degree of strategic complementarity from the occurrence of price adjustments. 3

1Examples of state-dependent models include Almeida and Bonomo (2002), Golosov and Lucas (2007), Gertler and Leahy(2008), Midrigan (2009), Nakamura and Steinson (2009). Time-dependent models consider cases in which prices are �xedbetween adjustments, as Taylor (1979), Calvo (1983), and Bonomo and Carvalho (2004, 2010), as well as models where priceschange continuously, as Mankiw and Reis (2004), and Reis (2006).

2Extreme cases are Caplin and Spulber (1987) model, for the state-dependent pricing rules and Calvo (1983) for time-dependent pricing rules. In the former case, the selection e¤ect is so strong that it generates money neutrality. In Calvo (1983)there is no selection e¤ect, since the �rms that change their prices are randomly drawn.

3The resulting econometric model for price-adjustments is a non-standard ordered probit model, where the emerging auto-correlation and heteroskedasticity in the residuals are treated. The parameter measuring strategic complementarity is obtained

2

Our results indicate that the parameter measuring the sensitivity of relative price to real aggregateoutput4 ranges from 0:03 to 0:10 for the economy as a whole, implying a substantial degree of strategiccomplementarity. In order to get a sense of the possible macro implications of this range of magnitude,notice that the values for the deep parameters assumed in Gertler and Leahy (2008) entail a degree ofstrategic complementarity of 0:08, inside the range of our estimations. In their state-dependent model, thismagnitude is enough to ensure that monetary shocks have signi�cant real e¤ects.5

The degree of strategic price complementarity depends on several features of a model structure. Higherdegree of strategic complementarity may result from the larger sensitivity of marginal cost to the �rm�sindividual output and lower sensitivity to aggregate output. As argued by Woodford (2003), this is favoredby short-run segmentation in labor and capital markets. The dependence of output on intermediate goods isanother factor that engenders strategic complementarities in prices (e.g. Basu 1995). Relaxing the constant-elasticity between di¤erentiated goods in favor of a formulation where the elasticity of demand decreases withthe proportion of the product in the aggregate (e.g. Kimball 1995) also increases price complementarities.In the current paper we are agnostic with respect to the relative importance of the di¤erent sources of

strategic complementarity. Our methodology allows us to directly measure it, and does not rely on partic-ular mechanisms. Thus, our work is complementary to those based on model simulation with the purposeof investigating whether some speci�c channel is consistent or not with some micro data set, among thoseKlenow and Willis (2006), Burstein and Hellwig (2007), and Krivtsov and Midrigan (2010). Klenow andWillis (2006) investigate whether price strategic complementarity engendered by variable desired markupsdue to Kimball�s preferences are consistent with CPI micro data on prices. They pointed out that recon-ciling a large degree of strategic complementarity with the large variability of relative prices would requirean implausibly large variance of idiosyncratic shocks. Burstein and Hellwig (2007) conclude that price andmarket share data from a large chain of supermarkets is consistent with a moderate degree of price comple-mentarity.67 Krivtsov and Midrigan (2012) found out that low variability of inventories at business cyclefrequencies are not easily reconcilable with marginal cost rigidity.Bils, Klenow and Malin (2009, 2012) approach is more related to ours, since they also proposed a method-

ology to disentangle the frequency of price adjustments from variations in the frictionless optimal price. Theyde�ne theoretical reset price, as the price each �rm would like to have in case of adjustment. It turns outthat if rules are two sided Ss, as we assume, their theoretical reset price corresponds to the frictionless opti-mal price (plus a constant). Their methodology does not allow them to recover the theoretical reset price,though. Then, they use BLS CPI micro data to construct a related auxiliary variable� empirical reset pricein�ation.8 Although this variable di¤ers from the theoretical counterpart, because of the selection e¤ect,they construct several statistics in order to evaluate simulated models. They conclude that adding strategiccomplementarities to a model does not help it to match reset price in�ation statistics.Our results di¤er markedly from those obtained in Bils, Klenon and Malin (2009), who concluded that a

state-dependent model without strategic complementarity �ts best their micro data. By expliciting assumingSs pricing rules, our methodology disentangles the e¤ect of strategic complementarity from selection. Anotherdistinctive feature of our approach is that it is based only on the supply side of the economy, and does notdepend on consumer Euler equation and monetary policy rule speci�cations. So, it should provide results

from parameter restrictions in the structural model.4 It corresponds to one minus the sensitivity of individual price to the average price.5Since we do not have a complete model, we refrain from evaluating numerically the aggregate implications of our estimation.6They assess that the value of the parameter measuring the sensitivity of an individual price to average price is about 0:6.7To obtain this result they also target the fraction of menu costs as a proportion of �rm revenues in 1%.8 It is the in�ation in empirical reset price. The latter is de�ned to be the new price set by adjusters, and it is imputed for

non-adjusters by adjusting their last period reset price by the in�ation in adjusters�reset price.

3

that are not a¤ected by possible mispeci�cations of the demand side.In order to investigate whether the di¤erence in methodology or in the data is driving the di¤erence

in results, we replicate their methodology with our data. We found a similar pattern of empirical resetin�ation with the Brazilian data. If we were to use their methodology with our data, we would arrive tothe same conclusion as them. However, our frictionless price estimation allow us to recover frictionless pricein�ation, which corresponds to theoretical reset price in�ation, and it is substantially more persistent thanthe empirical reset price in�ation. The impulse response function (IRF) of our frictionless price is similar totheir IRF for theoretical reset price simulated with base on a Ss model with strategic complementarities. Weconclude that the di¤erence must come from the methodology. As suggested by their own empirical results,the speci�cation of the demand side can make a big di¤erence.9 By contrast, our methodology focuses onestimating the frictionless optimal price equation, which does not depend on the demand side speci�cation.The remaining parts of the paper is organized as follows. In the next section we derive the equation

for the frictionless optimal price from a structural model. We also characterize the Ss pricing rule followedby �rms. In Section 3 we use the frictionless price equation and the pricing rule to derive the likelihood ofadjusments. Then, we present the identi�cation strategy used to estimate the parameter of interest. Section4 describes the data set. Results are presented in the next section: the degree of strategic complementarityand the pricing rules parameters are estimated. We also evaluate our results with base on statistics for thefrictionless in�ation, the empirical reset price in�ation and the actual in�ation process. The last sectionconcludes.

2 A model of state-dependent pricing

In this section we derive the model on which the econometric estimates are based and de�ne the way we mea-sure strategic complementarities. The model has three main elements. First, households obtain utility fromconsumption goods and disutility from supplying labor; �rms supply di¤erentiated goods in a monopolisti-cally competitive environment. In the (segmented) labor market, households and �rms behave competitively.Second, we assume that �rms must pay a �xed adjustment cost to change prices and, therefore, they willfollow state-dependent pricing rules. Third, there are aggregate shocks and idiosyncratic productivity shocks.

2.1 Frictionless optimal price10

The representative household seeks to maximize the discounted sum of utilities

E0

( 1Xt=0

�t

"UtC1��

�1

t

1� ��1 �Z 1

0

Vi;tL1+�i;t

1 + �di

#);

where Ct is a consumption index and Li;t is the quantity of labor of type i supplied, Ut is an aggregatedemand shock and Vi;t represents a shock in the disutility of type of work i. The consumption index overwhich utility is de�ned is given by

Ct =

�Z 1

0

C(��1)=�i;t di

��=��1: (1)

9When they allowed money supply to respond to real aggregate demand, the state-dependent model with strategic com-plementarity turned out to �t better the impulse response function of reset price in�ation than the model without strategiccomplementarity. However, they reject the model with endogenous money because it generates a too smooth in�ation process.10See Appendix A for a detailed derivation of the frictionless optimal price from fundamentals.

4

with � > 1, and Ci;t is the consumption of variety i.In this setting, the demand for an individual product has the familiar form:

Ci;t = Ct

�Pi;tPt

���; (2)

where Ct is the aggregate consumption and the price index is

Pt =

�Z 1

0

P 1��i;t di

�1=(1��): (3)

The optimal quantity of type i labor supplied satis�es:

Vi;tL�i;t

UtC� 1�

t

=Wi;t

Pt(4)

where Wi;t is the wage of type i labor at time t.There is a continuum of monopolistically competitive �rms supplying di¤erentiated goods. We assume

that each �rm has the production function:

Yi;t = Ai;tL�i;tM

(1��)t ; (5)

where Yi;t is the �rm�s product, Ai;t is a productivity factor andMt is a foreign input used in the productionprocess. The nominal exchange rate ~Et is assumed to be exogenous. We also assume that each of thedi¤erentiated goods uses a specialized labor input in its production.If price adjustment frictions were absent, a producer i would choose a pro�t maximizing price P �i;t that

satis�es the usual markup rule:P �i;tPt

= � (Yi;t; Yt; Et;Ut; Vi;t; Ai;t); (6)

where (:) is the real marginal cost, Yt is the aggregate output, Et is the real exchange rate and � � �=(��1)is the �rm�s desired markup.It is shown in the appendix that the real marginal cost function is given by:

(Yi;t; Yt; Et; �i;t; �t; Ai;t) = �A� 1+�1+�(1��)

i;t Y��

1+�(1��)it Y

���1(1+�(1��))t E

(1��)+�(1��)1+�(1��)

t V�

1+�(1��)i;t U

� �1+�(1��)

t (7)

where

� � �

�(1� �)�

��2�(1��)1+�(1��)

Substituting the marginal cost expression (7) into the markup rule (6) and taking the log of the resultingexpression yields the frictionless optimal price equation we want to estimate:

p�i;t = �+ �Yt + (1� �) pt + �et + ~�t + ~ai;t (8)

5

where lowercase variables represent log of the corresponding uppercase variables, Yt is the nominal aggreatedemand (given by pt+yt), and shocks and parameters are combinations of the original structural parametersand shocks:

� ���� + ��1

�1 + � [(1� �) + ��]

� � (1 + �) (1� �)1 + � (1� �) + ���

~ai;t � � 1 + �

1 + � (1� �) + ���ai;t +�

1 + � (1� �) + ���vi;t

~�t � � �

1 + � (1� �) + ���ut

� � 1 + � (1� �)1 + � (1� �) + ��� log �:

Observe that when � < 1, the optimal price the �rm would like to charge depends positively on the pricesset by its competitors (in our model represented by pt), which we refer to as strategic complementarities inprice setting. The parameter � measures real rigidities� the smaller is �, the larger is the degree of strategiccomplementarities in pricing decisions. The degree of strategic complementarity depends positively on �(elasticity of intertemporal substitution) and negatively on � (elasticity of product with respect to labor),and � (elasticity of substitution among alternative varieties) and positively on . Assuming ��1 � 1, as usualin the literature, guarantees that the degree of price complementarity is increasing in � (the inverse of theFrish elasticity of labor supply)11 .Notice that di¤erent detais of the structural model lead to di¤erent relations between the parameters of

equation (8) and the structural parameters. For example, if we have assumed homogenous labor markets,our parameter of interest � would be given by

� =��� + ��1

�1 + � (1� �)

which is larger than the one obtained under segmented labor market hypotheses if the structural parametervalues are the same. Although those alternative assumptions are important if one wants to assess the degreeof price complementarities with base on reasonable calibrated values for the structural parameters, they willnot in�uence our estimation of (8), and in particular our estimation of �.

