Manajemen | Fakultas Ekonomi Universitas Maritim Raja Ali Haji jbes%2E2011%2E09066
Journal of Business & Economic Statistics
ISSN: 0735-0015 (Print) 1537-2707 (Online) Journal homepage: http://www.tandfonline.com/loi/ubes20
Lumpy Price Adjustments: A Microeconometric
Analysis
Emmanuel Dhyne, Catherine Fuss, M. Hashem Pesaran & Patrick Sevestre
To cite this article: Emmanuel Dhyne, Catherine Fuss, M. Hashem Pesaran & Patrick Sevestre
(2011) Lumpy Price Adjustments: A Microeconometric Analysis, Journal of Business & Economic
Statistics, 29:4, 529-540, DOI: 10.1198/jbes.2011.09066
To link to this article: http://dx.doi.org/10.1198/jbes.2011.09066
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Supplementary materials for this article are available online. Please click the JBES link at http://pubs.amstat.org.
Lumpy Price Adjustments:
A Microeconometric Analysis
Emmanuel D HYNE
National Bank of Belgium, Boulevard de Berlaimont 14, 1000 Brussels, Belgium, and Warocqué School of
Business and Economics, Université de Mons, Place du Parc 20, 7000 Mons, Belgium ([email protected])
Catherine F USS
National Bank of Belgium, Boulevard de Berlaimont 14, 1000 Brussels, Belgium
M. Hashem P ESARAN
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Faculty of Economics, Cambridge University, Sidgwick Avenue, Cambridge, CB3 9DD, United Kingdom, and
Department of Economics, University of Southern California, 3620 South Vermont Avenue, Los Angeles, CA 90089
Patrick S EVESTRE
Paris School of Economics, Université Paris 1-Panthéon Sorbonne, 106-112 boulevard de l’Hôpital, 75013 Paris,
France, and Banque de France, 31 rue Croix des Petits Champs, 75001 Paris, France
Based on a reduced-form state-dependent pricing model with random thresholds, we specify and estimate
a nonlinear panel data model with an unobserved factor representing the common cost or demand components of the unobserved optimal price. Using this model, we are able to assess the relative importance of
common and idiosyncratic shocks in explaining the frequency and magnitude of price changes in the case
of a wide variety of consumer products in Belgium and France. We find that the mean level and variability of the random thresholds are key for explaining differences across products in the frequency of price
changes. We also find that the idiosyncratic shocks represent the most important driver of the magnitude
of price changes. Supplementary materials for this article are available online.
KEY WORDS: Idiosyncratic shock; Micro nonlinear panel; Sticky prices.
1. INTRODUCTION
Following the contributions of Cecchetti (1986) on newspaper prices, Kashyap (1995) on catalog prices (both using U.S.
data), and Lach and Tsiddon (1992) on meat and wine prices
in Israel, a recent wave of empirical research has provided
new evidence on the nature and sources of consumer and producer price stickiness at the micro level. These studies include
those of Bils and Klenow (2004), Klenow and Kryvstov (2008),
Nakamura (2008), and Nakamura and Steinsson (2008), who
analyzed U.S. consumer prices, and Dhyne et al. (2006), who
provided a synthesis of recent empirical analyses carried out
for the Euro area countries. Studies of producer prices include
those of Vermeulen et al. (2007), Cornille and Dossche (2008),
Loupias and Sevestre (2010), among others.
These previous works have pointed out several important features of price changes at the outlet/firm level. In particular, at
the microeconomic level, price changes tend to be infrequent
and not synchronized. The average size of the individual price
changes is also much larger than the overall inflation rate. As argued by Golosov and Lucas (2007) and Klenow and Kryvtsov
(2008), large idiosyncratic shocks are required to explain these
facts. Using U.S. price data, Hosken and Reiffen (2004) and
Nakamura (2008) have indeed shown that much of the observed
price changes can be attributed to idiosyncratic shocks at the
level of the outlet rather than to common shocks, which they assume to be well approximated by changes in wholesale prices.
The purpose of this article is to provide a further assessment
of the respective contributions of idiosyncratic and common
shocks to costs and/or outlets’ margin for the observed changes
in consumer prices. The article extends the results in the literature in several respects. First, we consider two CPI-based
data sets from Belgium and France that cover a wide range of
not only consumer goods, but also services, observed over a
relatively long time period. The two datasets combined cover
more than 180 different goods and services of all kinds observed at a monthly frequency from July 1994 to December
2003, and include energy, perishable food, nonperishable food,
nondurables, durables, and services. In addition, compared with
articles based on scanner data (e.g., Nakamura 2008), the price
reports available in the CPI datasets are less contaminated by
strategic price changes of large supermarkets (the relative advantages of CPI price data over the scanner data are analyzed
in Section 1.3 of the online supplementary materials). Second, using only price information, we propose a methodology based on an (S, s) model of sticky prices that allows us
to identify and evaluate the common and idiosyncratic components of the unobserved optimal price together with the parameters that characterize the adjustment of prices to their optimal
level. In this respect, we extend previous work by Ratfai (2006)
on meat product prices in Hungary; Bacon (1991), Borenstein,
Cameron, and Gilbert (1997), and Davis and Hamilton (2004)
for gasoline prices; Dutta, Bergen, and Levy (2002) for orange
529
© 2011 American Statistical Association
Journal of Business & Economic Statistics
October 2011, Vol. 29, No. 4
DOI: 10.1198/jbes.2011.09066
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530
juice; and Fougère, Gautier, and Le Bihan (2010) for restaurants, which considered only common cost shocks affecting the
raw material prices, wholesale prices, or wage costs. Third, our
methodology extends the work of Hosken and Reiffen (2004)
and Nakamura (2008) in that our (S, s) model explicitly accounts for the existence of periods of price stability between
two price changes in estimation of the common component of
price changes. Thus the (S, s) model that we consider represents
a nonlinear extension of the factor models used extensively in
the empirical finance and macroeconomic literature (e.g., Connor and Korajczyk 1986, 1988; Stock and Watson 1998, 2002;
Forni et al. 2000; Bai and Ng 2002, 2006). Fourth, our (S, s)
model and the proposed estimation procedures also allow for
random thresholds, which, as shown by Caballero and Engle
(1999), Dotsey, King, and Wolman (1999), and Costain and
Nakov (2011), may help explain the existence of both large and
small price changes.
Our main findings may be summarised as follows. The estimation results confirm the existence of a pervasive degree of
heterogeneity in the price behavior across products and outlets.
However, once we control for the product type, the behavior of
consumer prices in Belgium and France seems to be quite similar. The frequency of price changes appears to depend more
strongly on the characteristics of the price inaction band (i.e.,
its average width and its variance) than on the variability of
shocks to the optimal price. However, the latter are by far the
most important drivers of the magnitude of price changes, with
the idiosyncratic shocks playing a major role in this respect. Explaining small price changes appears to be more difficult. The
(S, s) model advanced in this article, which allows for stochastic inaction bands, is an improvement over (S, s) models with
a fixed band of inaction, asymmetric inaction bands, or bands
with seasonal adjustments in replicating the occurrence of small
price changes, but it does not fully succeed in predicting very
small price changes observed for some products.
The rest of the article is structured as follows. Section 2 sets
out the general (S, s) model of prices. Section 3 describes the
estimation procedures. Section 4 focuses on the presentation
and discussion of the estimation results, and Section 5 concludes.
2. AN (S, s) MODEL OF STICKY PRICES WHEN
ONLY PRICES ARE OBSERVED
It is now a well-established stylized fact that most consumer
prices remain unchanged for periods of up to several months
(see, e.g., Bils and Klenow 2004; Dhyne et al. 2006; Nakamura
and Steinsson 2008). Physical menu costs, fear of customer
anger, and implicit or explicit contracts are among the many
sources of price rigidity posited in the literature, which could
explain why retailers might not be willing to immediately adjust their prices to changes in their market conditions, such as
changes in wholesale prices, costs of distribution, or changes
in demand or local competition (see, e.g., Blinder et al. 1998).
Essentially three approaches have been proposed for modeling price stickiness. In time-dependent models, the probability of changing prices does not depend on the evolution of the
outlets’ economic environment (see Calvo 1983 for a prominent example of this). In state-dependent models advanced by,
Journal of Business & Economic Statistics, October 2011
among others, Sheshinski and Weiss (1977) and Dotsey, King,
and Wolman (1999), the probability of changing prices depends
on changes in the outlets’ economic environment. Following
Sims (2003) and Mankiw and Reis (2006), a third group of
models has recently emerged that explain price stickiness by
the costs of information acquisition and/or by the noise that
can affect the information collected by firms about their environment (e.g., see Eichenbaum and Fisher 2007; Klenow and
Willis 2007; Woodford 2009).
Whatever approach is adopted, assessing how consumer
prices react to changes in the outlets’ economic environment
and what model fits best to the “stylized facts” remains a largely
open issue. One of the main reasons for this is that we do not
have a fully satisfactory statistical measurement of the unobserved optimal prices targeted by outlets for the products they
sell. This problem has been addressed in the literature in various
ways. Some studies have considered how individual prices react
to general measures of inflation, considered at the national or
the industry level (see, e.g., Cecchetti 1986; Lach and Tsiddon
1992; Fougère, Le Bihan, and Sevestre 2007; Gagnon 2009).
Others have explicitly studied the link between individual price
changes and costs measured by wage costs or by wholesale
price variations; however, this has been carried out most often for a specific product or group of products (e.g., gasoline
in Peltzman 2000 and Davis and Hamilton 2004; orange juice
in Dutta, Bergen, and Levy 2002; meat in Ratfai 2006; grocery
products in Nakamura 2008; restaurants in Fougère, Gautier,
and Le Bihan 2010).
In this article we adopt a different approach to the identification and estimation of the optimal price and propose a statistical decomposition of changes in retail prices into common and
idiosyncratic shocks using a nonlinear factor model that explicitly allows for periods of no price changes. More specifically,
we consider the following decomposition of the (unobserved)
optimal log price, p∗jit , of outlet i for its product j at time t
p∗jit = x′jit β + fjt + vji + εjit ,
j = 1, 2, . . . , M, i = 1, 2, . . . , N, t = 1, 2, . . . , T,
(1)
where xjit is a vector of observable product and retail-specific
variables with the coefficients, β, and fjt represents the unobserved common cost or demand component of p∗jit at time t,
which is assumed to be the same across all outlets, i, for a given
product j. The remaining terms in (1) are intended to capture the
product and retail-specific, vji , or purely random differences,
εjit , in optimal prices across the outlets.
The elements of xjit measure the observed characteristics of
the product/outlet that might explain price-level differences of a
particular product across outlets, such as whether the product is
offered as part of sales promotion, outlet-specific features (e.g.,
hyper or supermarket vs. corner shop), and geographic location (city center vs. suburb). The elements of xjit could be timevarying, as in the case of sales promotion, or time-invariant, as
in the case of store type. The nature of the outlet (supermarket
or corner shop) is particularly important, because for a similar product, average prices tend to be lower in supermarkets
than in corner shops. The second component, fjt , is the common component of prices of a given product j, across outlets;
it is period-specific and is shared across all outlets selling a
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Dhyne et al.: Lumpy Price Adjustments
531
given fairly homogeneous product. From an economic standpoint, this component reflects the average marginal cost augmented with the average desired markup associated with this
particular product. From an econometric standpoint, we model
this as an unobserved common factor that may be estimated by
aggregating the nonlinear pricing equations across the outlets.
