Using Stationarity Tests in Antitrust Market Definition · Using Stationarity Tests in Antitrust Market Definition ... Massimo Ferrero, Michele Grillo, Mauro La Noce, Pierluigi Sabbatini

Using Stationarity Tests in AntitrustMarket Definition∗

Mario ForniUniversita di Modena and CEPR

June, 2002

Abstract

In this paper it is argued that, if two products or geographic areas belong inthe same market, their relative price must be stationary. Hence stationaritytests like the ADF and the KPSS can be helpful in delineating the relevantmarket for Antitrust purposes, particularly for abuses of dominant positions andagreements between competitors. The proposed procedure is strictly relatedwith cointegration analysis but is simpler and has more general validity. Anapplication to the Italian milk market illustrates the technique.

Key words: Antitrust market definition, merger guidelines, stationarity, Dickey-Fuller test, KPSS test.

JEL classification number: L41

∗Support from the Italian Antitrust Authority (AGCM) is gratefully acknowledged. The paperreflects the views of the author, not necessarily those of the AGCM. The author thanks Fabio MassimoEsposito, Massimo Ferrero, Michele Grillo, Mauro La Noce, Pierluigi Sabbatini for helpful comments.

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1 Introduction

A precise delineation of the market, both in the product and the geographicdimension, is a crucial preliminary step for antitrust analysis. In order to es-

tablish whether a merger will have adverse competitive effects, the antitrustagency should compare the pre- and post-merger concentration levels, which

may depend dramatically on the way in which the market boundaries have beentraced. Similarly, evaluating whether a company has a dominant position or

an agreement between competitors may have anticompetitive effects can only

be done once the market has been exactly delineated. If the market definitionis too large, a firm having a dominant position may look in competition with

several firms, while the opposite mistake can be made when the definition is toonarrow.

From a theoretical point of view, the concept of ‘antitrust market’ has beenaccurately defined in the 1992 Horizontal Merger Guidelines (US Department

of Justice and Federal Trade Commission, 1992). This definition has recentlybeen advocated by the European Commission (European Commission, 1997)

and is an important reference point for the economic debate and the antitrustauthorities of several countries. Unfortunately, the definition of the Merger

Guidelines cannot be easily translated into an econometric procedure, so thatthere is no general consensus among economists about the quantitative methods

which should be used to delineate antitrust markets. This lack of consensus isprobably a major reason why, in practice, antitrust agencies usually define the

relevant market by resorting to common sense and qualitative considerations.

The prevailing view about the most suitable statistical methodology is thatthe authority should decide on the basis of an estimate of the elasticity of the

residual demand function (Scheffman and Spiller, 1987, Baker and Bresnahan,1988, Kamerschen and Kohler 1993; for a review see Office of Fair Trading,

1999). We shall refer to this view as the ‘demand elasticity’ view. According toWerden and Froeb (1993), this is the only methodology which is fully consistent

with the Merger Guidelines. A shortcoming of this methodology is that it isdifficult to implement in general (Stigler and Sherwin, 1985) and impossible

in a large number of relevant antitrust cases, i.e. all cases of anticompetitiveagreements and abuses of dominant position (White 2000).

This lack of operational content is a strong argument in favor of a differ-ent line of research, according to which products should be grouped in a single

market when prices ‘move together’ in some well-defined sense. The basic ideabehind the latter approach, which could be called the ‘price comovement’ ap-

proach, is that a market is characterized by product homogeneity and arbitragewill prevent prices for the same good from moving independently of each other

(Stigler and Sherwin, 1985). Empirically, several kinds of tests have been pro-

posed, based on regression, contemporaneous correlation, Granger causality and

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cointegration (Horowitz, 1981; Stigler and Sherwin, 1985; Uri and Rifkin, 1985;Uri et al., 1985; Slade, 1986). Generally speaking, the methods proposed by

the price comovement literature are relatively easy to apply and data require-ments are minimal, since only the series of prices are needed. However, despite

the theoretical inspiration being essentially the same, the methods proposed inpractice are different and can lead to different results. Moreover, these methods

have been criticized because of their ambiguous relation with the concept of ‘an-titrust market’ (as opposed to ‘economic market’ or ’trade market’) as defined

in the Merger Guidelines (Sheffman and Spiller, 1987, Baker, 1987, Werden andFroeb, 1993).

In this paper, I propose a method which is very much in the spirit of thelatter line of thought but avoids some important criticisms. The method is

conceptually simple and involves only well-known econometric techniques. I

argue that a necessary condition for two products being in the same antitrustmarket is that the log of the price ratio is stationary. Consequently, I propose

to apply stationarity tests to the log of such ratio. If the tests indicate non-stationarity, we can conclude in favor of distinct markets. The proposed method

is closely related with cointegration techniques, but is simpler and has moregeneral validity.