2.2 The pricing rule

We would like to estimate equation (8) but we do not observe the frictionless optimal price p�t , but theactual price. To bridge this gap we need a pricing rule. We assume that �rms follow a Ss pricing policyparameterized by (s; c; S). The state variable is the discrepancy between the actual and the frictionlessoptimal price:

y�i;t � pi;t � p�i;t

11The price condition is that � < 1� �+ ��.

6

S

s

c

r*(t)=p (τ) ­p*(t)

Price decrease

Price increase

Time

Figure 1: State-dependent pricing rule

The parametes s and S are the lower and the upper bounds for y�i;t, and c is the target point. Although wedo not derive the pricing rule here, it can be rationalized by the existence of menu costs for price adjustmentand additional conditions for the p�i;t process (e.g. a brownian motion)

12 . For our purposes, knowing theformat of the pricing policy su¢ ces to derive the econometric model to be estimated.Figure 1 shows an example of path for y�i;t. While variable y

�i;t is inside the range (s; S), the �rm maintains

its price �xed. When y�i;t reaches the threshold s, however, p�i;t is su¢ ciently above the actual price charged

and the �rm increase its price. Similarly, when the threshold S is reached, p�i;t is su¢ ciently below pi;t andthe �rm decreases its price. In case of price changes, the variable y�i;t is set equal to c. Notice that the valueof c, is in general di¤erent from zero in optimized policies. As the �rm knows that it will maintain the actualprice �xed for some time interval, it may be optimal to set pi;t somewhat above p�i;t, if in�ation is positive.According to our pricing rule, an individual price will be adjusted by

c� s if y�it � s0 if s � y�it � S

S � c if y�it � S(9)

3 The empirical model

In this section we develop the empirical model for estimating the frictionless optimal price equation (8). Forthat purpose we use the frictionless optimal price equation (8) in the pricing rule (9), and assume a certain

12 In principle we could include the conditions that determine the optimal rule in the estimation, and estimate the menucost instead of the pricing rule. However since the p� for one �rm depends on other �rms�prices, this would involve solving acumbersome �x point problem in the estimation process.

7

structure for the idiosyncratic and aggregate shocks. The resulting model is an ordered probit model thatwe estimate by quasi-maximum likelihood.First notice that p�i;t is non-observable, and depends on the aggregate observables variables pt, Yt and et,

and on the unobservable aggregate and idiosyncratic shocks ~�t and ~ai;t. For convenience, we will rewrite (8)as

p�i;t = �+ x0t� +~�t + ~ai;t; (10)

where xt = (Yt; pt; et)0 represents the vector of observable aggregate variables and � = [�; (1 � �); �]0 thevector of their coe¢ cients.Now we use our model for p�it to arrive at an expression for the price discrepancy y

�i;t in terms of the

changes in the observable variables and in the shocks since the last adjustment date.

y�i;t � pi;t � p�i;t (11)

= pi;� i;t � p�i;t= c+ p�i;� i;t � p

�i;t

where � i;t is �rm�s i last adjustment date prior to t. The equality in the second line follows from the factthat, with price rigidity, the current price is the one of the last adjustment date. The one in the thirdline results from our Ss pricing rule assumption: under this type of rule adjustments are always made toa constant proportion of the frictionless optimal price. This last property is essential for our identi�cationstrategy, since it allows us to relate the price discrepancy to the change in the frictionless optimal price.Replacing the p� speci�cation (10) into the price discrepancy equation (11) yields:

y�i;t = c+��+ x0� i;t� +

~�� i;t + ~ai;� i;t

����+ x0t� +

~�t + ~ai;t

�= c+

�x� i;t � xt

�0� +

�~�� i;t � ~�t

�+�~ai;� i;t � ~ai;t

�= c� z

0

i;t� +�~�� i;t � ~�t

�+�~ai;� i;t � ~ai;t

�(12)

where zi;t � xt�x� . Notice that although only macroeconomic variables are contained in zi;t, this variableis not the same for all �rms at time t, since � i;t is the last adjustment date for �rm i prior to t. Since �rmschange their prices at di¤erent points in time, this accumulated variation will be di¤erent across �rms foreach time. And at the end, what matters for the individual �rm�s pricing decision is the accumulated changein those variables since the moment of its last price adjustment. It is precisely this di¤erence in the lengthof each price spell what enables us to estimate the model with only aggregate variables as regressors.In order to estimate the model we need to specify the aggregate and idiosyncratic processes. To make

the model tractable, we assume that the aggregate shock ~�t follows a random walk:

~�t =~�t�1 + vt; vt � IID(0; �2v): (13)

This assumption could be considered a reasonable approximation in view of the evidence in the recentliterature (e.g. Boivin, Giannoni and Mihov 2009) that the common component of in�ation is very persistent.In addition, we assume that the idiosyncratic shock ~ai;t is given by

~ai;t = ai + ai;t;

8

where ai is an individual �xed e¤ect component and13

ai;t = � + ai;t�1 + "i;t; "i;t � N(0; �2): (14)

Therefore, our econometric model also takes into account that there may be individual �xed speci�cities inthe price of each item, but since the idiosyncratic shock appears in di¤erence, these individual e¤ects willnot matter for the estimation.Substituting the speci�cation of shocks into equation (12), we obtain:

y�i;t = c� ��i;t � z0i;t� �tX

j=t��i;t+1�j � ui;t

= c� ��i;t � z0i;t� �TXj=1

jdi;t (j)� ui;t

where�i;t � (t� � i;t) ;

di;t (j) =

(1; if j 2 [t� �i;t + 1; t]0; otherwise

: (15)

and

ui;t �tX

j=t��i;t+1"i;j � N(0; �i;t�2):

Notice that we treat aggregate and idiosyncratic shocks di¤erently. We will be able to identify aggregateshocks � through the cross-section of prices. So, we de�ne a variable dummy speci�c to each �rm at eachtime which lights up during the period starting in the previous adjustment date up to the current period.But the coe¢ cient j does not depend on i or t, since the e¤ect of the aggregate shock will be the samefor all �rms that su¤er its impact. Those are the �rms that had their last adjustment before the shocktook place, i.e. �rms that have not incorporated the shock in their current prices. By controlling for thesecommon shocks in our model (in addition to the other observable regressors), we want to make sure thattwo �rms that change prices at the same time reprice together not because they were a¤ected by the sameshock, but because of some kind of strategic interaction between them.For the idiosyncractic shock, what matters is its distribution. We sum the idiosyncratic shocks "0i;js into

the new shock ui;t, which inherits its distribution properties from "0i;js. Also observe that u0i;ts are naturally

correlated. However, given our assumptions, we know the autocorrelation structure� ui;t is a moving averageMA(�i;t + 1) process� and we use this information in the estimation procedure.Our estimation is based on the observation of adjustment ocurrences and its signal. We create the

auxiliary variable ri;t which assumes values 0,1 and �1 if no-adjustment, positive or negative adjustmentsare observed:

ri;t =

8<: 1; if pi;t > pi;t�10; if pi;t = pi;t�1

�1; if pi;t < pi;t�1

: (16)

13 In the main speci�cation we assume that ai;t is a random walk with drift because we would like to take into account thatproductivity grows over time. But, given the recent evidence that idiosyncratic shock is short-lived, we also estimate the modelassuming that ai;t is a white-noise, i.e., ai;t = "i;t and "i;t � N(0; �2). See subsection 5.3.

9

De�ning wi;t = (�i;t;z0i;t;d

0i;t)

0, where di;t = (di;t (1) ; :::; di;t (T ))0, and using the pricing rule, we can

obtain the probability of observing a price increase:

Pr[ri;t = 1jwi;t] = Pr[y�i;t � sjwi;t]

= Pr[c� ��i;t � z0i;t� �TXj=1

jdi;t (j)� ui;t � sjwi;t] (17)

= Pr

"ui;tp�i;t�

�c� s� ��i;t � z0i;t� �

PTj=1 jdi;t (j)p

�i;t�

#

= 1� �

0@c� s�

1p�i;t

� �

�

p�i;t �

z0i;tp�i;t

�

��

TXj=1

j�jdi;t (j)p

�i;t

1A= 1� �

0@�1�1i;t � ~���i;t � �z0i;t~� � TXj=1

~ jj�di;t (j)

1Awhere variables with two dots represent the variables divided by

p�i;t, the parameters with tilde means that

the parameters is scaled by �, �(:) is the cumulative distribution function of a standard normal variable and�1 = (c� s)=�. In the third line we have used the fact that ui;t is independent of wi;t.Likewise, we can derive the probability of observing the other two possible outcomes for ri;t. This results

in the following ordered probit model for price changes:

Pr[ri;t = 1jwi;t] = 1� �

0@�1�1i;t � ~���i;t � �z0i;t~� � TXj=1

~ j�di;t (j)

1A

Pr[ri;t = 0jwi;t] = �

0@�1�1i;t � ~���i;t � �z0i;t~� � TXj=1

~ j�dj;t

1A� �0@�0�1i;t � ~���i;t � �z0i;t~� � TX

j=1

~ j�dj;t

1A (18)

Pr[ri;t = �1jwi;t] = �

0@�0�1i;t � ~���i;t � �z0i;t~� � TXj=1

~ j�dj;t

1Afor t = 1; :::; T and i = 1; :::; N , where �0 = c�S

� .The probit model is estimated by quasi-maximum likelihood method, which guarantees a consistent

estimation of the parameters. Inference is carried out using a robust variance-covariance matrix for auto-correlation and heteroskedasticity. Since we know the structure of the model in which the probit is basedon, given our assumptions, we can derive the variance and covariance of ui;t: We incorporate this piece ofinformation in the estimation process by dividing all variable by the standard deviation of ui;t,

p�i;t, as

already mentioned, and by using a robust variance-covariance matrix to account for serial correlation in thescores across t. Details are given in Appendix B.