In this respect, we go one step further than Hosken and Reiffen
(2004) and Nakamura (2008), who estimated this component
by averaging prices of a given product across outlets. Indeed,
as we show later, we explicitly account for periods of price inaction in the estimation of this common component. The third
component of p∗jit , vji , is an unobserved outlet-specific effect for
a given product j, which accounts for price differences due to
product differentiation, local competition conditions, and other
factors. The fourth component of the optimal price, εjit , is an
idiosyncratic term reflecting shocks that might affect the outletspecific optimal price in a given period (possibly due to outletspecific demand shocks or unexpected changes in costs at the
store level).
This statistical decomposition does not match the usual decomposition of the optimal price into a cost component and
a markup component. However, for each product j, it allows
estimation of the respective variances of aggregate (fjt ) and
idiosyncratic (εjit ) shocks and thus allows an assessment of
their respective impact on the frequency and magnitude of price
changes.
To link the unobserved optimal price components to observed
prices, a suitable price-setting decision rule that can explain infrequent but possibly large price changes is needed. One possibility is to assume the existence of a fixed price adjustment cost,
leading to an optimal price strategy of the (S, s) variety (see,
e.g., Sheshinski and Weiss 1977, 1983; Cecchetti 1986; Dixit
1991; Hansen 1999; Gertler and Leahy 2006). Indeed, several
previous studies have found evidence of fixed physical menu
costs of price adjustment (Levy et al. 1997; Blinder et al. 1998;
Zbaracki et al. 2004), although Zbaracki et al. (2004) argued
that in addition to these fixed physical menu costs, managerial and customer-related costs are convex in the price change.
A simple specification of a (S, s) model representing the pricing
rule followed by outlet i for its product j, can be written as
pji,t−1 if |p∗jit − pji,t−1 | ≤ sj ,
(2)
pjit =
if |p∗jit − pji,t−1 | > sj ,
p∗jit
where pjit is the (log) observed price of a product j in outlet i
at time t, p∗jit is the (log) optimal price as defined by (1), and sj
denotes the thresholds beyond which outlets find it profitable to
adjust their prices in response to a shock. This specification assumes that the pricing thresholds for price increases and price
decreases are equal on average and that there is no additional
downward price rigidity. In what follows, to simplify notation,
we drop the subscript j and refer to s as the “adjustment threshold” or “band of inaction.” We refer to
|p∗it − pi,t−1 | ≥ s,
(3)
as the “price change trigger” condition.
Assuming a common, time-invariant adjustment threshold
across all outlets might be considered too restrictive, because price setting may be strongly heterogeneous across outlets, even for relatively homogeneous product categories (see,
e.g., Aucremanne and Dhyne 2004; Fougère, Le Bihan, and
Sevestre 2007). At the outlet level, some price trajectories are
characterized by very frequent price changes, whereas others
are characterized by infrequent price changes. Moreover, as described by Campbell and Eden (2005), some price trajectories
at the micro level exhibit long periods of price stability followed
by periods of frenetic price changes. As noted by Caballero and
Engel (2007), this pattern of price changes suggests that the
range of price inaction is best modeled as a stochastic process.
Another argument for adopting such an approach lies in the evidence of small price changes (Klenow and Kryvstov 2008).
Thus we extend model (2) to allow for (random) time- and
outlet-varying pricing thresholds, considering the representation
pi,t−1 if |p∗it − pi,t−1 | ≤ sit ,
(4)
pit = ∗
if |p∗it − pi,t−1 | > sit ,
pit
and assume that sit are random draws from a common distribution. Other specifications of sit are considered and compared in
Section 6 of the online supplementary materials. Let I(A) denote an indicator function that takes the value of unity if A > 0
and 0 otherwise. Then model (4) can be written as
pit = pi,t−1 + (p∗it − pi,t−1 )I(p∗it − pi,t−1 − sit )
+ (p∗it − pi,t−1 )I(pi,t−1 − p∗it − sit ).
(5)
This specification is closely related to the models considered
by Rosett (1959) for the analysis of frictions in yield changes
and, more recently, by Tsiddon (1993), Willis (2006), Ratfai
(2006), and Fougère, Gautier, and Le Bihan (2010) in the sticky
price literature. But it departs from those models in several important respects. First, instead of using a producer price index
to proxy the common movements in consumer price trajectories
as done by Ratfai (2006) and Fougère, Gautier, and Le Bihan
(2010), we rely on an unobserved common component. This
allows us to conduct our analysis for products for which there
is no directly observable or easily identified explanatory variables. One important advantage of proceeding in this manner
is to ensure the coherency of this common component with the
underlying dynamics of micro price decisions as stated by our
model. Further, it avoids the drawback that if the observed variable fails to capture the common factor, then part of the common variation will show up in the error term.
Second, we also depart from the existing empirical literature
in the information used in our estimation procedure. Most of
the literature estimates state-dependent pricing models using binary response or duration models (Cecchetti 1986; Aucremanne
and Dhyne 2005; Campbell and Eden 2005; Ratfai 2006; Willis
2006; Fougère, Le Bihan, and Sevestre 2007) and thus neglects
the information contained in the magnitude of price changes.
However, this information is crucial for identifying the volatility of the idiosyncratic component and for disentangling the idiosyncratic component of the optimal price from the idiosyncratic threshold variable, sit . In a binary response model, where
only the occurrence of price changes is considered, the effects
of εit and sit cannot be identified separately. In our framework,
the observations on the price level, pit , allows the identification
of the idiosyncratic component, εit , from the optimal price, p∗it ,
because p∗it = pit when a price change occurs.
532
Journal of Business & Economic Statistics, October 2011
Third, our approach does not impose any restrictions on the
dynamics of the common factors, and it allows for possible
structural breaks in ft . In principle, it is also possible to allow
for the idiosyncratic shocks, εit , to be serially correlated. But to
simplify the exposition and for ease of estimation, in what follows we assume that εit are serially uncorrelated. The case of
serially correlated errors is considered in Section 4 of the online
supplementary materials, which uses Monte Carlo experiments
to show that neglecting (positive) serial correlation in the idiosyncratic shocks tends to result in overestimation of the band
of inaction. However, the bias is small for reasonable values of
the serial correlation coefficient.
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3. ALTERNATIVE ESTIMATION METHODS
Equations (1) and (5) can be combined to obtain the following econometric specification:
pit − pi,t−1 = (ft + x′it β + vi + εit − pi,t−1 )
× I(ft + x′it β + vi + εit − pi,t−1 − sit )
+ (ft + x′it β + vi + εit − pi,t−1 )
× I(pi,t−1 − ft − x′it β − vi − εit − sit ). (6)
There are essentially two groups of parameters to be estimated:
the unobserved common components, ft , which can also be
viewed as unobserved time effects, and the parameters that do
not vary over time, namely s and σs , which denote the mean
and standard deviation of sit ; σε , the standard deviation of the
idiosyncratic component εit ; σv , the standard deviation of the
firm-specific random effect, vi ; and β, the parameters associated with the observed explanatory variables, xit .
The estimation of the baseline model can be carried out in
two ways. One can use an iterative procedure that combines the
estimation of the ft ’s using the cross-sectional dimension of the
data with the maximum likelihood (ML) estimation of the remaining parameters, conditional on the first-stage estimate of
ft . Alternatively, one can use a standard ML procedure, where
the ft ’s are estimated simultaneously with the other parameters.
The two procedures lead to consistent estimates, provided that
N and T are sufficiently large. It is noteworthy that if N is small,
then one will face the well-known incidental parameters problem. The bias in estimating ft due to the limited size of the crosssectional dimension will contaminate the other parameter estimates. In the alternative situation where T happens to be small,
the problem of the initial observation will become an important
issue. Therefore, our estimation procedure is essentially valid
for relatively large N and T. Fortunately, in our context, the
prices of most of the products that we consider have been observed monthly over the period 1994:7–2003:2 (i.e., more than
100 months), and the number of outlets selling these products
also is relatively large, on average close to 300 in both Belgium
and France.
3.1 Estimation of f t Using Cross-Sectional Averages
As mentioned earlier, in practice ft is an unobserved time effect that needs to be estimated along with the other unknown
parameters. It reflects the common component in the optimal
prices for each particular product for which we estimate the
model. Moreover, because we are able to consider precisely defined types of products sold in a particular outlet, it is reasonable to assume that any remaining cross-sectional heterogeneity
in the price level can be modeled through the observable outletspecific characteristics, xit , and through random specific effects
(accounting for outlets unobserved characteristics).
Accordingly, we assume that, conditional on hit = (ft , x′it ,
pi,t−1 )′ , the random variables (sit , vi , εit )′ are distributed independently across i, and that sit and εit are serially uncorrelated.
Because of the nonlinear nature of the pricing process, and to
make the analysis tractable, we also assume that
⎛ ⎞
⎛⎛ ⎞ ⎛ 2
⎞⎞
sit
0
0
s
σs
⎝ vi ⎠ hit ∽ iid N ⎝⎝ 0 ⎠ , ⎝ 0 σv2 0 ⎠⎠ .
0
0
0 σε2
εit
The assumption of zero covariances across the errors is made
for convenience and can be relaxed.
Before discussing the derivation of ft we state the following
lemma (established in Section 3 of the online supplementary
materials), which provides a few results needed later.
Lemma 1. Suppose that y ∽ N(μ, σ 2 ); then
a+μ
a+μ
E[yI(y + a)] = σ φ
+ μ
,
σ
σ
y+a
a+μ
b
E φ
φ √
=√
,
b
b2 + σ 2
b2 + σ 2
a+μ
y+a
= √
,
Ey
b
b2 + σ 2
where φ(·) and (·) are, respectively, the density and the cumulative distribution function of the standard normal variate,
and I(A) is the indicator function defined earlier.
Let dit = ft + x′it β − pi,t−1 , ξit = vi + εit ∽ N(0, σξ2 ), and note
that σξ2 = σv2 + σε2 . Now consider the baseline model, (6), and
using the foregoing, write it as
pit = (dit + ξit )I(dit + ξit − sit ) + (dit + ξit )I(−dit − ξit − sit )
or
pit = (dit + ξit ) + (dit + ξit )[I(dit + ξit − sit ) − I(dit + ξit + sit )].
Denote the unknown parameters of the model by θ =
(s, β ′ , σs2 , σv2 , σε2 )′ , and note that E(pit |hit , θ) = dit + git ,
where git = g1,it + g2,it , with
g1,it = dit E[I(dit + ξit − sit ) − I(dit + ξit + sit )|hit , θ]
and
g2,it = E[ξit I(dit + ξit − sit ) − ξit I(dit + ξit + sit )|hit , θ].
Also, under our assumptions,
2
sit
s
σ
, s
hit ∽ iid N
0
0
ξit
0
σv2 + σε2
.
It is easily seen that
E[I(dit + ξit − sit ) − I(dit + ξit + sit )|hit , θ]
dit + s
dit − s
−
.
=
σs2 + σξ2
σs2 + σξ2
Dhyne et al.: Lumpy Price Adjustments
533
Using the results in Lemma 1 and noting that ξit |hit , θ ∽
N(0, σξ2 ), we have
dit − sit
.