The technique is illustrated by means of an empirical application to themarket of fresh milk in Italy. I use series of prices obtained by elaborating on

weekly data provided by AcNielsen, relative to different kinds of fresh milk indifferent areas during the period March 1999-March 2001. I conclude that the

Italian regions are distinct geographic markets.The paper is organized as follows. Section 2 illustrates the prevailing defini-

tion of Antitrust market. Section 3 describes the difficulties of a direct empiricalimplementation of the merger-guidelines definition. Section 4 explains the pro-

posed stationarity test procedure. Section 5 discusses the relations with the

literature. The empirical illustration is reported in Section 6. The last sectionconcludes.

2 The ‘Hypothetical monopolist’ definition and

the notion of ‘Antitrust market’

According to the 1992 version of the Horizontal Merger Guidelines, jointly pro-duced by the US Department of Justice and the Federal Trade Commission,

[...] a market is defined as a product or group of products and a geographic areain which it is produced or sold such that a hypothetical profit-maximizing firm,not subject to price regulation, that was the only producer or seller of thoseproducts in that area likely would impose at least a ‘small but significant and

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nontransitory’ increase in price, assuming the terms of sale of all other productsare held constant. A relevant market is a group of products and a geographic areawhich is no bigger than necessary to satisfy this test (Federal Trade Commissionand the US Department of Justice, 1992, Section 1.0).

This definition is closely related to the notion of ‘market power’. Accord-ing to the Merger Guidelines, “mergers should not be permitted to create or

enhance market power,” defined as “the ability profitably to maintain pricesabove competitive levels for a significant period of time” (ibidem, Section 0.1).

If contrasting market power is the aim of the Antitrust Agency, it is quite naturalto define the market as the smallest context in which the creation of a significant

amount of market power is possible in principle. Clearly, if the relevant market

were defined too narrowly, in such a way as to exclude very close substitutes,a hypothetic single producer could not raise prices above the competitive level

without losing his customers, so that a merger could not create market power.On the other hand, if the market were defined too largely, the ability of a merger

to enhance market power could be substantially underestimated.The market definition of the Merger Guidelines is also adopted for non-

merger antitrust cases: the recent Antitrust Guidelines for Collaboration amongCompetitors states that “in general, for goods markets affected by a competitor

collaboration, the Agencies approach relevant market definition as described inSection 1 of the Horizontal Merger Guidelines” (Federal Trade Commission and

the US Department of Justice, 2000, Section 3.32 (a)). Moreover, a similardefinition is given by the Office of Fair Trading (OFT), UK (OFT 1998, Section

2.8). In the ‘Commission Notice on the Definition of Relevant Market’, theEuropean Commission is somewhat more vague:

A relevant product market comprises all those products and/or services whichare regarded as interchangeable or substitutable by the consumer, by reason ofthe products’ characteristics, their prices and their intended use (EC 1997, p.2).

However, in the discussion which follows the sentence above there is a refer-ence to the profitability of “an hypothetical small [...], permanent relative price

increase in the products and areas being considered.” (ibidem, p.4).

3 A basic problem with the ‘demand elasticity’

view

While the ‘hypothetical monopolist’ definition is reasonable from a theoretical

point of view, its direct practical implementation involves serious difficulties(Froeb and Werden, 1991, Sherwin, 1993).

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In order to see whether the hypothetical monopolist would rise prices, weshould be able to evaluate profits at different price levels. This requires estima-

tion of the cost function of our hypothetical firm or cartel, which in most casesis impossible. Without the cost function, we cannot be sure that an increase in

price will reduce profits.Luckily, there are cases in which we can be sure that an increase in price raises

profits. This is when it raises revenues (because costs cannot rise). Since weknow that revenues increase if and only if the own-price elasticity of demand is

smaller than one, we can get important indications by estimating such elasticity,which is feasible in principle. However, we should recognize that such estimation

involves technical problems and requires a lot of information. Moreover, weshould keep in mind that we have only a sufficient condition for profitability. If

we find an elasticity greater than one, the result is inconclusive.

But the main problem with the demand elasticity strategy is that demandelasticity depends on price. Typically, we shall find figures smaller than one for

low prices and larger than one for high prices. Hence the question is: whichprice should we look at?

The definition of market adopted in the Guidelines is not explicit on this.But the very notion of market power reported above implies that the reference

point should be the price that would prevail in a competitive situation, whichunfortunately is not observable. This conclusion is explicit in the document by

the UK Agency (OFT 1998, Section 2.8). Moreover, it is indirectly confirmedby the following passage of the Guidelines:

In the above analysis, the Agency will use prevailing prices of the products ofthe merging firms [...] unless premerger circumstances are strongly suggestive ofcoordinated interaction, in which case the Agency will use a price more reflectiveof the competitive price (Federal Trade Commission and the US Department ofJustice, 1992, Section 1.11).