10

3.1 Identi�cation of strategic complementarity and pricing rule parameters

The strategy to recover the structural parameter � measuring strategic complementarity in the model arisesnaturally from the restrictions on the frictionless optimal price parameters and the ordered probit model.Notice that from equation (8) and the probit model, we have:

~�1 =�

�(19)

and~�2 =

1� ��

: (20)

Then, by dividing equation (19) by (20) and isolating � we obtain:

� =~�1

~�1 +~�2: (21)

Similarly, we can estimate the variance of shocks hitting the �rms. To obtain �, substitute equation (21)into (19):

� =1

~�1 +~�2: (22)

Since we are able to write � and � as a function of the probit model�s parameters, we can use the Deltamethod to construct con�dence intervals for them. Details are given in Appendix B.Parameters �0 and �1 of the probit model are combinations of the pricing rule�s parameters and the

variance of shocks. Even though we cannot obtain all parameters of the pricing rule in isolation, once weestimate the variance of shocks we can estimate the adjustment sizes and the inaction band size. They areobtained respectively as

c� s = �1� and c� S = �0�: (23)

S � s = (�1 � �0)� (24)

4 Data

In this section we describe the data set that we use in our estimations and calculate some statistics of pricechanges, such as frequency, duration, average price change etc. These statistics will be useful as a guidelineto our methodology. For instance, we compare some of these statistics in the data to those estimated by theprobit model, and this exercise can provide some sense of goodness of �t of our model. We also relate someof them to the estimated princing rule.

4.1 Data set description

The data set used here consists of primary information of price quotes of individual products collected andused by the Brazilian Institute of Economics of the Getulio Vargas Foundation (FGV-IBRE) to computethe consumer price index (CPI-FGV). The CPI-FGV has wide coverage and it is one of the oldest Brazilianprice indices, being calculated since 1944.

11

The data set contains information on prices at the �rm level collected in the 12 largest Brazilian metropol-itan regions, even though the coverage has changed during our sample period14 .Currently, the CPI-FGV comprises 456 products and services grouped into seven di¤erent sectors: Ap-

parel; Education and Recreation; Food; Housing; Medical and Personal Care; Transportation; and OtherGoods and Services. The weight of each product or service used in computing the index mirrors the expendi-ture by households receiving income up to 33 times the minimum monthly wage, obtained from a householdconsumption survey� Pesquisa de Orcamento Familiar (POF)� also conducted by FGV.The data are systematically collected by FGV employees. Some data collections are made every ten days,

while for other products prices are collected on a monthly basis. We refer to the most disaggregated levelof data as an item. Each item is identi�ed by a set of very speci�c characteristics, such as brand, model,packaging, type/variety that is sold in a particular outlet, in a speci�c city15 . Each price also includes theexact date of collection. However, while the outlets are identi�ed by codes, for con�dential reasons no otherfeature of the �rms, such as size, number of employees, revenues and costs amounts etc is available.The number of items collected each month is not constant. There is variation due to item exclusions

and inclusions over time. However, there is never substitution of an item by a similar one. Therefore, it ispossible to follow the same item over time. When the price of a speci�c item is no longer collected, its pricetrajectory continues to be registered as missing. Thus, one can be assured that each change recorded is dueto an actual price change.

4.2 Data sample and some statistics

In this subsection we describe the sample and calculate some simple statistics of price changes in the data.We have a very representative sample of the overall CPI-FGV (around 85%), containing 243 categories ofproducts and services, whose prices are accompanied by a period of approximately 11 years, from 1996 to2006. The seven sectors are very well represented.We start from the original full sample with approximately 7.4 million price quotes and 122 thousand

quote lines. The original sample was treated in order to have a set of information more suited to our goals.First, recently Eichenbaum, Jaimovich, Rebelo and Smith (2012) raised the possibility that most of papers

dealing with microdata of prices potentially su¤er from problems of measurement errors due to problematicprice quotes. In particular, they argue that the evidence suporting the conventional view that �rms makemany small price changes is illusory, determined by a variety of measurement erros. Aiming to avoid this typeof problem, we treat our original dataset excluding the products identi�ed with potential problematic pricequotes16 . Detailed analysis of the procedures adopted when prices in our database are collected identi�ed Xbroad categories of potential sources of measurement errors. Products identi�ed as falling into any of thesecategories were excluded from our sample. Appendix D documents the X categories of potential sources ofmeasurement errors and lists the products excluded from our sample. After these exclusions we were leftwith 178 products and services into the seven sectors, representing 40.4% of the CPI-FGV. For robustness,in subsection 5.3 we also estimate our model without making these itens exclusions.Second, we would like to have a data set with monthly data. Thus, for products whose prices are collected

every 10 days, in the aggregation we choose to keep always the �rst price quote in each month. Third, the very

14Up to the end of 2000, the coverage comprised only the metropolitan regions of Rio de Janeiro and São Paulo. AfterJanuary 2001, ten other cities were included in the survey: Belo Horizonte, Brasília, Porto Alegre, Recife, Salvador, Belém,Curitiba, Florianópolis, Fortaleza and Goiania. At the beginning of 2005 the last �ve cities were dropped.15For example, type I black beans of the Combrasil brand, sold in a 1kg package in the outlet number 16,352, in Belém.16 It is worth emphasizing that these potential measurement errors do not a¤ect the calculation of the CPI-FGV, but are

important if one needs to be sure that the price trajectory refers to the same item along the time.

12

short price trajectories were excluded. Trajectories with less than 18 observations or with more than 30%of missing observations were dropped. We further re�ned our sample by treating speci�cally the remaininggaps. The gaps with only one missing observation were �lled using the price collected in the immediatelypreceding month. Also, gaps with up to three missing observations, when preceded and succeeded by thesame price, were �lled with the last price available. In the case of four or more missing observations, wemaintained the longest uninterrupted part of the price trajectory.Fourth, retail prices are characterized by a signi�cant number of temporary price decreases (sales or

promotions). Following the recent literature, as Golosov and Lucas (2007), Midrigan (2006) and others, wetreated observations identi�ed as sales prices. Because sales are not explicitly classi�ed in our database,we used the following algorithm to identify them. If a price decrease was large enough and reversed inthe following months to levels near the one prevailing before the change, we classi�ed that observation asa price sale17 . Following Midrigan (2006), to deal with large price changes in a V shape, we repeated thealgorithm three times. When a �sale�was identi�ed, the price was replaced by the price collected in theimmediately preceding period. For robustness, because there is also considerable discussion in the literatureabout whether one should or not treat the temporary price decreases (sales or promotions) when workingon models of price dynamics, in subsection 5.3 we also estimate our model without treating promotions.Outlier were also treated. An observation was classi�ed as an outlier when it was 10 times larger or 10

times smaller than the previous observation.Finally, to avoid the problem of left-censored spells, for all price trajectories we dropped the observations

before the �rst observed price change.After this treatment we were are left with a �nal data sample with approximately 2.2 million observations

and 46.2 thousand price trajectories into the seven sectors of the CPI-FGV.

Table 1: Some statistics of price changes

Statistics Estimated valueAverage price duration (months) 3.99Average frequency of price change 0.49Average price change (%) 15.3Standard deviation of the price change distribution 18.0Kurtosis of the price change distribution 3.8Average price increase (%) 15.3Average price decrease (%) 16.0Percentage of price decrease 44.1Percentage of small price changes 37.7

Notes: 1) p is de�ned as the natural logarithm of the item price.2) All statistics are calculated based on unweighted price changes.3) Kurtosis is calculated excluding the top and bottom 1% of observations.4) Small price change is de�ned as that whose value is lower than 0.5 of the mean of �p.

Table 1 presents some statistics related to price changes estimated in the data. All statisticas are calcu-lated using unweighted price changes. The average price duration is estimated in almost 4 months. Noticethat our estimate of price duration in Brazil is at the upper limit of the range estimated by Gouvea (2007)

17Formally, if(pt�1�pt)

pt�1> 25% and pt+1 � pt

�1 +

pt�1�ptpt�1

�. One can always say that 25%is an arbitrary value, and/or

that it is high for some sectors and/or low for some others. But we decided to keep a single rule to minimize arbitrariness.Changing this value does not signi�cantly change our main results.

13

and much higher than the inverted duration estimated by Barros et al. (2009)18 . This result is consistentwith the reasoning that the treatment of the potential measurement error in the data, as advocated byEichenbaum, Jaimovich, Rebelo and Smith (2012), should increase the statistics measuring price rigidity.We also �nd that price changes are, on average, large (approximately 15%), even though there is a great frac-tion of changes that are small. Interestingly, price decreases are estimated to be larger than price increases.While the average price increase is 15.3%, the average price cut is 16%.

5 Estimation and results

In this section we apply our methodology to the data set of individual prices described in the previoussection, combined with aggregate data, to estimate the parameter measuring strategic complementaritiesin �rms�pricing decisions. We also estimate other parameters of the structural model. In particular, weestimate the variance of shocks and some parameters of the pricing policy: the size of positive and negativeprice adjustments.