E[ξit I(dit + ξit − sit )|hit , sit , θ ] = σξ φ
σξ
Thus, taking expectations with respect to sit , we have
dit − sit
E[ξit I(dit + ξit − sit )|hit , θ] = σξ E φ
hit , θ .
σξ
Again using the results in Lemma 1, we have
σξ
dit − s
dit − sit
φ
,
E φ
hit , θ =
σξ
σs2 + σξ2
σs2 + σξ2
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and thus
E[ξit I(dit + ξit − sit )|hit , θ] =
Similarly,
E[ξit I(dit + ξit + sit )|hit , θ] =
σξ2
dit − s
φ
.
σs2 + σξ2
σs2 + σξ2
σξ2
σs2 + σξ2
dit + s
φ
.
σs2 + σξ2
Collecting the various results, we obtain
dit − s
dit + s
g1,it = dit
−
σs2 + σξ2
σs2 + σξ2
and
g2,it =
σξ2
dit − s
dit + s
φ
−φ
.
σs2 + σξ2
σs2 + σξ2
σs2 + σξ2
Here g1,it and g2,it are nonlinear functions of ft and depend on i
only through the observable, pi,t−1 and xit . Thus, it is possible
to compute ft for each t in terms of pi,t−1 , xit and θ . Then, following Pesaran (2006), the cross-sectional average estimator of
ft , denoted by f˜t , can be obtained as the solution to the following
nonlinear equation:
(7)
p¯ t = f˜t + x¯ ′t β + g¯ t (f˜t ),
N
N
¯t =
where p¯ t =
¯ t (ft ) =
i=1 wit pit , x
i=1 wit xit , and g
N
a predeteri=1 wit git , and {wit , i = 1, 2, . . . , N} represent
2
mined set of weights such that wit = O(N −1 ) and N
i=1 wit =
−1
O(N ).
For a given value of θ and each t, (7) provides a nonlinear
function in f˜t . This equation clearly shows that unlike the linear
models considered by Pesaran (2006), here the solution to the
common component ft does not reduce to an average of (log)
prices. In particular, f˜t also accounts for the dynamic feature of
the price-setting behavior through the g¯ t component, which depends on pi,t−1 . The Monte Carlo simulations provided in Section 4 of the online supplementary materials show that taking
into account the nonlinear component, g¯ t , substantially reduces
the root mean squared error (RMSE) of estimating ft by f˜t compared with using the linear cross-section approximation given
by p¯ t . As expected, the RMSE of f˜t relative to p¯ t declines as the
frequency of price changes diminishes, namely as price lumpiness increases. Equation (7) has a unique solution as long as
s > 0. A proof of this is provided in Section 2 of the online
supplementary materials. It is also easily seen that under the
cross-sectional independence of vi and εit , g¯ t (ft ) → E(git ) and
p
f˜t − ft → 0, as N → ∞. Note that for the sake of simplicity, here
we assume that the panel data sample is balanced; however, the
result can be easily generalized to unbalanced panels assuming
that Nt → ∞ for each t, where Nt denotes the number of outlets
in period t.
3.2 Conditional Likelihood Estimation Without
Random Effects
In this section we derive the ML estimator (MLE) of the
structural parameters, θ , conditional on ft and assuming that
there are no firm-specific effects, so that σv2 = 0, and thus in this
case θ = (s, β ′ , σs2 , σε2 )′ . Given the distributional assumptions
stated in Section 3.1, and defining ζit as sit − s, our baseline
model can be rewritten as
pit = dit + εit + (dit + εit ){I[dit + εit − ζit − s]
− I[dit + εit + ζit + s]},
where
2
ζit
0
σ
∽ iid N
, s
0
0
εit
0
σε2
for i = 1, 2, . . . , N, t = 1, 2, . . . , T.
Equivalently
pit = dit + εit + (dit + εit ){I[dit − s + ε1it ] − I[dit + s + ε2it ]},
where ε1it = εit − ζit and ε2it = εit + ζit , with
⎛
⎞
⎛⎛ ⎞ ⎛ 2
σε + σs2 σε2 − σs2
ε1it
0
⎝ ε2it ⎠ ∼ iid N ⎝⎝ 0 ⎠ , ⎝ · · ·
σε2 + σs2
0
εit
···
···
Let
τ1it =
1 if pit = 0,
0 otherwise,
τ2it =
1 if pit > 0,
0 otherwise,
τ3it =
1 if pit < 0,
0 otherwise
⎞⎞
σε2
σε2 ⎠⎠ .
σε2
for i = 1, 2, . . . , N and t = 1, 2, . . . , T.
Then, conditional on ft , t = 1, 2, . . . , T, and the initial value pi0 ,
the log-likelihood function of the model for each i can be written as
Li (θ|f) = Pr(pi1 |pi0 ) Pr(pi2 |pi0 , pi1 )
× Pr(pi,T |pi0 , pi1 , . . . , pi,T−1 ) × Pr(pi0 ),
where f = (f1 , f2 , . . . , fT )′ . In view of the model’s first-order
Markovian property, we have
Li (θ |f) = Pr(pi1 |pi0 ) Pr(pi2 |pi1 ) Pr(pi,T |pi,T−1 ) × Pr(pi0 ).
When T is small, the contribution of Pr(pi0 ) could be important.
In what follows, we assume that pi0 is given and T is reasonably
534
Journal of Business & Economic Statistics, October 2011
large so that the contribution of the initial observations to the
log-likelihood function can be ignored.
To derive Pr(pit |pi,t−1 , ft ), we distinguish among cases
where pit = 0, pit > 0, and pit < 0, noting that
Pr(pit = 0|pi,t−1 , ft )
= Pr(ε1it ≤ s − dit ; ε2it ≥ −s − dit )
= Pr(ε1it ≤ s − dit ) − Pr(ε1it ≤ s − dit ; ε2it ≤ −s − dit )
s − dit
=
σε2 + σs2
s − dit
−s − dit σε2 − σs2
− 2
;
;
σε2 + σs2
σε2 + σs2 σε2 + σs2
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= π1it ,
where 2 (x; y; ρ) is the cumulative distribution function of the
standard bivariate normal. Similarly,
Pr(pit > 0|pi,t−1 , ft )
= Pr(εit = pit − dit ) Pr(ε1it ≥ s − dit ; ε2it > −s − dit |εit )
−s + pit
−s − pit
1
pit − dit
−
= φ
σε
σε
σs
σs
= π2it
3.3 Conditional ML Estimation With Random Effects
Now consider the random-effects specification where p∗it =
ft + x′it β + vi + εit , and note that
Cov(p∗it , p∗it′ |hit , hit′ ) = σv2
for all t and t′ , t
= t′ .
Under this model, the probability of no price change in a given
period, conditional on the previous price, pi,t−1 , will not be independent of episodes of no price changes in the past. Thus we
need to consider the joint probability distribution of successive
unchanged prices. For example, suppose that prices for outlet i
have remained unchanged over the period t and t + 1; then the
relevant joint events of interest are
Ait : {−s − ζit − dit ≤ εit + vi ≤ s + ζit − dit },
Ai,t+1 : {−s − ζi,t+1 − di,t+1 ≤ εi,t+1 + vi ≤ s + ζit − di,t+1 }.
An explicit derivation of the joint distribution of Ait and Ait+1
seems rather difficult. An alternative strategy is to use the conditional independence property of successive price changes and
note that for each i, and conditional on v = (v1 , v2 , . . . , vN )′
and f, the likelihood function will be given by
L(θ, v, f) =
N
T
i=1 t=1
[π1it (vi )]τ1it [π2it (vi )]τ2it [π3it (vi )]τ2it ,
where
and
Pr(pit < 0|pi,t−1 , ft )
= Pr(εit = pit − dit ) Pr(ε1it < s − dit ; ε2it ≤ −s − dit |εit )
−s − pit
−s + pit
pit − dit
1
−
= φ
σε
σε
σs
σs
= π3it .
Thus,
ℓ(θ , f) =
=
N
ln Li (θ , f)
i=1
N
T
[τ1it ln(π1it ) + τ2it ln(π2it ) + τ3it ln(π3it )].
(8)
i=1 t=1
The MLE of θ is given by
θˆ ML (f) = arg max ℓ(θ , f)
θ
and for N and T sufficiently large, we have
√
a
NT(θˆ ML (f) − θ ) ∽ N(0, Vθ ),
where Vθ is the asymptotic variance of the MLE and can be
estimated consistently using the second derivatives of the loglikelihood function.
Remark 1. In the case where ft , t = 1, 2, . . . , T, are estimated, the MLE will continue to be consistent as both N and
T tend to infinity. However, the asymptotic distribution of the
MLE is likely to be subject to the generated regressor problem. The importance of the generated regressor problem in the
present application could be investigated using a bootstrap procedure.
s − vi − dit
π1it (vi , ft ) =
σε2 + σs2
s − vi − dit −s − vi − dit σε2 − σs2
− 2
,
;
; 2
σε2 + σs2
σε2 + σs2 σε + σs2
pit − vi − dit
1
π2it (vi , ft ) = φ
σε
σε
−s + pit
−s − pit
×
−
,
σs
σs
1
pit − vi − dit
π3it (vi , ft ) = φ
σε
σε
−s + pit
−s − pit
−
.
×
σs
σs
The random effects can now be integrated out with respect to
the distribution of vi , assuming vi ∼ N(0, σv2 ), for example, and
then the integrated log-likelihood function, Ev ℓ(θ , v, f)), maximized with respect to θ .
3.4 Full ML Estimation
In the case where N and T are sufficiently large, the incidental parameters problem does not arise, and the effects of the
initial distributions, Pr(pi0 ), on the likelihood function can be
ignored. Then the MLEs of θ and f can be obtained as the solution to the following maximization problem:
(ˆfML , θˆ ML ) = arg max
f,θ
T
N
[τ1it ln(π1it )
t=1 i=1
+ τ2it ln(π2it ) + τ3it ln(π3it )].
(9)
Dhyne et al.: Lumpy Price Adjustments
Note that for a given value of θ the MLE of ft can be obtained
as
fˆt (θ ) = arg max
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ft
N
[τ1it ln(π1it ) + τ2it ln(π2it ) + τ3it ln(π3it )],
i=1
and will be consistent as N → ∞, because, conditional on θ
and ft , the elements in the foregoing sum are independently distributed. Also, for a given estimate of f, the optimization problem defined by (9) will yield a consistent estimate of θ as N
and T → ∞. Iterating between the solutions of the two optimization problems will deliver consistent estimates of θ and
f1 , f2 , . . . , fT , even though the number of incidental parameters,
ft , t = 1, 2, . . . , T, is rising without bounds as T → ∞. This
is analogous to the problem of estimating time and individual
fixed effects in standard linear panel data models. Individual
fixed effects can be consistently estimated from the time dimension, and time effects can be estimated from the cross-section
dimension.
To evaluate the performance of these estimation methods,
Section 4 of the online supplementary materials reports a number of Monte Carlo simulations. We evaluate ML estimation
with and without random effects. These lead to roughly qualitatively similar results. We also report a set of ML results
for alternative values of the parameters and frequency of price
changes. We then perform a set of Monte Carlo simulations
to evaluate the robustness of the model under deviations from
the underlying assumptions. We first examine the small-sample
properties of our estimator. We then consider the case of serially correlated idiosyncratic shocks. Finally, we investigate the
impact of cross-sectional dependence on the estimates of the
model parameters.
The results of these simulations may be summarized as follows. Estimation of the common component is adversely affected only if the cross-section dimension is relatively small.
Ignoring serial correlation of the idiosyncratic component leads
to a positive bias in the estimates of s and σs . However, the bias
becomes substantial only as the serial correlation coefficient of
the idiosyncratic errors approaches unity. For the level of error
serial correlation estimated by Ratfai (2006) for meat (at 0.34),
our simulations suggest that the upward bias in the estimates
of s should be below 8 percent. Finally, as is the case with linear factor models, estimates of the common components are not
adversely affected by the presence of weak cross-sectional dependence in the idiosyncratic shocks.
4. EMPIRICAL RESULTS
The model presented earlier was estimated using individual
consumer price quotes compiled by the Belgian and French
statistical institutes for the computation of their respective consumer price indices. Each dataset contains more than 10 million observations referring to monthly price quotes of individual products sold in a particular outlet. For each product category, the price in a given outlet is computed as the logarithm
of sales per unit of product, so that promotions in quantities are
captured in our analysis. (For further details of the data sets, see
Section 1 of the online supplementary materials; Aucremanne
535
and Dhyne 2004; Baudry et al. 2007). Given the monthly frequency of our data sets, the price effects of short-lived promotions will not be adequately captured in our analysis compared
with what would be possible with scanner data used in some
studies (see, e.g., Nakamura 2008). But it is perhaps important
to note that CPI datasets are less contaminated by short-term
strategic price changes used by outlets. For example, Baudry
et al. (2007) showed that the proportion of price changes associated with sales or promotions in the CPI data is quite low
in France compared with results reported for the United States.
Furthermore, because one of our goals is to extract the common component of the individual price trajectories, temporary
price changes associated with promotions and strategic pricing
would not provide additional information and would only increase the idiosyncratic component of price changes. Thus, this
may be considered to provide a lower bound for the impact of
idiosyncratic shocks on price changes. There are other advantages to using CPI based micro data sets. First, CPI datasets
have a much wider coverage than the scanner data sets, both
in terms of products (from energy products to services through
perishable and nonperishable food and durable and nondurable
manufactured goods) and in terms of outlet types and chains
(e.g., large and small supermarkets from different chains, corner shops, department stores, service outlets). Second, despite
the fact that the period covered was restricted to the intersection of the two databases (i.e., July 1994–February 2003), it
covers 10 years of monthly observations. In contrast, the scanner data set used by Nakamura (2008), for example, covers only
12 months in 2004. Having panels of price data on reasonably
homogeneous product categories that cover relatively long periods is important for consistent estimation of the common versus the idiosyncratic components of price movements. The CPI
data sets allow us to group the price series into narrowly defined product categories with a sufficient number of price series
in each group. The number of price series for each product is
typically large, often exceeding 200. There are 368 such product categories for Belgium and 305 for France. However, because the computations of the various nonlinear estimation procedures is quite time-consuming, we conducted the estimation
on a subset of randomly selected product categories, restricting
ourselves to price trajectories at least 20 months long. As a result, we ended up estimating our baseline model for 94 product
categories in Belgium and 88 product categories in France.
4.1 Simulated and Realized Frequency
of Price Changes
For each of the 182 products, we estimated the (S, s) model,
(6), with a stochastic band of inaction by the full ML method
described in Section 3.4. To allow for possible differences in
prices between supermarkets and corner shops, xit is chosen to
be a dummy variable that takes the value of 1 if the product is
sold in a supermarket and 0 otherwise. For each product, the unobserved common components, ft , for t = 1, 2, . . . , T; the mean
adjustment threshold, s; its standard deviation, σs ; the volatility
of the idiosyncratic component, σε ; and the volatility of outlet
specific random effects, σv , were estimated simultaneously.
To evaluate the model’s goodness of fit, we simulated price
trajectories for all products, using the estimated model parameters (details of this simulation exercise are provided in Section 5
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536
Journal of Business & Economic Statistics, October 2011
Figure 1. Observed and simulated frequencies of price changes. The online version of this figure is in color.
of the online supplementary materials). The scatterplots of the
realized and simulated frequencies for the 94 product categories
in the Belgian CPI and the 88 product categories for the French
CPI are presented in Figure 1.
The complete set of results by individual product is provided in Tables 5 and 6 of the online supplementary materials, along with a number of summary statistics, including the
average number of price trajectories per month, the correlation
coefficient of fˆt with the corresponding product category price
index, and the frequency and average size (in absolute terms) of
price changes. As can be seen from Figure 1, except for a small
number of products (8 out of the 94 for the Belgian CPI and 2
out of 88 for the French CPI), the simulated frequency of price
changes matches the observed ones quite closely. The exceptions tended to be products with relatively rigid prices. These
products are “dining room oak furniture,” “cup and saucer,”
“parking spot in a garage,” “fabric for dress,” “wallet,” “small
anorak,” “men’s T shirt,” and “hair spray 400 ml” in Belgium,
and “classic lunch in a restaurant” and “pasta” in France. For
these 10 products, our simulations overestimate the frequency
and underestimate the average size of price changes. In what
follows, we exclude these products and focus on the remaining
172 products that seem to fit the observed price changes reasonably well.
4.2 Characteristics of Price Changes
by Product Categories
The main statistics regarding the price changes observed for
these 172 products, grouped into six broad categories (energy,
perishable food, nonperishable food, nondurable manufactured
goods, durable manufactured goods, and services) are provided
in Table 1. These statistics show that the patterns of consumer
price changes are essentially similar in Belgium and in France,
and are in line with the previous empirical evidence regarding
price changes in the Euro area (see, e.g., Dhyne et al. 2006).
Energy product prices are changed very frequently but by small
amounts, whereas services exhibit small but quite infrequent
Table 1. Descriptive statistics by broad product categories—CPI weighted averages
Freq
|p|
% small p
No. of products
Freq
|p|
% small p
No. of products
Energy
Perishable
food
0.723
0.039
31%
0.315
0.139
34%
3
23
0.799
0.022
36%
0.247
0.119
50%
2
13
Nonperishable
food
Belgium
0.127
0.102
32%
12
France
0.204
0.064
47%
11
Nondurable
goods
Durable
goods
Services
0.145
0.083
33%
0.056
0.072
38%
0.041
0.056
36%
15
18
15
0.124
0.166
41%
0.134
0.083
44%
0.077
0.047
43%
31
13
16
NOTE: Freq is the observed frequency of price changes. |p| is the observed average absolute value of price changes. % small
p is, following Midrigan (2011), the fraction of price changes of magnitude less than half of the average price change (in
absolute value) in the product category.
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Dhyne et al.: Lumpy Price Adjustments
537
price changes. Somewhere in between, the frequency and magnitude of price changes for food products are both quite high,
whereas those for other manufactured goods are of lower frequency and magnitude. Clearly, there is a significant degree of
heterogeneity in the price setting behavior across these products.
It is also interesting that if we exclude energy products, then
the ranking of the product categories by the frequency of price
changes is the same as that by the average size of these price
changes. All product categories also display a significant fraction of small price changes. It is difficult to explain both of
these features with a standard (S, s) model in which the band
of price inaction is fixed across outlets and products. Thus it is
reasonable to expect that our more general specification of the
(S, s) model, in which the band of inaction is allowed to vary
across outlets and over time, could better fit the wide variety of
outcomes observed across different products. This conjecture
is supported by the additional empirical evidence provided in
Section 6 of the online supplementary materials on the ability
of alternative state-dependent pricing models to generate small
price changes.
4.3 Parameter Estimates by Product Categories
A summary of the ML estimates of the main parameters of
interest is given in Table 2. A full set of estimates by individual commodities is provided by Tables 5 and 6 of the online
supplementary materials. To compute σˆ ω2 , the estimated variance of common shocks, we assume that ft follows a general
autoregressive process possibly with a linear trend. Therefore,
for each product category, we use the estimates fˆ1 , fˆ2 , . . . , fˆT to
fit an AR(K) model, defined as
fˆt = β0 + β1 t +
K
k=1
ρk fˆt−k + ωt ,
ωt ∽ iid (0, σω2 ),
where for each product category, K is selected using the Akaike
information criterion applied to autoregressions with the maximum value of K set to 12.
Not surprisingly, given the proximity of the price change
characteristics in the two countries, the parameter estimates
show similar qualitative patterns in France and Belgium. Infrequent and small price changes in the case of services are associated with large estimates of inaction bands and relatively
low estimates of shock volatilities. Indeed, wages are the most
important cost component for the production of services, and
their variations tend to be rather infrequent and limited (see,
e.g., Heckel, Le Bihan, and Montornès 2008). This explains in
part why, despite the relatively large inaction band estimates
obtained for services, service prices change by rather limited
amounts; the magnitude of the variations in the underlying costs
is indeed quite small.
Now consider the estimates for the energy prices, which tend
to exhibit opposite characteristics to those of service prices. The
estimated thresholds appear to be quite small. Moreover, the
estimates of the variances of the shocks, although quite small in
magnitude, are larger than the estimated mean thresholds, thus
explaining the observed high frequency and small magnitude of
price changes. Taken together, these results imply that energy
prices are flexible.
Regarding the other categories of goods, it can be seen that
the higher frequency of price changes for food products compared with manufactured products seems to stem from smaller
inaction bands rather than from larger shocks. Also noteworthy is the strong link between the mean inaction band, s, and
its variability, σs ; the correlation between the estimates of these
two parameters is 0.95 at the product level. Gautier and Le Bihan (2011) explained why these two parameters are strongly
positively linked. To explain the coexistence of small and large
price changes, both parameters must take large values; that is,
the variance of s must increase with s to allow the same proportion of small price changes when price changes are larger on
average.
Overall, the larger the s¯ˆ (the weighted average estimate of
the mean of the inaction band), the smaller the frequency of
price changes. But the magnitude of shocks also plays a role in
explaining these low frequencies.
Table 2. Parameter estimates by broad product categories—CPI weighted averages
Energy
Perishable
food
sˆ¯
σˆ¯ s
σˆ¯ ε
σˆ¯ ω
σˆ¯ v
0.013
0.004
0.020
0.032
0.029
0.219
0.118
0.108
0.036
0.140
s¯ˆ
σˆ¯ s
σˆ¯ ε
σˆ¯ ω
σˆ¯ v
0.004
0.003
0.023
0.017
0.135
0.215
0.148
0.106
0.015
0.260
Nonperishable
food
Nondurable
goods
Durable
goods
Services
Belgium
0.304
0.170
0.080
0.016
0.199
0.367
0.182
0.076
0.018
0.326
0.522
0.254
0.074
0.016
0.211
0.378
0.171
0.046
0.009
0.152
France
0.203
0.135
0.074
0.063
0.233
0.396
0.206
0.104
0.037
0.416
0.304
0.172
0.074
0.028
0.366
0.308
0.153
0.053
0.015
0.213
NOTE: s¯ˆ is the average estimate of the mean of the price inaction band. σˆ¯ s is the average estimate of the standard deviation
of the price inaction band. σˆ¯ ε is the average estimated standard deviation of the idiosyncratic component. σˆ¯ ω is the average
estimated standard deviation of the common shock. σˆ¯
ISSN: 0735-0015 (Print) 1537-2707 (Online) Journal homepage: http://www.tandfonline.com/loi/ubes20
Lumpy Price Adjustments: A Microeconometric
Analysis
Emmanuel Dhyne, Catherine Fuss, M. Hashem Pesaran & Patrick Sevestre
To cite this article: Emmanuel Dhyne, Catherine Fuss, M. Hashem Pesaran & Patrick Sevestre
(2011) Lumpy Price Adjustments: A Microeconometric Analysis, Journal of Business & Economic
Statistics, 29:4, 529-540, DOI: 10.1198/jbes.2011.09066
To link to this article: http://dx.doi.org/10.1198/jbes.2011.09066
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Date: 11 January 2016, At: 23:20
Supplementary materials for this article are available online. Please click the JBES link at http://pubs.amstat.org.