5.1 The estimated models

We combine the set of microdata just described with aggregate data to estimate the probit model. Thedependent variable is the observed ri;t de�ned in equation (16). Following the theoretical model, our baselinespeci�cation has �ve explanatory variables and the whole vector of dummies, which are constructed, for eachitem in our data set, as given in equation (??). The �rst two variables are, respectively, the square root ofthe time interval between t and � , and its inverse. These two piece of information are directly obtained fromthe pricing history of each item in the data set of microdata. The last three variables are, respectively, theaccumulated change in the aggregate price level, �pi;t = log(Pt=P� ), in the nominal expenditure, �Yi;t =log(Yt=Y� ), and in the exchange rate, �ei;t = log(Et=E� ), since the last price adjustment in time � , dividedby the square root of the time interval between t and � ,

p�i;t. Therefore, for each item in our database,

in every time t, we calculate the changes in these aggregate variables between the moments t and � . Aspreviously pointed out, it is exactly this di¤erence in the length of price spells what allows us to estimatethe model using basically aggregate variables as regressors. Once �rms reprice at di¤erent points in time,the accumulated changes in those variables since the last adjustment will be di¤erent. We must divide thevariables by

p�i;t to account for the heteroskedasticity that appears in the residuals. It is worth noting that

two products with the same price adjustment history will have the same regressors.The baseline speci�cation incorporates the random walk assumption for the idiosyncratic shock and is

estimated using the whole sample period, but with exclusions of products with problematic price quotes.For robustness, we also estimate several other speci�cations, such as speci�cations considering a white-noiseassumption for the idiosyncratic shock; using di¤erent sample periods; without including the exchange ratevariable; without making the treatment of temporary price decreases (sales or promotions); and withoutmaking itens exclusions due to problematic price quotes.The measure of nominal expenditure we use in our estimations is the monthly nominal GDP series

calculated by the Central Bank of Brazil using the quarterly national accounts of the Brazilian Institute ofGeography and Statistics �IBGE. Our measure of aggregate price is the Consumer Price Index �IPC-FGV,

18Gouvea (2007) estimates the aggregate duration between 2.7 and 3.8 months, depending on the calculation method. Barroset al. (2009) also �nd a very di¤erent measure by calculating the frequency of price changes for each particular item and thenaggregating the durations implied by these frequencies that were obtained in the more disaggregate level� 8.5 months.

14

which is calculated by the FGV from the set of microdata described in the previous section. The exchangerate is the monthly Real/US$ exchange rate (for sale) series provided by the Central Bank of Brazil.

5.2 Baseline speci�cation

5.2.1 Goodness of �t

Before examining the estimation results regarding strategic complementarities, we provide some informationabout the �t of the model. To measure the model performance we compare the probabilities of price changesestimated using the baseline speci�cation to the frequencies of price changes calculated directly from thedata. The probability of a price decrease in the model is obtained by calculating the probability of a pricedecrease for each observation in the dataset, Pr (ri;t = �1jwi;t), and then calculating the average for all i andt, E [Pr (ri;t = �1jwi;t)]. Similar calculation is done for price increases. Table 2 reports the results.The probitmodel estimates the probability of a price decrease in 25%, while the average frequency obtained directlyfrom the data is 23%. Considering price increases, the respective values are 34% and 25%. Therefore, eventhough the average probability of a price increase estimated by the probit model is somewhat greater thanthe frequency calculated in the data, in the case of price decreases the model estimates the frequency quitewell, and in general seems to be capturing important features of the data. This is a good performance, if weconsider that our theoretical model is very simple.

Table 2: Probability and frequency of price change, baseline speci�cation

E [Pr (ri;t = �1jwi;t)] Freq. of price decrease E [Pr (ri;t = 1jwi;t)] Freq. of price increase0.25 0.23 0.34 0.25

5.2.2 Strategic complementarity

We now proceed to estimate the degree of strategic complementarity in �rm�s price decisions using thestrategy outlined in subsection 3.1. Table 3 reports the estimated parameters of the probit model necessaryto compute the parameter of strategic complementarities, using the baseline speci�cation. Detailed resultsof the probit model are presented in Appendix C.Before presenting the result regarding strategic complementarities, two points are worth noting. First,

the parameters of the probit model are all signi�cant at any of the usual signi�cance levels. They also havethe expected signs based on the theoretical model. Therefore, nominal expenditure, in�ation and exchangerate a¤ect positively and signi�cantly pricing decisions of �rms.Second, the signi�cance of the probit model parameters means that pricing decisions have a state-

dependent component. As already pointed out in the introduction, Barros et al. (2009) have shown usingthe same set of microdata we use in our estimations that the frequency of price change is signi�cantly andpositively related to in�ation, output and exchange rate depreciation. In a state-dependent pricing modelthis connection should result from the relation of the frictionless optimal price with those same variables,and this is what our probit results are capturing.We infer the parameter of strategic complementarity from this relationship between the frictionless op-

timal price and the macroeconomic variables derived from the microfounded model. The mapping betweenthe probit model parameters and � is made through equation (21). Table 3 reports point estimation for �,as well as 95% con�dence intervals obtained using the Delta method.

15

The �rst important piece of evidence that emerges from Table 3 is that the value of � is much less than one.This result implies that �rms�pricing decisions are strategic complementary rather than substitute. Actually,the upper bound of the con�dence interval is far from 1� the lower limit for strategic substitutability.Second, our results suggest that not only pricing decisions are complementary, but also that the degree

of strategic complementarity is substantial in the data. The estimated value of 0.06 for � implies a strongpositive interaction between the frictionless optimal price of a given �rm and the prices of its competitors.

Table 3: Probit parameters and strategic complementarity, baseline speci�cation

Parameter of Yt Parameter of pt Parameter of et Strat. complementarity � Conf. interval for �0:60(0:077)

9:57(0:049)

0:11(0:049)

0:06(0:012)

0:05� 0:07

Notes: The robust standard deviations are in parenthesis. The con�dence interval is 95% of con�dence.The standard deviation of � was obtained by the Delta method. .

5.2.3 The estimated pricing rules

Here we explore the results of the probit model regarding the pricing rule. Table 4 shows the estimatedparameters using the baseline speci�cation. As can be seen, the intercepts19 �1 and �0 are statisticallysigni�cant at 1%. Actually, their standard deviations are very small. One fact that may help to explain thislarge signi�cance is that we take into account the variability in the price spells duration� observe that allvariables in the probit model are divided by the square root of the time interval between t and � ,

p�i;t.

Objectively, this means that the intercepts are changing over time, which may help to improve the �t of themodel.Since we are able to estimate the parameter �, from the intercepts of the probit model we can estimate

the pairwise distances between S, s and c using the equations (23)� even though we cannot isolate the ownparameters S, s and c, individually. These results are reported in Table 4.An interesting evidence that emerges from these results is that the estimated Ss band is larger at the

top than at the bottom, i.e., the estimated distance between S and c is larger than the distance between cand s: while c� s is 0.06, S � c is equal to 0.09. This means that, if c were zero, on average an individual�rm would wait the frictionless optimal price p�i;t deviate 6% from the actual price before it increasing, anddeviate 9% before it reducing its price. Unfortunately, we are not able to estimate the individual value ofc. But the estimated adjustments size are smaller than in the sample. One possible explanation is that ourestimation does not use adjustment size, but frequency of adjustments. And there are nonlinearities in therelation between size and frequency of price adjustments.The results in Table 4 also indicate that a typical �rm seems to be more tolerant with shocks that decrease

its frictionless optimal price than with shocks that increase it. This result is compatible with the evidenceraised from the data that, on average, price decreases are larger than price increases. Such evidence canbe rationalized by the positive average in�ation during the period covered by our data. When in�ation ispositive, if a �rm is hit by a small negative shock and want to decrease its price, by maintaining its nominalprice unchanged the �rm will actually have a lower price in real terms. When the distance from its optimalprice is su¢ ciently large, however, the �rm reprices and the size of the adjustment will be possibly larger.Finally, the standard deviation of the idiosyncratic shocks are estimated in 10%, which is equal to that

estimated by Klenow and Willis (2006) in the model without strategic complementarity.

19We are calling them intercepts, but observe that our probit is not standard and the variable 1=p�i;t changes over time.

16

Table 4: Estimated parameters of the pricing rule, baseline speci�cation�0 =

c�S� �1 =

c�s� Std. deviation

�Top band

S�cBottom band

c�sBand width

S�s

�0:92(0:01)

0:62(0:01)

0:10(0:01)

0:09 0:06 0:15

5.3 Robustness

We also carried out a number of estimations for robustness check, both for strategic complementarity as forpricing rule. First, we split the sample in two time periods: from 1997 to 2000 and from 2001 to 2004. The�rst and second rows of Table 5 show the results. They are consistent with the previous estimations� theestimated � is 0.03 in the 1997-2000 period and 0.10 in the 2001-2004 period. We also estimated the modelusing data of each year separately during the whole sample period. Results are not reported, but they arealso consistent with those already presented.Second, we estimated the model without including the exchange rate variable in the frictionless optimal

price equation. This formulation can be obtained from our theoretical model if we do not include the foreigninput in the production function. But this speci�cation should be viewed with caution, since the exchangerate is very important for price dynamics in Brazil. Results for the aggregate economy are presented inthe third row of Table 5. Even though the estimated � is much larger than those obtained in the previousestimations, it is still well bellow 1 and also supports the evidence of signi�cant real rigidities in the data.Third, taking into account the evidence in the recent literature showing that idiosyncratic shocks are

typically short-lived, we changed the assumption in equation (??) and estimated the models assuming thatthe idiosyncratic shock is a white-noise process: ai;t = "i;t and "i;t � N(0; �2). The fourth row of Table 5presents the results: the estimated value for � is 0.07, very close to that of the baseline speci�cation.Fourth, there is also considerable discussion in the literature about whether one should or not treat

the temporary price decreases (sales or promotions) when working on models of price dynamics. Thus,for robustness check of our results in the baseline speci�cation we estimate a model without treating theobservations identi�ed as sales/promotions. In this case, the value obtained for � was 0.06, the same as inthe baseline speci�cation.Finally, we estimated a model considering our complete sample, i.e., without excluding the itens with

potential problematic price quotes, as raised by Eichenbaum, Jaimovich, Rebelo and Smith (2012). The lastrow of Table 5 reports the results. They also show that our results of strong real rigidities are very robust.In summary, considering all models we have estimated, usually the point estimates of � for the aggregate

economy is between 0.03 and 0.10.Regarding the pricing rule, in almost all models we have estimated (splitting the sample in two parts,

estimating the model year by year, excluding the exchange rate variable, without treating promotions or evenwithout itens exclusions) the parameters are very close to those presented in Table 4 (see detailed resultsin Appendix C). The only speci�cation producing slightly di¤erent parameters of the pricing rule is thatusing the white noise assumption for the idiosyncratic shocks. But even in this speci�cation the estimatedSs rule keeps the main characteristics of that obtained in the baseline speci�cation. Another important issueabout the pricing policy is whether it should or not be constant over time. Remember that our assumptionis that the pricing policy is constant over time and does not depend, for instance, on the macroeconomicenvironment. Regarding this issue, the estimations for di¤erent time periods, like those for period 1997-2000and period 2001-2004, show that the assumption of a constant pricing policy is not too restrictive.