Lumpy Price Adjustments:
A Microeconometric Analysis
Emmanuel D HYNE
National Bank of Belgium, Boulevard de Berlaimont 14, 1000 Brussels, Belgium, and Warocqué School of
Business and Economics, Université de Mons, Place du Parc 20, 7000 Mons, Belgium ([email protected])
Catherine F USS
National Bank of Belgium, Boulevard de Berlaimont 14, 1000 Brussels, Belgium
M. Hashem P ESARAN
Downloaded by [Universitas Maritim Raja Ali Haji] at 23:20 11 January 2016
Faculty of Economics, Cambridge University, Sidgwick Avenue, Cambridge, CB3 9DD, United Kingdom, and
Department of Economics, University of Southern California, 3620 South Vermont Avenue, Los Angeles, CA 90089
Patrick S EVESTRE
Paris School of Economics, Université Paris 1-Panthéon Sorbonne, 106-112 boulevard de l’Hôpital, 75013 Paris,
France, and Banque de France, 31 rue Croix des Petits Champs, 75001 Paris, France
Based on a reduced-form state-dependent pricing model with random thresholds, we specify and estimate
a nonlinear panel data model with an unobserved factor representing the common cost or demand components of the unobserved optimal price. Using this model, we are able to assess the relative importance of
common and idiosyncratic shocks in explaining the frequency and magnitude of price changes in the case
of a wide variety of consumer products in Belgium and France. We find that the mean level and variability of the random thresholds are key for explaining differences across products in the frequency of price
changes. We also find that the idiosyncratic shocks represent the most important driver of the magnitude
of price changes. Supplementary materials for this article are available online.
KEY WORDS: Idiosyncratic shock; Micro nonlinear panel; Sticky prices.
1. INTRODUCTION
Following the contributions of Cecchetti (1986) on newspaper prices, Kashyap (1995) on catalog prices (both using U.S.
data), and Lach and Tsiddon (1992) on meat and wine prices
in Israel, a recent wave of empirical research has provided
new evidence on the nature and sources of consumer and producer price stickiness at the micro level. These studies include
those of Bils and Klenow (2004), Klenow and Kryvstov (2008),
Nakamura (2008), and Nakamura and Steinsson (2008), who
analyzed U.S. consumer prices, and Dhyne et al. (2006), who
provided a synthesis of recent empirical analyses carried out
for the Euro area countries. Studies of producer prices include
those of Vermeulen et al. (2007), Cornille and Dossche (2008),
Loupias and Sevestre (2010), among others.
These previous works have pointed out several important features of price changes at the outlet/firm level. In particular, at
the microeconomic level, price changes tend to be infrequent
and not synchronized. The average size of the individual price
changes is also much larger than the overall inflation rate. As argued by Golosov and Lucas (2007) and Klenow and Kryvtsov
(2008), large idiosyncratic shocks are required to explain these
facts. Using U.S. price data, Hosken and Reiffen (2004) and
Nakamura (2008) have indeed shown that much of the observed
price changes can be attributed to idiosyncratic shocks at the
level of the outlet rather than to common shocks, which they assume to be well approximated by changes in wholesale prices.
The purpose of this article is to provide a further assessment
of the respective contributions of idiosyncratic and common
shocks to costs and/or outlets’ margin for the observed changes
in consumer prices. The article extends the results in the literature in several respects. First, we consider two CPI-based
data sets from Belgium and France that cover a wide range of
not only consumer goods, but also services, observed over a
relatively long time period. The two datasets combined cover
more than 180 different goods and services of all kinds observed at a monthly frequency from July 1994 to December
2003, and include energy, perishable food, nonperishable food,
nondurables, durables, and services. In addition, compared with
articles based on scanner data (e.g., Nakamura 2008), the price
reports available in the CPI datasets are less contaminated by
strategic price changes of large supermarkets (the relative advantages of CPI price data over the scanner data are analyzed
in Section 1.3 of the online supplementary materials). Second, using only price information, we propose a methodology based on an (S, s) model of sticky prices that allows us
to identify and evaluate the common and idiosyncratic components of the unobserved optimal price together with the parameters that characterize the adjustment of prices to their optimal
level. In this respect, we extend previous work by Ratfai (2006)
on meat product prices in Hungary; Bacon (1991), Borenstein,
Cameron, and Gilbert (1997), and Davis and Hamilton (2004)
for gasoline prices; Dutta, Bergen, and Levy (2002) for orange
529
© 2011 American Statistical Association
Journal of Business & Economic Statistics
October 2011, Vol. 29, No. 4
DOI: 10.1198/jbes.2011.09066
Downloaded by [Universitas Maritim Raja Ali Haji] at 23:20 11 January 2016
530
juice; and Fougère, Gautier, and Le Bihan (2010) for restaurants, which considered only common cost shocks affecting the
raw material prices, wholesale prices, or wage costs. Third, our
methodology extends the work of Hosken and Reiffen (2004)
and Nakamura (2008) in that our (S, s) model explicitly accounts for the existence of periods of price stability between
two price changes in estimation of the common component of
price changes. Thus the (S, s) model that we consider represents
a nonlinear extension of the factor models used extensively in
the empirical finance and macroeconomic literature (e.g., Connor and Korajczyk 1986, 1988; Stock and Watson 1998, 2002;
Forni et al. 2000; Bai and Ng 2002, 2006). Fourth, our (S, s)
model and the proposed estimation procedures also allow for
random thresholds, which, as shown by Caballero and Engle
(1999), Dotsey, King, and Wolman (1999), and Costain and
Nakov (2011), may help explain the existence of both large and
small price changes.
Our main findings may be summarised as follows. The estimation results confirm the existence of a pervasive degree of
heterogeneity in the price behavior across products and outlets.
However, once we control for the product type, the behavior of
consumer prices in Belgium and France seems to be quite similar. The frequency of price changes appears to depend more
strongly on the characteristics of the price inaction band (i.e.,
its average width and its variance) than on the variability of
shocks to the optimal price. However, the latter are by far the
most important drivers of the magnitude of price changes, with
the idiosyncratic shocks playing a major role in this respect. Explaining small price changes appears to be more difficult. The
(S, s) model advanced in this article, which allows for stochastic inaction bands, is an improvement over (S, s) models with
a fixed band of inaction, asymmetric inaction bands, or bands
with seasonal adjustments in replicating the occurrence of small
price changes, but it does not fully succeed in predicting very
small price changes observed for some products.
The rest of the article is structured as follows. Section 2 sets
out the general (S, s) model of prices. Section 3 describes the
estimation procedures. Section 4 focuses on the presentation
and discussion of the estimation results, and Section 5 concludes.
2. AN (S, s) MODEL OF STICKY PRICES WHEN
ONLY PRICES ARE OBSERVED
It is now a well-established stylized fact that most consumer
prices remain unchanged for periods of up to several months
(see, e.g., Bils and Klenow 2004; Dhyne et al. 2006; Nakamura
and Steinsson 2008). Physical menu costs, fear of customer
anger, and implicit or explicit contracts are among the many
sources of price rigidity posited in the literature, which could
explain why retailers might not be willing to immediately adjust their prices to changes in their market conditions, such as
changes in wholesale prices, costs of distribution, or changes
in demand or local competition (see, e.g., Blinder et al. 1998).
Essentially three approaches have been proposed for modeling price stickiness. In time-dependent models, the probability of changing prices does not depend on the evolution of the
outlets’ economic environment (see Calvo 1983 for a prominent example of this). In state-dependent models advanced by,
Journal of Business & Economic Statistics, October 2011
among others, Sheshinski and Weiss (1977) and Dotsey, King,
and Wolman (1999), the probability of changing prices depends
on changes in the outlets’ economic environment. Following
Sims (2003) and Mankiw and Reis (2006), a third group of
models has recently emerged that explain price stickiness by
the costs of information acquisition and/or by the noise that
can affect the information collected by firms about their environment (e.g., see Eichenbaum and Fisher 2007; Klenow and
Willis 2007; Woodford 2009).
Whatever approach is adopted, assessing how consumer
prices react to changes in the outlets’ economic environment
and what model fits best to the “stylized facts” remains a largely
open issue. One of the main reasons for this is that we do not
have a fully satisfactory statistical measurement of the unobserved optimal prices targeted by outlets for the products they
sell. This problem has been addressed in the literature in various
ways. Some studies have considered how individual prices react
to general measures of inflation, considered at the national or
the industry level (see, e.g., Cecchetti 1986; Lach and Tsiddon
1992; Fougère, Le Bihan, and Sevestre 2007; Gagnon 2009).
Others have explicitly studied the link between individual price
changes and costs measured by wage costs or by wholesale
price variations; however, this has been carried out most often for a specific product or group of products (e.g., gasoline
in Peltzman 2000 and Davis and Hamilton 2004; orange juice
in Dutta, Bergen, and Levy 2002; meat in Ratfai 2006; grocery
products in Nakamura 2008; restaurants in Fougère, Gautier,
and Le Bihan 2010).
In this article we adopt a different approach to the identification and estimation of the optimal price and propose a statistical decomposition of changes in retail prices into common and
idiosyncratic shocks using a nonlinear factor model that explicitly allows for periods of no price changes. More specifically,
we consider the following decomposition of the (unobserved)
optimal log price, p∗jit , of outlet i for its product j at time t
p∗jit = x′jit β + fjt + vji + εjit ,
j = 1, 2, . . . , M, i = 1, 2, . . . , N, t = 1, 2, . . . , T,
(1)
where xjit is a vector of observable product and retail-specific
variables with the coefficients, β, and fjt represents the unobserved common cost or demand component of p∗jit at time t,
which is assumed to be the same across all outlets, i, for a given
product j. The remaining terms in (1) are intended to capture the
product and retail-specific, vji , or purely random differences,
εjit , in optimal prices across the outlets.
The elements of xjit measure the observed characteristics of
the product/outlet that might explain price-level differences of a
particular product across outlets, such as whether the product is
offered as part of sales promotion, outlet-specific features (e.g.,
hyper or supermarket vs. corner shop), and geographic location (city center vs. suburb). The elements of xjit could be timevarying, as in the case of sales promotion, or time-invariant, as
in the case of store type. The nature of the outlet (supermarket
or corner shop) is particularly important, because for a similar product, average prices tend to be lower in supermarkets
than in corner shops. The second component, fjt , is the common component of prices of a given product j, across outlets;
it is period-specific and is shared across all outlets selling a
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Dhyne et al.: Lumpy Price Adjustments
531
given fairly homogeneous product. From an economic standpoint, this component reflects the average marginal cost augmented with the average desired markup associated with this
particular product. From an econometric standpoint, we model
this as an unobserved common factor that may be estimated by
aggregating the nonlinear pricing equations across the outlets.