17

Table 5: Probit parameters and strategic complementarity (Robustness)Model Yt pt et Strat. comp. � Conf. interval for �Period 1997 - 2000 0:15

(0:160)4:93(0:146)

0:62(0:024)

0:03(0:031)

-0:03� 0:09

Period 2001 - 2004 1:29(0:094)

11:07(0:054)

�0:40(0:015)

0:10(0:007)

0:09� 0:12

Without Exchange Rate 2:93(0:074)

3:43(0:037)

� 0:46(0:008)

0:44� 0:48

Idiosyncratic White Noise 0:90(0:112)

12:72(0:072)

0:05(0:018)

0:07(0:008)

0:05� 0:09

Without treating promotions 0:53(0:079)

8:90(0:050)

0:02(0:013)

0:06(0:008)

0:04� 0:07

Without itens exclusions 0:44(0:065)

9:79(0:044)

0:03(0:011)

0:04(0:006)

0:03� 0:05

Notes: The robust standard deviations are in parenthesis. The con�dence interval is 95% of con�dence.The standard deviation of � was obtained by the Delta method. .

5.4 Actual, frictionless and reset price in�ation

Our main result indicating substantial degree of strategic complementarities in the data is di¤erent from theusual results found in previous studies. In this section we compare our method to those proposed in theliterature to estimate real rigidities and provide some other results. In particular we compare our method,using the baseline speci�cation, to that of Bils, Klenow and Malin (2009). In order to investigate whetherthe di¤erence in results is due to the di¤erent data sets, we reproduce their method with our data.First of all, we emphasize that our procedure is di¤erent from those in the literature. Since �de-

sired�prices, marginal costs and markups (variables to which strategic complementarities are related) arenot directly observable in practice, or there is no data about them available, generally the studies rely oncalibrated models and/or statistical properties of speci�c observable variables to indirectly infer the presenceof real rigidities in the data. For instance, Klenow and Willis (2006), Burstein and Hellwig (2007), Bils,Klenow and Malin (2009), Kryvtsov and Midrigan (2009), Gopinath and Itskhoki (2010) all rely on indirectmethods. In contrast, our method, by modeling the frictionless optimal price p�i;t as a latent variable inan ordered probit model and by deriving the relationship between p�i;t and pt from the price-setting model,makes the measurement and interpretation of strategic complementarities more direct and independent ofother details of the model. Regarding this point, the comparison with Bils, Klenow and Malin (2009) willbe very informative.Bils, Klenow and Malin (2009) (hereafter BKM) develop a statistical measure called �reset price in�ation�

to indirectly infer the role for strategic complementarities in the data. They use a general equilibriumprice-setting model to simulate the behavior of reset price in�ation and compare the simulated results to ameasure of this variable constructed using the microdata underpinning the U.S. CPI. They report that astate-dependent model with strategic complementarity is fundamentally at odds with the data� the modeldisplays unrealistically high persistence and low volatility of reset price in�ation. Their empirical proxy haslow (in fact negative) serial correlation and high standard deviation20 .In order to compare our method to that developed by BKM we review and simulate their measure of reset

price in�ation in our data. They de�ne theoretical reset price for an individual seller as that price whichthe seller would choose if he/she implemented a price change in the current period. Observe that this takes

20They also found that TDP models with or without strategic complementarity generate reset price in�ation series that aretoo persistent.

18

into consideration that the changed price is likely to last for several periods. Therefore, it di¤ers from thede�nition of frictionless optimal price p�it we use. The theoretical reset price in�ation, �

�i;t, is the weighted

average change of all reset prices, including those of current price changers and non-changers alike.Their empirical measure of reset price for an item i at time t, p�i;t, is given by:

p�i;t =

�pi;t; if pi;t 6= pi;t�1

p�i;t�1 + ��t ; if pi;t = pi;t�1

; (25)

where ��t , the reset price in�ation, is given by:

b��t �Pi

!i;t�pi;t � p�i;t�1

�Ii;tP

i

!i;tIi;t;

where Ii;t is equal to 1 if pi;t 6= pi;t�1 and zero otherwise.Notice that since one cannot observe the reset price for those not changing prices, their empirical method

updates reset prices for them using the reset in�ation b��t . That measure, in turn, is de�ned by the weightedaverage of the di¤erence between new prices in t and reset prices de�ned in t � 1. The motivation forconstructing such measure is similar to ours: to construct a measure that reveals strategic complementarity.As BKM acknowledge, �rms that change price may have stronger incentive to do so, which means that thereis a selection e¤ect. As a consequence, the theoretical reset price in�ation, which is based on desired pricesfor changers and non-changers, may di¤er markedly from its empirical counterpart when there is selectione¤ect. In our methodology the selection e¤ect is taken into account by the probit model. Firms whose pricesare more out of line are those with larger probabilities of changing prices in the probit.BKM construct several statistics based on their empirical measure of reset price in�ation in order to

discriminate among alternative models of price-setting with and without strategic complementarities. Theysimulate the models, closing them with an exogenous process for the money supply. They compute stan-dard deviation and serial correlation of reset price in�ation and in�ation. They also show impulse responsefunctions for reset price in�ation with base on univariate AR(6) regressions. They conclude that the state-dependent model without strategic complementarity �ts better the proposed set of statistics than the com-peting model. Since the reason for the di¤erence between our �ndings and their �ndings could be that weuse Brazilian microdata and they use US microdata, we repeat their experiment with our data.21

Also notice that in our methodology we can construct a measure of frictionless optimal price in�ation inthe data. The comparison of the statistical properties of the BKM�s reset price in�ation to the propertiesof our frictionless optimal in�ation can be very informative. In particular, this exercise can verify if theestimated frictionless optimal in�ation has the properties that BKM were looking for. After all, strategiccomplementarities should also make frictionless optimal in�ation, like reset price in�ation, persistent andstable.To carry out this exercise we use the results of the probit model in the baseline speci�cation. Thus, to

construct an empirical measure for frictionless optimal price in�ation we use the following formula:

��t =Xi

!i;t��i;t; (26)

where !i;t is the weight of each item in the CPI-FGV, and the individual frictionless in�ation for item i isestimated by21We do not have access to disaggregated data on prices collected by the U.S. Bureau of Labor Statistics. Otherwise, we

could use these data in our estimations.

19

��i;t = p�i;t � p�i;t�1 = �x0t� + (~�t � ~�t�1) + (~ai;t � ~ai;t�1) (27)

= � +�x0t� + (~�t � ~�t�1) + "i;t;

where � and � here are the parameters of the probit model estimated in the baseline speci�cation. The term~�t � ~�t�1 can be constructed using the parameter of the dummy variables which control for the aggregateshocks. The only term for which we do not have estimates is "i;t. But we have an estimate for the variance ofthe idiosyncratic shock, obtained using the equation (22). Thus, to construct a measure of frictionless optimalprice in�ation we carry out a Monte Carlo experiment. For each individual item in our data set we simulateone trajectory for f"i;tg using the distribution N(0; �2), where �2 is the estimated variance of idiosyncraticshocks. We then aggregate the individual frictionless optimal price in�ations from (27) using equation (26),to have ��t . We repeat this experiment one thousand times and compare the statistical properties of theseseries to those of the reset price in�ation.Table 6 reports the standard deviation and the serial correlation for reset, frictionless and actual price

in�ation in our sample. Figure 5 in Appendix C plots the series. For frictionless optimal price in�ation thestatistics reported correspond to the average obtained across the one thousand simulations. Notice that thestatistics for reset price in�ation are similar to those reported by BKM for US: the standard deviation is2.30% and the serial correlation coe¢ cient is -0.35. In fact their simulated state-dependent model withoutstrategic complementarity generates a standard deviation of 1.79% and a serial correlation of -0.31, �ttingour data set even better than theirs. It �ts also our in�ation statistics at least as well as the one in theirsample, although not as close. In turn, the estimated frictionless optimal price in�ation is much less volatile�with standard deviation equal to 0.72%� , and has dynamics closer to that of actual in�ation. Regardingpersistence, frictionless optimal price in�ation is much more persistent than reset price in�ation, with �rst-order autocorrelation equal to 0.21. These results do not change if we use seasonally adjusted series or not,as the last rows of Table (6) show.

Table 6: Summary statistics for frictionless, reset and actual in�ation after exclusionsSeries Average Std deviation PersistenceActual In�ationI PC-FGV 0.57% 0.76% 0.51Frictionless In�ation 0.52% 0.72% 0.21Reset Price In�ation 0.62% 2.30% �0:35Actual In�ation IPC-FGV, Seas. Adj. 0.57% 0.69% 0.51Frictionless In�ation, Seas. Adj. 0.52% 0.63% 0.26Reset Price In�ation, Seas. Adj. 0.62% 2.07% �0:41Notes: 1) The statistics are calculated using the period Jun/1996-Aug/2005. 2) Frictionless in�ation is obtained by aggregating

��i;t=�+�x0t�+(

~�t�~�t�1)+"i;t, where � and � are the estimated parameters in the baseline model. Aggregation uses thegood weights in IPC-FGV. The terms "i;t come from a Monte Carlo experiment with 1000 simulation. 3) Persistence ismeasuredby the �rst-order autocorrelation. For frictionless in�ation, average, standard deviation and persistence are theaverages of the values obtained in the simulation.

We also display in Figures 2, 3 and 4, respectively, the impulse response function (IRF) (in level) foractual, frictionless and for reset price in�ation in our data. Figure 3 plots the average response of frictionlessprice and a fan-chart with various levels of signi�cance obtained in the simulation. It shows that, aftera shock, frictionless optimal price has an upward sloping trajectory. The shape is similar to the impulse

20

0

1

2

3

4

5

6

1 2 3 4 5 6 7 8 9 10 11 12

Per

cent

age

poin

ts

Note: Accumulated response of actual inflation to one unit innovation.

Months

Figure 2: Impulse response of actual in�ation

response for our actual in�ation, presented in Figure 2, and also similar to that found by BKM for the impulseresponse of theoretical reset prices generated by their model with strategic complementarities. Moreover,the frictionless optimal price response is consistent with our previous results of strong degree of strategiccomplementarities in the data: the impact is initially small (as price setters wait for the average price torespond), but accumulates over time as more �rms change prices.On the other hand, visual inspection of the IRF for BKM�s simulated model reinforces the conclusion

that their state-dependent model without strategic complementarity �ts reset price in�ation from our database even more closely than theirs. Figure 4 shows that the response of reset prices is much greater onimpact than over time, exactly like those in BKM. Therefore, if we were to use the same methodology asthem in our data set, we would arrive at that same conclusion: the state-dependent model without strategiccomplementarities �ts better the data than the state-dependent model with strategic complementarity. If itis not the di¤erent data set, what could be the reason for the di¤erent results?The di¤erence must come from the methodology. Their exercise is based on model simulation. Although

their focus in on the supply side� price setting and strategic complementarity,� in order to simulate themodel they must close it with a demand side. As their last exercise shows, the speci�cation of the demandside can make a big di¤erence: when they allowed money supply to respond to real aggregate demand thestate-dependent model with strategic complementarity turned out to �t better the reset price in�ation IRSthan the model without strategic complementarity22 . This result suggests that in simulated models thedemand side speci�cation could be as important as the supply side.By contrast, our methodology focus on estimating the frictionless optimal price equation. The parameters

of this equation do not depend on the frequency of price adjustments or on the demand side speci�cation.They are not a¤ect by the selection of price changes either. Of course, we could be subject to other problemsas, for example, a misspeci�cation of the supply shocks.

22They do not select this model as the best because it generates a in�ation rate process that is too smooth.

21

Figure 3: Impulse response of estimated frictionless optimal price, baseline speci�cation

0.0

0.2

0.4

0.6

0.8

1.0

1 2 3 4 5 6 7 8 9 10 11 12

Note: Accumulated response of reset price inflation to one unit innovation.Months

Per

cent

age

poin

ts

Figure 4: Impulse response of estimated reset price, using BKM�s methodology

22

6 Conclusions

In this paper we have developed an econometric methodology to directly estimate the structural parametermeasuring the degree of strategic complementarities in pricing decisions in a state-dependent pricing model.At the micro level, �rms face idiosyncratic shocks and �xed costs of adjusting prices. We obtain the �rm�sfrictionless optimal price from fundamentals and, based on the Ss pricing rule, we derive a structural, non-standard ordered probit model. We use a quasi-maximum likelihood method that takes into account theautocorrelation and the heteroskedasticity that emerge from the microfounded model to estimate the probitusing the microdata underpinning the CPI-FGV in Brazil for the 1996-2006 period.Unlike the results in the literature, we �nd a substantial degree of strategic complementarity in �rms�

pricing decisions. For the aggregate economy we estimate the parameter � measuring strategic complemen-tarities ranging from 0.03 to 0.10. Therefore, we do not �nd that a state-dependent pricing model withstrategic complementarities is fundamentally in contradiction with the data, as suggested by Bills, Klenowand Malin (2009). The di¤erence in the methodology, and not in the data, is responsible for the distinctresults. As their methodology depends on the speci�cation of the demand side, a more realistic speci�cationof this part of their model could help to reconcile our results to theirs.A limitation of our results is that although they seem robust to the speci�cation of the demand side, they

are not certainly robust to the type of pricing rule. If the right model is time-dependent, strategic comple-mentarities found in the estimated state-dependent model could be a compensation for the mispeci�cationin the type of pricing rule. Thus, a natural next step should be to develop a similar methodology to estimatethe degree of strategic complementarities in a time-dependent pricing model and verify whether the sameresults are obtained.The methodology also allows us to estimate some characteristics of the pricing rules. Even though we

cannot separately identify the parameters S, s and c, we can easily estimate some characteristics of thepricing rules, like their widths, the length of the top and the bottom bands etc. The variance of shocks isalso estimated. All the results are quite consistent with the statistics on price changes calculated directlyfrom the data.

References

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[3] Barros, R., M. Bonomo, C. Carvalho and S. Matos (2009), �Price Setting in a Variable Macroeco-nomic Environment: Evidence from Brazilian CPI", unpublished paper, Getulio Vargas Foundationand Federal Reserve Bank of New York.

[4] Basu, S. (1995), �Intermediate Goods and Business Cycle: Implication for Productivity and Welfare",American Economic Review , 85 (March): 512-31.

[5] Bils, M. and P. Klenow (2004) "Some Evidence on the Importance of Sticky Prices," Journal of PoliticalEconomy 112, October 2004, 947-985.

23

[6] Bils, M., P. Klenow, and B. Malin (2009), �Reset Price In�ation and the Impact of Monetary PolicyShocks", NBER Working Paper 14787.

[7] Blanchard, O. and J. Gali (2007), �Real Wage Rigidities and the New Keynesian Model", Journal ofMoney, Credit and Banking , 39: 35-65.

[8] Boivin, J., M. Giannoni and I. Mihov (2009), �Sticky Prices and Monetary Policy: Evidence fromDisaggregated U.S. Data", American Economic Review 99 (March): 350-384.

[9] Bonomo, M. (1992), �Dynamic Pricing Models", Ph.D. thesis, Princeton University.

[10] Bonomo, M. and C. V. Carvalho (2010), �Imperfectly Credible Disin�ation under Endogenous Time-Dependent Pricing", Journal of Money, Credit and Banking , Forthcoming.

[11] Bonomo, M. and C. V. Carvalho (2004), �Endogenous Time-Dependent Rules and In�ation Inertia",Journal of Money, Credit and Banking , vol. 36, no. 6: 1015-1041.

[12] Burstein, A. and C. Hellwig (2007),�Prices and Market Shares in a Menu Cost Model", mimeo.

[13] Calvo, G. A. (1983), �Staggered Prices in a Utility-Maximizing Framework", Journal of MonetaryEconomics , 12 (September): 383-98.

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[15] Carvalho, C. (2006), �Heterogeneity in Price Stickiness and the Real E¤ects of MonetaryShocks",Frontiers of Macroeconomics : Vol.2: Iss. 1, Article 1.

[16] Danziger, L. �A Dymanic Economy with Costly Price Adjustment", American Economics Review 89(September): 878-901.

[17] Dhyne, E., L. Alvarez, H. Le Bihan, G. Veronese, D. Dias, J. Ho¤man, N. Jonker, P. Lunnemann, F.Rumler and J. Vilmunen (2006), �Price Changes in the Euro Area and the United States: Some Factsfrom Individual Consumer Price Data", Journal of Economic Perspectives 20: 171-192.

[18] Dias, D., R. Marques, and J. Santos Silva (2007), �Time- or state-dependent price setting rules? Evi-dence from micro data", European Economic Review 51: 1589-1613.

[19] Dotsey, M., R. King and A. Wolman (1999), �State-Dependent Pricing and the General EquilibriumDynamics of Money and Output", Quarterly Journal of Economics 114 (May): 655-90.

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[23] Gopinath, G. and O. Itskhoki (2010), �In Search of Real Rigidities", NBER Working Papers 16065.

24

[24] Gouvea, S. (2007), �Price Rigidity in Brazil: Evidence from CPI Micro Data", Central Bank of BrazilWorking Paper 143.

[25] Kimball, M. (1995), �The Quantitative Analytics of the Basic Neomonetarist Model", Journal of Money,Credit and Banking , 27: 1241-77.

[26] Klenow, P. and B. Malin (2010), �Microeconomic Evidence on Price-Setting", NBER Working Paper15826.

[27] Klenow, P. and J. Willis (2006), �Real Rigidities and Nominal Price Changes", mimeo.

[28] Kryvtsov, O. and V. Midrigan(2009), �Inventories, Markups and Real Rigidities in Menu Cost Models,"mimeo.

[29] Mankiw, G., and R. Reis (2002), �Sticky Information versus Sticky Prices: A Proposal to Replace theNew Keynesian Phillips Curve", Quarterly Journal of Economics 117 (November): 1295-1328.

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[31] Nakamura, E. and J. Steinsson (2008), �Five Facts About Prices: A Reevaluation of Menu Cost Models",Quarterly Journal of Economics , 123 (November): 1415-1464.

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[34] Sheshinski, E. and Y. Weiss (1983), �Optimum Pricing Policy under Stochastic In�ation", Reviw ofEconomics Studies , 50 (July): 513-29.

[35] Woodford, M. (2003), Interest and Prices: Foundations of a Theory of Monetary Policy , PrincetonUniversity Press.