In this respect, we go one step further than Hosken and Reiffen
(2004) and Nakamura (2008), who estimated this component
by averaging prices of a given product across outlets. Indeed,
as we show later, we explicitly account for periods of price inaction in the estimation of this common component. The third
component of p∗jit , vji , is an unobserved outlet-specific effect for
a given product j, which accounts for price differences due to
product differentiation, local competition conditions, and other
factors. The fourth component of the optimal price, εjit , is an
idiosyncratic term reflecting shocks that might affect the outletspecific optimal price in a given period (possibly due to outletspecific demand shocks or unexpected changes in costs at the
store level).
This statistical decomposition does not match the usual decomposition of the optimal price into a cost component and
a markup component. However, for each product j, it allows
estimation of the respective variances of aggregate (fjt ) and
idiosyncratic (εjit ) shocks and thus allows an assessment of
their respective impact on the frequency and magnitude of price
changes.
To link the unobserved optimal price components to observed
prices, a suitable price-setting decision rule that can explain infrequent but possibly large price changes is needed. One possibility is to assume the existence of a fixed price adjustment cost,
leading to an optimal price strategy of the (S, s) variety (see,
e.g., Sheshinski and Weiss 1977, 1983; Cecchetti 1986; Dixit
1991; Hansen 1999; Gertler and Leahy 2006). Indeed, several
previous studies have found evidence of fixed physical menu
costs of price adjustment (Levy et al. 1997; Blinder et al. 1998;
Zbaracki et al. 2004), although Zbaracki et al. (2004) argued
that in addition to these fixed physical menu costs, managerial and customer-related costs are convex in the price change.
A simple specification of a (S, s) model representing the pricing
rule followed by outlet i for its product j, can be written as
pji,t−1 if |p∗jit − pji,t−1 | ≤ sj ,
(2)
pjit =
if |p∗jit − pji,t−1 | > sj ,
p∗jit
where pjit is the (log) observed price of a product j in outlet i
at time t, p∗jit is the (log) optimal price as defined by (1), and sj
denotes the thresholds beyond which outlets find it profitable to
adjust their prices in response to a shock. This specification assumes that the pricing thresholds for price increases and price
decreases are equal on average and that there is no additional
downward price rigidity. In what follows, to simplify notation,
we drop the subscript j and refer to s as the “adjustment threshold” or “band of inaction.” We refer to
|p∗it − pi,t−1 | ≥ s,
(3)
as the “price change trigger” condition.
Assuming a common, time-invariant adjustment threshold
across all outlets might be considered too restrictive, because price setting may be strongly heterogeneous across outlets, even for relatively homogeneous product categories (see,
e.g., Aucremanne and Dhyne 2004; Fougère, Le Bihan, and
Sevestre 2007). At the outlet level, some price trajectories are
characterized by very frequent price changes, whereas others
are characterized by infrequent price changes. Moreover, as described by Campbell and Eden (2005), some price trajectories
at the micro level exhibit long periods of price stability followed
by periods of frenetic price changes. As noted by Caballero and
Engel (2007), this pattern of price changes suggests that the
range of price inaction is best modeled as a stochastic process.
Another argument for adopting such an approach lies in the evidence of small price changes (Klenow and Kryvstov 2008).
Thus we extend model (2) to allow for (random) time- and
outlet-varying pricing thresholds, considering the representation
pi,t−1 if |p∗it − pi,t−1 | ≤ sit ,
(4)
pit = ∗
if |p∗it − pi,t−1 | > sit ,
pit
and assume that sit are random draws from a common distribution. Other specifications of sit are considered and compared in
Section 6 of the online supplementary materials. Let I(A) denote an indicator function that takes the value of unity if A > 0
and 0 otherwise. Then model (4) can be written as
pit = pi,t−1 + (p∗it − pi,t−1 )I(p∗it − pi,t−1 − sit )
+ (p∗it − pi,t−1 )I(pi,t−1 − p∗it − sit ).
(5)
This specification is closely related to the models considered
by Rosett (1959) for the analysis of frictions in yield changes
and, more recently, by Tsiddon (1993), Willis (2006), Ratfai
(2006), and Fougère, Gautier, and Le Bihan (2010) in the sticky
price literature. But it departs from those models in several important respects. First, instead of using a producer price index
to proxy the common movements in consumer price trajectories
as done by Ratfai (2006) and Fougère, Gautier, and Le Bihan
(2010), we rely on an unobserved common component. This
allows us to conduct our analysis for products for which there
is no directly observable or easily identified explanatory variables. One important advantage of proceeding in this manner
is to ensure the coherency of this common component with the
underlying dynamics of micro price decisions as stated by our
model. Further, it avoids the drawback that if the observed variable fails to capture the common factor, then part of the common variation will show up in the error term.
Second, we also depart from the existing empirical literature
in the information used in our estimation procedure. Most of
the literature estimates state-dependent pricing models using binary response or duration models (Cecchetti 1986; Aucremanne
and Dhyne 2005; Campbell and Eden 2005; Ratfai 2006; Willis
2006; Fougère, Le Bihan, and Sevestre 2007) and thus neglects
the information contained in the magnitude of price changes.
However, this information is crucial for identifying the volatility of the idiosyncratic component and for disentangling the idiosyncratic component of the optimal price from the idiosyncratic threshold variable, sit . In a binary response model, where
only the occurrence of price changes is considered, the effects
of εit and sit cannot be identified separately. In our framework,
the observations on the price level, pit , allows the identification
of the idiosyncratic component, εit , from the optimal price, p∗it ,
because p∗it = pit when a price change occurs.
532
Journal of Business & Economic Statistics, October 2011
Third, our approach does not impose any restrictions on the
dynamics of the common factors, and it allows for possible
structural breaks in ft . In principle, it is also possible to allow
for the idiosyncratic shocks, εit , to be serially correlated. But to
simplify the exposition and for ease of estimation, in what follows we assume that εit are serially uncorrelated. The case of
serially correlated errors is considered in Section 4 of the online
supplementary materials, which uses Monte Carlo experiments
to show that neglecting (positive) serial correlation in the idiosyncratic shocks tends to result in overestimation of the band
of inaction. However, the bias is small for reasonable values of
the serial correlation coefficient.
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3. ALTERNATIVE ESTIMATION METHODS
Equations (1) and (5) can be combined to obtain the following econometric specification:
pit − pi,t−1 = (ft + x′it β + vi + εit − pi,t−1 )
× I(ft + x′it β + vi + εit − pi,t−1 − sit )
+ (ft + x′it β + vi + εit − pi,t−1 )
× I(pi,t−1 − ft − x′it β − vi − εit − sit ). (6)
There are essentially two groups of parameters to be estimated:
the unobserved common components, ft , which can also be
viewed as unobserved time effects, and the parameters that do
not vary over time, namely s and σs , which denote the mean
and standard deviation of sit ; σε , the standard deviation of the
idiosyncratic component εit ; σv , the standard deviation of the
firm-specific random effect, vi ; and β, the parameters associated with the observed explanatory variables, xit .
The estimation of the baseline model can be carried out in
two ways. One can use an iterative procedure that combines the
estimation of the ft ’s using the cross-sectional dimension of the
data with the maximum likelihood (ML) estimation of the remaining parameters, conditional on the first-stage estimate of
ft . Alternatively, one can use a standard ML procedure, where
the ft ’s are estimated simultaneously with the other parameters.
The two procedures lead to consistent estimates, provided that
N and T are sufficiently large. It is noteworthy that if N is small,
then one will face the well-known incidental parameters problem. The bias in estimating ft due to the limited size of the crosssectional dimension will contaminate the other parameter estimates. In the alternative situation where T happens to be small,
the problem of the initial observation will become an important
issue. Therefore, our estimation procedure is essentially valid
for relatively large N and T. Fortunately, in our context, the
prices of most of the products that we consider have been observed monthly over the period 1994:7–2003:2 (i.e., more than
100 months), and the number of outlets selling these products
also is relatively large, on average close to 300 in both Belgium
and France.
3.1 Estimation of f t Using Cross-Sectional Averages
As mentioned earlier, in practice ft is an unobserved time effect that needs to be estimated along with the other unknown
parameters. It reflects the common component in the optimal
prices for each particular product for which we estimate the
model. Moreover, because we are able to consider precisely defined types of products sold in a particular outlet, it is reasonable to assume that any remaining cross-sectional heterogeneity
in the price level can be modeled through the observable outletspecific characteristics, xit , and through random specific effects
(accounting for outlets unobserved characteristics).
Accordingly, we assume that, conditional on hit = (ft , x′it ,
pi,t−1 )′ , the random variables (sit , vi , εit )′ are distributed independently across i, and that sit and εit are serially uncorrelated.
Because of the nonlinear nature of the pricing process, and to
make the analysis tractable, we also assume that
⎛ ⎞
⎛⎛ ⎞ ⎛ 2
⎞⎞
sit
0
0
s
σs
⎝ vi ⎠ hit ∽ iid N ⎝⎝ 0 ⎠ , ⎝ 0 σv2 0 ⎠⎠ .
0
0
0 σε2
εit
The assumption of zero covariances across the errors is made
for convenience and can be relaxed.
Before discussing the derivation of ft we state the following
lemma (established in Section 3 of the online supplementary
materials), which provides a few results needed later.
Lemma 1. Suppose that y ∽ N(μ, σ 2 ); then
a+μ
a+μ
E[yI(y + a)] = σ φ
+ μ
,
σ
σ
y+a
a+μ
b
E φ
φ √
=√
,
b
b2 + σ 2
b2 + σ 2
a+μ
y+a
= √
,
Ey
b
b2 + σ 2
where φ(·) and (·) are, respectively, the density and the cumulative distribution function of the standard normal variate,
and I(A) is the indicator function defined earlier.
Let dit = ft + x′it β − pi,t−1 , ξit = vi + εit ∽ N(0, σξ2 ), and note
that σξ2 = σv2 + σε2 . Now consider the baseline model, (6), and
using the foregoing, write it as
pit = (dit + ξit )I(dit + ξit − sit ) + (dit + ξit )I(−dit − ξit − sit )
or
pit = (dit + ξit ) + (dit + ξit )[I(dit + ξit − sit ) − I(dit + ξit + sit )].
Denote the unknown parameters of the model by θ =
(s, β ′ , σs2 , σv2 , σε2 )′ , and note that E(pit |hit , θ) = dit + git ,
where git = g1,it + g2,it , with
g1,it = dit E[I(dit + ξit − sit ) − I(dit + ξit + sit )|hit , θ]
and
g2,it = E[ξit I(dit + ξit − sit ) − ξit I(dit + ξit + sit )|hit , θ].
Also, under our assumptions,
2
sit
s
σ
, s
hit ∽ iid N
0
0
ξit
0
σv2 + σε2
.
It is easily seen that
E[I(dit + ξit − sit ) − I(dit + ξit + sit )|hit , θ]
dit + s
dit − s
−
.
=
σs2 + σξ2
σs2 + σξ2
Dhyne et al.: Lumpy Price Adjustments
533
Using the results in Lemma 1 and noting that ξit |hit , θ ∽
N(0, σξ2 ), we have
dit − sit
.