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[37] Wooldridge, J. (2002), Econometric Analysis of Cross Section and Panel Data , MIT Press.

A Derivation of real marginal cost equation

Marginal Cost. The optimal quantities of labor and foreign input required to produce the quantity Yi;t ofgood i is given by

Li;t =

�

(1� �)~EtWit

!1��Yi;tAi;t

; (28)

25

Mt =

�(1� �)�

Wi;t

~Et

��Yi;tAi;t

: (29)

Then, the real cost of supplying the quantity Yi;t is

wi;tLi;t + EtMt = ���(1� �)(��1)w�i;ttE

(1��)t Yi;t

Ai;t

= �w�i;tE

(1��)t Yi;t

Ai;t; (30)

where � = ���(1� �)(��1), wi;t is the real wage of type i labor, and Et is the real exchange rate.Since we assume that each labor market is competitive, each �rm producing variety of type i, take type

i wage as given when choosing how much to produce. Thus, its marginal cost is given by:

�w�i;tE

(1��)t

Ai;t

In order to eliminate wi;t from the marginal cost equation, �rst note that by equation (4), the quantityof labor is positively related to the �rm�s product. Therefore, we will rewrite equation (4) as

wi;t =Vi;tL

�i;t

UtC� 1�

t

=

Vi;t

���

(1��)Etwi;t

�1��Yi;tAi;t

��UtY

� 1�

t

(31)

Then, we can write the real wage of type i as

w1+�(1��)i;t =

�(1� �)�

��(1��)�YitAi;t

��Y

1�t E

�(1��)t Vi;tU

�1t

wi;t =

�(1� �)�

� ��(1��)1+�(1��)

�YitAi;t

� �1+�(1��)

Y1

�(1+�(1��))t E

�(1��)1+�(1��)t V

11+�(1��)i;t U

� 11+�(1��)

t

Substituting the real wage expression into marginal cost equation yields:

(Yi;t; Yt; Et;Vi;t; Ut; Ai;t) = �w�i;tE

(1��)t

Ai;t=�Yi;tAi;t

�(1� �)�

��2�(1��)1+�(1��)

�YitAi;t

� ��1+�(1��)

Y�

�(1+�(1��))t E

��(1��)1+�(1��)t E

(1��)t V

�1+�(1��)i;t U

� �1+�(1��)

t

= �

�(1� �)�

��2�(1��)1+�(1��)

A� 1+�1+�(1��)

i;t Y�

�(1+�(1��))t Y

��1+�(1��)it E

(1��)+�(1��)1+�(1��)

t V�

1+�(1��)i;t U

� �1+�(1��)

t

26

B Econometric Estimation

Main Assumptions

Consider the following assumptions.

Assumption 1 The log aggregated price index, pt, is given by the following weighted sum:

pt =NXn=1

!n;tpn;t; (32)

where pn;t is the individual log price index and 0 < !n;t < 1 is a weight such that

!n;t �! 0 asN �!1;8 t: (33)

Assumption 2 The frictionless price follows the model

p�i;t = �+ �Yt + (1� �)pt +(1� �)1 + �!�

et + ~�t + ~ai;t; (34)

where Yt is the nominal expenditure, pt is the aggregated price index, � and � are combinations of deepparameters of the economy, ~�t is the aggregate shock and ~ai;t is the idiosyncratic shock.

Assumption 3 The aggregated shock ~�t follows a random walk process

~�t =~�t�1 + vt;

where ~�0 = Op(1),

vt =1Xj=0

'j�t�j ;

and1Xj=0

jj'j j <1:

Furthermore, f�tg1t=�1 is a sequence of independent and identically distributed random variables with zeromean such that E

��2t�<1 and E

��4+�t

�<1, for a � > 0.

Assumption 4 For each individual i, the idiosyncratic shocks, ~ai;t, follows one of the following processes:

1. ~ai;t = ai + ~ai;t�1 + "i;t, where ~�0 = Op(1); or

2. ~ai;t = ai + "i;t.

In both cases ai is a �xed e¤ect and "i;t can be written as

"i;t =1Xj=0

i;jei;t�j ; (35)

1Xj=0

jj j;tj <1:

Furthermore, fei;tg1t=�1 is a sequence of independent and identically distributed random variables with zeromean such that E

�e2i;t�<1 and E

�e4+�i;t

�<1, for a � > 0. Finally, E (ei;tej;t) = 0, 8 t and j 6= i.

27

Derivation of the Econometric Model

Under Assumption 2 we have:

r�i;t � pi;� � p�i;t = c� z0i;t� � (~�t � ~�� )� (~ai;t � ~ai;� ):

Under Assumption 3 and iterating ~�t backward we get

~�t � ~�� = vt + � � �+ vt��i;t+1:

Let

vt =TXj=1

j~dj;t

where

~dj;t =

(1; if t = j;

0; if t 6= j:

Hence,

~�t � ~�� =vt + vt�1 + � � �+ vt��i;t+1

=TXj=1

j~dj;t +

TXj=1

j~dj;t�1 + � � �+

TXj=1

j~dj;t��i;t+1

=TXj=1

jdj;t;

where

dj;t =

(1; if j 2 [t; t� �i;t + 1]0; if otherwise

Now, under Assumption 4 we have:

~ai;t � ~ai;� =(� + � � �+ �) + ("i;t + � � �+ "i;t��i;t+1)=��i;t + ui;t;

where �i;t is the time interval between t and � , and ui;t = "i;t+� � �+"i;t��i;t+1 is a moving average MA(�i;t�1)process. Therefore, ui;t � N(0; �i;t�2).Then, putting all parts together we obtain:

r�i;t � pi;� � p�i;t = c� ��i;t � z0i;t� �TXj=1

jdj;t � ui;t:

28

Likelihood and Robust Variance-Covariance Matrix

For notational reason, write equations in (18) in the following simpler way:

Pr[ri;t = 1jwi;t] = 1� ���1�1i;t � ~w0

i;t��

Pr[ri;t = 0jwi;t] = ���1�1i;t � ~w0

i;t��� �

��0�1i;t � ~w0

i;t��;

P r[ri;t = �1jwi;t] = ���0�1i;t � ~w0

i;t��;

where ~wi;t = (��i;t; �z0i;t; �d

0t)0 and � =

�~�; ~�

0; ~ 1; : : : ; ~ T

�0.

Then, the partial log-likelihood of an observation i at time t is given by:

li;t( ) = Ifri;t = 1g log�1� �

��1�1i;t � ~w0

i;t���+

+ Ifri;t = 0g log����1�1i;t � ~w0

i;t��� �

��0�1i;t � ~w0

i;t���+

+ Ifri;t = �1g log����0�1i;t � ~w0

i;t���;

where = (�1; �0;�0)0 and If�g is an indicator function.

Set

�1(:) � ���1�1i;t � ~w0

i;t��;

�0(:) � ���0�1i;t � ~w0

i;t��;

�1(:) � ���1�1i;t � ~w0

i;t��;

�0(:) � ���0�1i;t � ~w0

i;t��:

Then, the score of an observation i at time t is given by:

si;t( ) =

264@li;t@�1@li;t@�0@li;t@�

375 =24 s1;i;ts2;i;ts3;i;t

35 ;where,

s1;i;t =

�Ifri;t = 0g�1(:)� �0(:)

� Ifri;t = 1g1� �1(:)

��1(:)�1i;t;

s2;i;t =

�Ifri;t = �1g

�0(:)� Ifri;t = 0g�1(:)� �0(:)

��0(:)�1i;t;

s3;i;t =

�Ifri;t = 1g�1(:)1� �1(:)

+Ifri;t = 0g�1(:)� �0(:)

[��1(:) + �0(:)]�Ifri;t = �1g�0(:)

�0(:)

�~wi;t:

Asymptotic Results

Lemma 1 Let "i;t be de�ned as in Assumption 4. Under Assumption 1, E ("i;tpt) �! 0 as N �! 1,forall t.

29

Proof : Note that

E ("i;tpt) = E

"it

NXn=1

!n;tpn;t

!= !i;tE("i;tpi;t):

If �1 < E("i;tpi;t) <1, for all i and t, under Assumption 1, !i;tE("i;tpi;t) �! 0 as N �!1. �De�ne as the vector containing the parameters to be estimated and b the quasi maximum likelihood

estimator b = argmax 2

NXi=1

TXt=1

li;t( ): (36)

Theorem 1 First, assume that the true parameter vector is in the interior of , a compact parameterspace. Under Assumptions 1�4, b p�! and

pN�b � � d�! N

�0;A�1BA�1� :

Furthermore, A and B can be consistently estimated by

bA =1

N

NXi=1

TXt=1

si;t( )si;t( )0

and bB =1

N

NXi=1

TXt=1

si;t( )si;t( )0 +

1

N

NXi=1

TXt=1

Xr 6=t

si;r( )si;t( )0;

Con�dence interval for � and for �

Theorem (Delta Method): Suppose that � is an estimator of the Px1 vector � 2 � and that

pN(� � �) d�! N(0;V );

where V is a PxP positive de�nite matrix. Let c : � ! RQ be a continuous di¤erentiable function onthe parameter space � � RP , where Q � P , and assume that � is in the interior of the parameter space.De�ne C(�) � r�c(�) and the QxP Jacobian of c. Then

pN [c(�)� c(�)] d�! N[0;C(�)V C(�)0]: (37)

De�ne C � C(�). Then plim C = C(�). If plim V = V , then

N [c(�)� c(�)]0[CV C 0]�1[c(�)� c(�)] d�! �2Q: (38)

Proof : See Wooldridge (2002), pp. 44-45 �

Once � and � are given respectively by � =~�1

~�1+~�2and � = 1

~�1+~�2, we can use the theorem above to

obtain

30

pN [� � �] d�! N[0;C�V 1C

0� ] (39)

and

pN [� � �] d�! N[0;C�V 1C

0�]; (40)

where C� = [C1� C2� ]0, C� = [C1� C2�]

0 and C1� =1

~�1+~�2� ~�1

~�1+~�2, C2� = � ~�1

(~�1+~�2)

2, C1� = � 1

~�1+~�2,

C2� = � 1~�1+

~�2. V 1 is the respective variance-covariance matrix of ~�1 and ~�2.