E[ξit I(dit + ξit − sit )|hit , sit , θ ] = σξ φ
σξ
Thus, taking expectations with respect to sit , we have
dit − sit
E[ξit I(dit + ξit − sit )|hit , θ] = σξ E φ
hit , θ .
σξ
Again using the results in Lemma 1, we have
σξ
dit − s
dit − sit
φ
,
E φ
hit , θ =
σξ
σs2 + σξ2
σs2 + σξ2
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and thus
E[ξit I(dit + ξit − sit )|hit , θ] =
Similarly,
E[ξit I(dit + ξit + sit )|hit , θ] =
σξ2
dit − s
φ
.
σs2 + σξ2
σs2 + σξ2
σξ2
σs2 + σξ2
dit + s
φ
.
σs2 + σξ2
Collecting the various results, we obtain
dit − s
dit + s
g1,it = dit
−
σs2 + σξ2
σs2 + σξ2
and
g2,it =
σξ2
dit − s
dit + s
φ
−φ
.
σs2 + σξ2
σs2 + σξ2
σs2 + σξ2
Here g1,it and g2,it are nonlinear functions of ft and depend on i
only through the observable, pi,t−1 and xit . Thus, it is possible
to compute ft for each t in terms of pi,t−1 , xit and θ . Then, following Pesaran (2006), the cross-sectional average estimator of
ft , denoted by f˜t , can be obtained as the solution to the following
nonlinear equation:
(7)
p¯ t = f˜t + x¯ ′t β + g¯ t (f˜t ),
N
N
¯t =
where p¯ t =
¯ t (ft ) =
i=1 wit pit , x
i=1 wit xit , and g
N
a predeteri=1 wit git , and {wit , i = 1, 2, . . . , N} represent
2
mined set of weights such that wit = O(N −1 ) and N
i=1 wit =
−1
O(N ).
For a given value of θ and each t, (7) provides a nonlinear
function in f˜t . This equation clearly shows that unlike the linear
models considered by Pesaran (2006), here the solution to the
common component ft does not reduce to an average of (log)
prices. In particular, f˜t also accounts for the dynamic feature of
the price-setting behavior through the g¯ t component, which depends on pi,t−1 . The Monte Carlo simulations provided in Section 4 of the online supplementary materials show that taking
into account the nonlinear component, g¯ t , substantially reduces
the root mean squared error (RMSE) of estimating ft by f˜t compared with using the linear cross-section approximation given
by p¯ t . As expected, the RMSE of f˜t relative to p¯ t declines as the
frequency of price changes diminishes, namely as price lumpiness increases. Equation (7) has a unique solution as long as
s > 0. A proof of this is provided in Section 2 of the online
supplementary materials. It is also easily seen that under the
cross-sectional independence of vi and εit , g¯ t (ft ) → E(git ) and
p
f˜t − ft → 0, as N → ∞. Note that for the sake of simplicity, here
we assume that the panel data sample is balanced; however, the
result can be easily generalized to unbalanced panels assuming
that Nt → ∞ for each t, where Nt denotes the number of outlets
in period t.
3.2 Conditional Likelihood Estimation Without
Random Effects
In this section we derive the ML estimator (MLE) of the
structural parameters, θ , conditional on ft and assuming that
there are no firm-specific effects, so that σv2 = 0, and thus in this
case θ = (s, β ′ , σs2 , σε2 )′ . Given the distributional assumptions
stated in Section 3.1, and defining ζit as sit − s, our baseline
model can be rewritten as
pit = dit + εit + (dit + εit ){I[dit + εit − ζit − s]
− I[dit + εit + ζit + s]},
where
2
ζit
0
σ
∽ iid N
, s
0
0
εit
0
σε2
for i = 1, 2, . . . , N, t = 1, 2, . . . , T.
Equivalently
pit = dit + εit + (dit + εit ){I[dit − s + ε1it ] − I[dit + s + ε2it ]},
where ε1it = εit − ζit and ε2it = εit + ζit , with
⎛
⎞
⎛⎛ ⎞ ⎛ 2
σε + σs2 σε2 − σs2
ε1it
0
⎝ ε2it ⎠ ∼ iid N ⎝⎝ 0 ⎠ , ⎝ · · ·
σε2 + σs2
0
εit
···
···
Let
τ1it =
1 if pit = 0,
0 otherwise,
τ2it =
1 if pit > 0,
0 otherwise,
τ3it =
1 if pit < 0,
0 otherwise
⎞⎞
σε2
σε2 ⎠⎠ .
σε2
for i = 1, 2, . . . , N and t = 1, 2, . . . , T.
Then, conditional on ft , t = 1, 2, . . . , T, and the initial value pi0 ,
the log-likelihood function of the model for each i can be written as
Li (θ|f) = Pr(pi1 |pi0 ) Pr(pi2 |pi0 , pi1 )
× Pr(pi,T |pi0 , pi1 , . . . , pi,T−1 ) × Pr(pi0 ),
where f = (f1 , f2 , . . . , fT )′ . In view of the model’s first-order
Markovian property, we have
Li (θ |f) = Pr(pi1 |pi0 ) Pr(pi2 |pi1 ) Pr(pi,T |pi,T−1 ) × Pr(pi0 ).
When T is small, the contribution of Pr(pi0 ) could be important.
In what follows, we assume that pi0 is given and T is reasonably
534
Journal of Business & Economic Statistics, October 2011
large so that the contribution of the initial observations to the
log-likelihood function can be ignored.
To derive Pr(pit |pi,t−1 , ft ), we distinguish among cases
where pit = 0, pit > 0, and pit < 0, noting that
Pr(pit = 0|pi,t−1 , ft )
= Pr(ε1it ≤ s − dit ; ε2it ≥ −s − dit )
= Pr(ε1it ≤ s − dit ) − Pr(ε1it ≤ s − dit ; ε2it ≤ −s − dit )
s − dit
=
σε2 + σs2
s − dit
−s − dit σε2 − σs2
− 2
;
;
σε2 + σs2
σε2 + σs2 σε2 + σs2
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= π1it ,
where 2 (x; y; ρ) is the cumulative distribution function of the
standard bivariate normal. Similarly,
Pr(pit > 0|pi,t−1 , ft )
= Pr(εit = pit − dit ) Pr(ε1it ≥ s − dit ; ε2it > −s − dit |εit )
−s + pit
−s − pit
1
pit − dit
−
= φ
σε
σε
σs
σs
= π2it
3.3 Conditional ML Estimation With Random Effects
Now consider the random-effects specification where p∗it =
ft + x′it β + vi + εit , and note that
Cov(p∗it , p∗it′ |hit , hit′ ) = σv2
for all t and t′ , t
= t′ .
Under this model, the probability of no price change in a given
period, conditional on the previous price, pi,t−1 , will not be independent of episodes of no price changes in the past. Thus we
need to consider the joint probability distribution of successive
unchanged prices. For example, suppose that prices for outlet i
have remained unchanged over the period t and t + 1; then the
relevant joint events of interest are
Ait : {−s − ζit − dit ≤ εit + vi ≤ s + ζit − dit },
Ai,t+1 : {−s − ζi,t+1 − di,t+1 ≤ εi,t+1 + vi ≤ s + ζit − di,t+1 }.
An explicit derivation of the joint distribution of Ait and Ait+1
seems rather difficult. An alternative strategy is to use the conditional independence property of successive price changes and
note that for each i, and conditional on v = (v1 , v2 , . . . , vN )′
and f, the likelihood function will be given by
L(θ, v, f) =
N
T
i=1 t=1
[π1it (vi )]τ1it [π2it (vi )]τ2it [π3it (vi )]τ2it ,
where
and
Pr(pit < 0|pi,t−1 , ft )
= Pr(εit = pit − dit ) Pr(ε1it < s − dit ; ε2it ≤ −s − dit |εit )
−s − pit
−s + pit
pit − dit
1
−
= φ
σε
σε
σs
σs
= π3it .
Thus,
ℓ(θ , f) =
=
N
ln Li (θ , f)
i=1
N
T
[τ1it ln(π1it ) + τ2it ln(π2it ) + τ3it ln(π3it )].
(8)
i=1 t=1
The MLE of θ is given by
θˆ ML (f) = arg max ℓ(θ , f)
θ
and for N and T sufficiently large, we have
√
a
NT(θˆ ML (f) − θ ) ∽ N(0, Vθ ),
where Vθ is the asymptotic variance of the MLE and can be
estimated consistently using the second derivatives of the loglikelihood function.
Remark 1. In the case where ft , t = 1, 2, . . . , T, are estimated, the MLE will continue to be consistent as both N and
T tend to infinity. However, the asymptotic distribution of the
MLE is likely to be subject to the generated regressor problem. The importance of the generated regressor problem in the
present application could be investigated using a bootstrap procedure.
s − vi − dit
π1it (vi , ft ) =
σε2 + σs2
s − vi − dit −s − vi − dit σε2 − σs2
− 2
,
;
; 2
σε2 + σs2
σε2 + σs2 σε + σs2
pit − vi − dit
1
π2it (vi , ft ) = φ
σε
σε
−s + pit
−s − pit
×
−
,
σs
σs
1
pit − vi − dit
π3it (vi , ft ) = φ
σε
σε
−s + pit
−s − pit
−
.
×
σs
σs
The random effects can now be integrated out with respect to
the distribution of vi , assuming vi ∼ N(0, σv2 ), for example, and
then the integrated log-likelihood function, Ev ℓ(θ , v, f)), maximized with respect to θ .
3.4 Full ML Estimation
In the case where N and T are sufficiently large, the incidental parameters problem does not arise, and the effects of the
initial distributions, Pr(pi0 ), on the likelihood function can be
ignored. Then the MLEs of θ and f can be obtained as the solution to the following maximization problem:
(ˆfML , θˆ ML ) = arg max
f,θ
T
N
[τ1it ln(π1it )
t=1 i=1
+ τ2it ln(π2it ) + τ3it ln(π3it )].
(9)
Dhyne et al.: Lumpy Price Adjustments
Note that for a given value of θ the MLE of ft can be obtained
as
fˆt (θ ) = arg max
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ft
N
[τ1it ln(π1it ) + τ2it ln(π2it ) + τ3it ln(π3it )],
i=1
and will be consistent as N → ∞, because, conditional on θ
and ft , the elements in the foregoing sum are independently distributed. Also, for a given estimate of f, the optimization problem defined by (9) will yield a consistent estimate of θ as N
and T → ∞. Iterating between the solutions of the two optimization problems will deliver consistent estimates of θ and
f1 , f2 , . . . , fT , even though the number of incidental parameters,
ft , t = 1, 2, . . . , T, is rising without bounds as T → ∞. This
is analogous to the problem of estimating time and individual
fixed effects in standard linear panel data models. Individual
fixed effects can be consistently estimated from the time dimension, and time effects can be estimated from the cross-section
dimension.
To evaluate the performance of these estimation methods,
Section 4 of the online supplementary materials reports a number of Monte Carlo simulations. We evaluate ML estimation
with and without random effects. These lead to roughly qualitatively similar results. We also report a set of ML results
for alternative values of the parameters and frequency of price
changes. We then perform a set of Monte Carlo simulations
to evaluate the robustness of the model under deviations from
the underlying assumptions. We first examine the small-sample
properties of our estimator. We then consider the case of serially correlated idiosyncratic shocks. Finally, we investigate the
impact of cross-sectional dependence on the estimates of the
model parameters.
The results of these simulations may be summarized as follows. Estimation of the common component is adversely affected only if the cross-section dimension is relatively small.
Ignoring serial correlation of the idiosyncratic component leads
to a positive bias in the estimates of s and σs . However, the bias
becomes substantial only as the serial correlation coefficient of
the idiosyncratic errors approaches unity. For the level of error
serial correlation estimated by Ratfai (2006) for meat (at 0.34),
our simulations suggest that the upward bias in the estimates
of s should be below 8 percent. Finally, as is the case with linear factor models, estimates of the common components are not
adversely affected by the presence of weak cross-sectional dependence in the idiosyncratic shocks.
4. EMPIRICAL RESULTS
The model presented earlier was estimated using individual
consumer price quotes compiled by the Belgian and French
statistical institutes for the computation of their respective consumer price indices. Each dataset contains more than 10 million observations referring to monthly price quotes of individual products sold in a particular outlet. For each product category, the price in a given outlet is computed as the logarithm
of sales per unit of product, so that promotions in quantities are
captured in our analysis. (For further details of the data sets, see
Section 1 of the online supplementary materials; Aucremanne
535
and Dhyne 2004; Baudry et al. 2007). Given the monthly frequency of our data sets, the price effects of short-lived promotions will not be adequately captured in our analysis compared
with what would be possible with scanner data used in some
studies (see, e.g., Nakamura 2008). But it is perhaps important
to note that CPI datasets are less contaminated by short-term
strategic price changes used by outlets. For example, Baudry
et al. (2007) showed that the proportion of price changes associated with sales or promotions in the CPI data is quite low
in France compared with results reported for the United States.
Furthermore, because one of our goals is to extract the common component of the individual price trajectories, temporary
price changes associated with promotions and strategic pricing
would not provide additional information and would only increase the idiosyncratic component of price changes. Thus, this
may be considered to provide a lower bound for the impact of
idiosyncratic shocks on price changes. There are other advantages to using CPI based micro data sets. First, CPI datasets
have a much wider coverage than the scanner data sets, both
in terms of products (from energy products to services through
perishable and nonperishable food and durable and nondurable
manufactured goods) and in terms of outlet types and chains
(e.g., large and small supermarkets from different chains, corner shops, department stores, service outlets). Second, despite
the fact that the period covered was restricted to the intersection of the two databases (i.e., July 1994–February 2003), it
covers 10 years of monthly observations. In contrast, the scanner data set used by Nakamura (2008), for example, covers only
12 months in 2004. Having panels of price data on reasonably
homogeneous product categories that cover relatively long periods is important for consistent estimation of the common versus the idiosyncratic components of price movements. The CPI
data sets allow us to group the price series into narrowly defined product categories with a sufficient number of price series
in each group. The number of price series for each product is
typically large, often exceeding 200. There are 368 such product categories for Belgium and 305 for France. However, because the computations of the various nonlinear estimation procedures is quite time-consuming, we conducted the estimation
on a subset of randomly selected product categories, restricting
ourselves to price trajectories at least 20 months long. As a result, we ended up estimating our baseline model for 94 product
categories in Belgium and 88 product categories in France.
4.1 Simulated and Realized Frequency
of Price Changes
For each of the 182 products, we estimated the (S, s) model,
(6), with a stochastic band of inaction by the full ML method
described in Section 3.4. To allow for possible differences in
prices between supermarkets and corner shops, xit is chosen to
be a dummy variable that takes the value of 1 if the product is
sold in a supermarket and 0 otherwise. For each product, the unobserved common components, ft , for t = 1, 2, . . . , T; the mean
adjustment threshold, s; its standard deviation, σs ; the volatility
of the idiosyncratic component, σε ; and the volatility of outlet
specific random effects, σv , were estimated simultaneously.
To evaluate the model’s goodness of fit, we simulated price
trajectories for all products, using the estimated model parameters (details of this simulation exercise are provided in Section 5
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536
Journal of Business & Economic Statistics, October 2011
Figure 1. Observed and simulated frequencies of price changes. The online version of this figure is in color.
of the online supplementary materials). The scatterplots of the
realized and simulated frequencies for the 94 product categories
in the Belgian CPI and the 88 product categories for the French
CPI are presented in Figure 1.
The complete set of results by individual product is provided in Tables 5 and 6 of the online supplementary materials, along with a number of summary statistics, including the
average number of price trajectories per month, the correlation
coefficient of fˆt with the corresponding product category price
index, and the frequency and average size (in absolute terms) of
price changes. As can be seen from Figure 1, except for a small
number of products (8 out of the 94 for the Belgian CPI and 2
out of 88 for the French CPI), the simulated frequency of price
changes matches the observed ones quite closely. The exceptions tended to be products with relatively rigid prices. These
products are “dining room oak furniture,” “cup and saucer,”
“parking spot in a garage,” “fabric for dress,” “wallet,” “small
anorak,” “men’s T shirt,” and “hair spray 400 ml” in Belgium,
and “classic lunch in a restaurant” and “pasta” in France. For
these 10 products, our simulations overestimate the frequency
and underestimate the average size of price changes. In what
follows, we exclude these products and focus on the remaining
172 products that seem to fit the observed price changes reasonably well.
4.2 Characteristics of Price Changes
by Product Categories
The main statistics regarding the price changes observed for
these 172 products, grouped into six broad categories (energy,
perishable food, nonperishable food, nondurable manufactured
goods, durable manufactured goods, and services) are provided
in Table 1. These statistics show that the patterns of consumer
price changes are essentially similar in Belgium and in France,
and are in line with the previous empirical evidence regarding
price changes in the Euro area (see, e.g., Dhyne et al. 2006).
Energy product prices are changed very frequently but by small
amounts, whereas services exhibit small but quite infrequent
Table 1. Descriptive statistics by broad product categories—CPI weighted averages
Freq
|p|
% small p
No. of products
Freq
|p|
% small p
No. of products
Energy
Perishable
food
0.723
0.039
31%
0.315
0.139
34%
3
23
0.799
0.022
36%
0.247
0.119
50%
2
13
Nonperishable
food
Belgium
0.127
0.102
32%
12
France
0.204
0.064
47%
11
Nondurable
goods
Durable
goods
Services
0.145
0.083
33%
0.056
0.072
38%
0.041
0.056
36%
15
18
15
0.124
0.166
41%
0.134
0.083
44%
0.077
0.047
43%
31
13
16
NOTE: Freq is the observed frequency of price changes. |p| is the observed average absolute value of price changes. % small
p is, following Midrigan (2011), the fraction of price changes of magnitude less than half of the average price change (in
absolute value) in the product category.
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Dhyne et al.: Lumpy Price Adjustments
537
price changes. Somewhere in between, the frequency and magnitude of price changes for food products are both quite high,
whereas those for other manufactured goods are of lower frequency and magnitude. Clearly, there is a significant degree of
heterogeneity in the price setting behavior across these products.
It is also interesting that if we exclude energy products, then
the ranking of the product categories by the frequency of price
changes is the same as that by the average size of these price
changes. All product categories also display a significant fraction of small price changes. It is difficult to explain both of
these features with a standard (S, s) model in which the band
of price inaction is fixed across outlets and products. Thus it is
reasonable to expect that our more general specification of the
(S, s) model, in which the band of inaction is allowed to vary
across outlets and over time, could better fit the wide variety of
outcomes observed across different products. This conjecture
is supported by the additional empirical evidence provided in
Section 6 of the online supplementary materials on the ability
of alternative state-dependent pricing models to generate small
price changes.
4.3 Parameter Estimates by Product Categories
A summary of the ML estimates of the main parameters of
interest is given in Table 2. A full set of estimates by individual commodities is provided by Tables 5 and 6 of the online
supplementary materials. To compute σˆ ω2 , the estimated variance of common shocks, we assume that ft follows a general
autoregressive process possibly with a linear trend. Therefore,
for each product category, we use the estimates fˆ1 , fˆ2 , . . . , fˆT to
fit an AR(K) model, defined as
fˆt = β0 + β1 t +
K
k=1
ρk fˆt−k + ωt ,
ωt ∽ iid (0, σω2 ),
where for each product category, K is selected using the Akaike
information criterion applied to autoregressions with the maximum value of K set to 12.
Not surprisingly, given the proximity of the price change
characteristics in the two countries, the parameter estimates
show similar qualitative patterns in France and Belgium. Infrequent and small price changes in the case of services are associated with large estimates of inaction bands and relatively
low estimates of shock volatilities. Indeed, wages are the most
important cost component for the production of services, and
their variations tend to be rather infrequent and limited (see,
e.g., Heckel, Le Bihan, and Montornès 2008). This explains in
part why, despite the relatively large inaction band estimates
obtained for services, service prices change by rather limited
amounts; the magnitude of the variations in the underlying costs
is indeed quite small.
Now consider the estimates for the energy prices, which tend
to exhibit opposite characteristics to those of service prices. The
estimated thresholds appear to be quite small. Moreover, the
estimates of the variances of the shocks, although quite small in
magnitude, are larger than the estimated mean thresholds, thus
explaining the observed high frequency and small magnitude of
price changes. Taken together, these results imply that energy
prices are flexible.
Regarding the other categories of goods, it can be seen that
the higher frequency of price changes for food products compared with manufactured products seems to stem from smaller
inaction bands rather than from larger shocks. Also noteworthy is the strong link between the mean inaction band, s, and
its variability, σs ; the correlation between the estimates of these
two parameters is 0.95 at the product level. Gautier and Le Bihan (2011) explained why these two parameters are strongly
positively linked. To explain the coexistence of small and large
price changes, both parameters must take large values; that is,
the variance of s must increase with s to allow the same proportion of small price changes when price changes are larger on
average.
Overall, the larger the s¯ˆ (the weighted average estimate of
the mean of the inaction band), the smaller the frequency of
price changes. But the magnitude of shocks also plays a role in
explaining these low frequencies.
Table 2. Parameter estimates by broad product categories—CPI weighted averages
Energy
Perishable
food
sˆ¯
σˆ¯ s
σˆ¯ ε
σˆ¯ ω
σˆ¯ v
0.013
0.004
0.020
0.032
0.029
0.219
0.118
0.108
0.036
0.140
s¯ˆ
σˆ¯ s
σˆ¯ ε
σˆ¯ ω
σˆ¯ v
0.004
0.003
0.023
0.017
0.135
0.215
0.148
0.106
0.015
0.260
Nonperishable
food
Nondurable
goods
Durable
goods
Services
Belgium
0.304
0.170
0.080
0.016
0.199
0.367
0.182
0.076
0.018
0.326
0.522
0.254
0.074
0.016
0.211
0.378
0.171
0.046
0.009
0.152
France
0.203
0.135
0.074
0.063
0.233
0.396
0.206
0.104
0.037
0.416
0.304
0.172
0.074
0.028
0.366
0.308
0.153
0.053
0.015
0.213
NOTE: s¯ˆ is the average estimate of the mean of the price inaction band. σˆ¯ s is the average estimate of the standard deviation
of the price inaction band. σˆ¯ ε is the average estimated standard deviation of the idiosyncratic component. σˆ¯ ω is the average
estimated standard deviation of the common shock. σˆ¯