Now the con�dence interval for the desired signi�cance level can be constructed as usual.

31

C Additional estimation results

Table 7: Detailed results of the probit model, baseline speci�cationBaseline speci�cationLog-likelihoood: -3.0634e+06 Number of obs: 2; 243; 642Parameter Estimates Std. error t-stat 95% conf. interval(c� s)=� 0.62 0.01 71.59 0.61 0.64(c� S)=� -0.92 0.01 -106.19 -0.93 -0.90p

�i;t -0.39 1.63 -0.24 -3.58 2.79Yt 0.60 0.08 7.76 0.45 0.75pt 9.57 0.05 194.12 9.48 9.67et 0.11 0.01 9.12 0.09 0.13� 0.06 0.01 - 0.04 0.07� 0.10 0.01 - 0.08 0.11

32

­10

­5

0

5

10

15

20

1996 1997 1998 1999 2000 2001 2002 2003 2004 2005

IPC­FGV Frictionless Reset

Perc

enta

gepo

ints

Note: Frictionless inflation corresponds to the average obtained across the 1000 simulations.          Actual inflation is calculated among products left after exclusions. Reset inflation is          obtained using BKM's methodology.

Figure 5: Estimated frictionless, estimated reset and actual infation

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Table 8: Detailed results of probit models, speci�cations for robustnessSpeci�cation with sample 1997-2000Log-likelihoood: -8.9567e+05 Number of obs: 666,948Parameter Estimates Std. error t-stat 95% conf. interval(c� s)=� 0.53 0.02 22.52 0.48 0.58(c� S)=� -0.76 0.02 -32.68 -0.81 -0.72p

�i;t -0.02 0.02 -1.19 -0.06 0.02Yt 0.15 0.16 0.91 -0.17 0.46pt 4.93 0.15 33.67 4.64 5.21et 0.62 0.02 25.53 0.57 0.67� 0.03 0.03 - -0.03 0.09� 0.19 0.03 - 0.13 0.26

Speci�cation with sample 2001-2004Log-likelihoood: -1.6960e+06 Number of obs: 1,277,736Parameter Estimates Std. error t-stat 95% conf. interval(c� s)=� 0.54 0.01 42.45 0.51 0.56(c� S)=� -0.93 0.01 -75.12 -0.96 -0.91p

�i;t -0.05 0.01 -4.25 -0.08 -0.03Yt 1.29 0.09 13.66 1.10 1.47pt 11.07 0.05 204.29 10.97 11.18et -0.40 0.02 -26.35 -0.43 -0.37� 0.10 0.01 - 0.09 0.12� 0.08 0.01 - 0.07 0.09

Speci�cation without exchange rateLog-likelihoood: -3.0698e+06 Number of obs: 2,243,642Parameter Estimates Std. error t-stat 95% conf. interval(c� s)=� 0.60 0.01 107.17 0.58 0.61(c� S)=� -0.94 0.01 -171.69 -0.95 -0.93p

�i;t -6.10 0.08 -74.61 -6.26 -5.94Yt 2.93 0.07 39.51 2.78 3.07pt 3.43 0.04 93.05 3.36 3.50� 0.46 0.01 - 0.45 0.48� 0.16 0.01 - 0.14 0.18

Speci�cation with idiosyncratic white noiseLog-likelihoood: -2.2784e+06 Number of obs: 2,243,642Parameter Estimates Std. error t-stat 95% conf. interval(c� s)=� 0.82 0.01 82.19 0.80 0.84(c� S)=� -1.19 0.01 -118.74 -1.21 -1.17

Yt 0.90 0.11 7.99 0.68 1.12pt 12.72 0.07 175.79 12.58 12.87et 0.05 0.02 2.72 0.01 0.08� 0.07 0.01 - 0.05 0.08� 0.07 0.01 - 0.06 0.09

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Table 9: Detailed results of probit models, speci�cations for robustness (continuation)Speci�cation without treating promotionsLog-likelihoood: -3.0779e+06 Number of obs: 2; 243; 642Parameter Estimates Std. error t-stat 95% conf. interval(c� s)=� 0.59 0.01 67.35 0.57 0.60(c� S)=� -0.88 0.01 -101.90 -0.90 -0.86p

�i;t -0.04 0.31 -0.13 -0.64 0.56Yt 0.53 0.08 6.75 0.38 0.69pt 8.90 0.05 176.56 8.80 9.00et 0.02 0.01 1.80 0.00 0.05� 0.06 0.01 - 0.04 0.07� 0.11 0.01 - 0.09 0.12

Speci�cation without itens exclusionsLog-likelihoood: -3.8638e+06 Number of obs: 2; 851; 318Parameter Estimates Std. error t-stat 95% conf. interval(c� s)=� 0.63 0.01 76.81 0.61 0.64(c� S)=� -0.93 0.01 -113.99 -0.94 -0.91p

�i;t -0.35 0.06 -5.90 -0.47 -0.23Yt 0.44 0.07 6.79 0.31 0.57pt 9.79 0.04 224.11 9.70 9.88et 0.03 0.01 3.07 0.01 0.05� 0.04 0.01 - 0.03 0.05� 0.10 0.01 - 0.09 0.11

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D Products excluded because of potential measurement errors

In this appendix we document the rationale for excluding some products from our database due to problematicprice quotes. We also list those products excluded in the estimation of the baseline speci�cation. Byproblematic price quotes we mean those prices in which there is evidence that, for technical reasons (themethod used to measure and/or collect prices), the recorded values not necessarily refer to the same individualproduct along the time, i.e., we are not sure that the prices recorded in the database are e¤ectively microprices in the sense we need in our estimations. In all cases, by maintaining these items, we could becomputing a price change when it actually did not happen. These problematic price quotes arise from avariety of measurement problems, which we organize in �ve broad categories.

D.1 Micro prices hard to collect:

There are products whose prices, for several reasons, are hard to collect and, consequently, there are doubtsif they are actually micro prices. In some cases the di¢ culty arises from the fact that is hard to �nd thesame product, with the same characteristics, along the time. In this group, "used cars" is a good example.In other cases, the method of price collection changes among itens and/or over time. Here "�sh" is anexample. Sometimes the price of �sh is collected per pound, sometimes by package and also by piece. Whencalculating the CPI these changes are handled, but we cannot identify them in the dataset.Products excluded: Shrimp (cod 112507), Corvina �sh (cod 112515), Fresh sardine (cod 112529),

European hake (cod 112536), Salmon (cod 112562), Videotape (cod 530753), Compact disc (cod 530755),Ticket for concerts (cod 530905), Ticket for theater (cod 530907), Ticket for soccer and other sports events(cod 531303), Airfare (cod 531505), Used cars (cod 620103), Telephone card (cod 720101)

D.2 Index number

Prices of some products are not individually registered in the dataset, but only in the form of an index.For residential rent, for example, the database does not record the price of the rent for each apartmentindividually. Instead, the price collectors visit many apartments with the same characteristics (e.g. two-bedroom apartments) and calculate the average price. This "rental index for two-bedroom apartments" iswhat is recorded in the dataset. Similar procedure is used for other products, such as condominium fees andhousemaids.Products excluded: Residential rents (cod 210101), Condominium fees (cod 210103), Daily housemaid

(cod 280101), Monthly housemaid (cod 280103)

D.3 Prices of a basket of goods

Because of their very distinct characteristics, for some goods the procedure involves collecting prices fora basket of goods with similar characteristics and recording only the average price. Typical goods in thiscategory are the items of clothing. For instance, it is very hard to collect prices for the same shirt alongthe time. In this case, prices of a basket of shirts with very similar characteristics (e.g. men�s shirts) arecollected and the average price is recorded in the dataset.Products excluded: Residence furniture (cod 230111), Mattresses (cod 230703), Blankets (cod 240101),

Sheets and pillowcase (cod 240103), Towels (cod 240105), Curtains and blinds (cod 260101), Rug and carpet(cod 260115), Drinking glass (cod 260305), Pots (cod 260309), Men�s shorts (cod 310101), Men�s pants (cod310105), Men�s shirts (cod 310111), Men�s underwear (cod 310117), Women�s shirts (cod 310307), Women�s

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pants (cod 310309), Women�s underwear (cod 310313), Dresses and skirts (cod 310331), Children�s shorts(cod 310503), Children�s shirts (cod 310505), Children�s pants (cod 310507), Men�s shoes (cod 320105), Men�stennis shoes (cod 320107), Women�s sandal (cod 320303), Women�s tennis shoes (cod 320305), Women�s shoes(cod 320307), Children�s tennis shoes (cod 320511), Eyeglasses (cod 420303)

D.4 Prices collected annually or semiannually

Educational fees and tuition are known to change on an annual or semiannual basis and, therefore, pricecollection is generally made in the same frequency.Products excluded: Elementary education (cod 510101), High school education (cod 510103), Kinder-

garten (cod 510105), College education (cod 510107), Day-care (cod 510153), Graduate education (cod510317)

D.5 Administered prices

This group of prices does not necessarily su¤er from problematic price quotes, but we decided to excludethem from our estimations because they are administered by contract or monitored, i.e., they are relativelyinsensitive to supply and demand conditions or are in someway regulated by a public agency.Products excluded: Water and sewer service charge (cod 210305), Residential electricity rates (cod

220101), Cylinder gas (cod 220103), Residential gas (cod 220105), Home phone service (cod 220109), Anal-gesic and antipyretic (cod 420111), Anti-infective drugs (cod 420112), Antiin�ammatory (cod 420114), An-titussive (cod 420115), Anti-allergy drugs (cod 420116), Blood pressure medication (cod 420117), Antide-pressants (cod 420118), Contraceptive pills (cod 420123), Subway fare (cod 610103), Bus fare (cod 610105),Intercity bus fare (cod 610303), Gasoline (cod 620503), Toll tari¤s (cod 620909), Lottery game (cod 720305)

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