In other words, the competitive price is the correct benchmark in theory,but in practice it can be replaced by the current price, whenever the pre-merger

situation is not too far from perfect competition. The exception is necessary

because at the monopoly price demand cannot be rigid, otherwise the monopolistwould not be profit maximizing. Put another way, a hypothetical monopolist

would not increase price, if the monopoly price is already prevailing. Hence,at the monopoly price, the elasticity test would not make any sense. This

observation has become known in the Antitrust literature as the ‘Cellophanefallacy’. The US Department of Justice claimed that Du Pont had a dominant

position in the cellophane market, while Du Pont claimed that the relevantmarket was much larger (flexible wrapping materials). The Court decided in

favor of Du Pont, not taking into account that other flexible wrapping materials

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were highly competitive with respect to cellophane, precisely because of the highprice of cellophane fixed by Du Pont (Stocking and Mueller, 1955).

Going back to the Guidelines, it seems that, in order to define the market,we should know in advance the competitive price; or, at least, we should be able

to exclude that the market is characterized by a ‘coordinated interaction’. Butevaluating the existing degree of competition could be precisely the reason why

we need a definition of the relevant market! Clearly, there is a logical difficultyhere.

In merger cases, the problem is somewhat less disturbing. After all, thecurrent price is what really matters, because, when evaluating mergers, the

target of the Antitrust Agency is to prevent the creation of a new firm havingthe power to increase prices above the current level (Posner, 1976).

But in many important cases the goal of the Agency is to identify an already

existing monopoly or collusion among competitors. In such cases, the demandelasticity strategy leads to a inescapable circular reasoning. The embarrassment

related to this logical difficulty is transparent in the following passage of theAntitrust Guidelines for collaborations among competitors:

To determine the relevant market, the Agencies [...] consider the likely reac-tion of buyers to a price increase [...] over prevailing price levels. However, whencircumstances strongly suggest that the prevailing price exceeds what likely wouldhave prevailed absent the relevant agreement, the Agencies use a price more re-flective of the price that likely would have prevailed (Federal Trade Commissionand the US Department of Justice, 2000, Section 3.32 (a)).

In such monopoly cases the Merger Guidelines market definition and its‘demand elasticity’ econometric implementation are simply unfeasible (Sherwin

1993, White 2000)and different methods are needed.

4 Stationarity tests and the proposed method-

ology

There is little doubt that, if two products or geographic areas belong tothe same market, their prices will not move indefinitely far from each other in

the long run. The economic intuition is a simple arbitrage argument: if theproducts are very close substitutes, either on the demand or on the supply side,

their prices cannot move too far apart, since either consumers or producers willshift between them in such a way as to eliminate the more expensive one from

the market.

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This consideration can be rendered precise and operational by resorting tothe statistical concept of stationarity.

Time series are said to be covariance stationary, or wide-sense stationary,if their moments up to the second order do not depend on time. Hence, for

instance, the mean must be constant over time. An important property ofstationary series is that they frequently cross their mean line and exhibit a

tendency to revert to it. In this sense, we say that the shocks affecting stationaryseries have only temporary effects. For this reason, the long-run prediction of

the series is equal to the mean.By contrast, a series is said to be difference-stationary (DS) if its first dif-

ference is stationary, but the series itself is not. A property of DS series is thatthey do not have necessarily a constant mean and do not frequently cross any

horizontal line. The variance of a DS series grows with time without limit and

the shocks affecting it are permanent, i.e. their effects last forever. For thesereasons, if the difference between two log prices is DS, the log prices can move

far from each other without limit as time goes on.Figure 4.1 can be useful to get an idea of stationarity by comparing a sta-

tionary series with a non-stationary one. Part a shows a series following thestationary AR(1) xt = 0.5xt−1 + εt, with εt i.i.d standard normal (generated

with the Matlab routine ’randn’). Part b shows an example of a DS process, i.e.the random walk xt = xt−1 + εt.

Figure 4.1: A stationary (a) and a non-stationary series (b)

0 20 40 60 80 100−10

−8

−6

−4

−2

0

2

4

6

8

10

0 20 40 60 80 100−10

−8

−6

−4

−2

0

2

4

6

8

10

a b

Several stationarity tests have been proposed in the literature. Two widely

used ones are the Dickey-Fuller test (Dickey and Fuller, 1979), both in thesimple (DF) and the ‘augmented’ (ADF) version, and the KPSS test, proposed

by Kwiatkowski et al. (1992). A precise description of these tests is given in theAppendix. Here we mention only two features of the tests that are necessary to

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understand the empirical application. First, both tests depend on a parameterwhich must be chosen in advance: we shall write ADF(k) and KPSS(w). Second,

the null hypothesis for the ADF(k) test is non-stationarity, so that rejectionsuggests stationarity; by contrast, the null for the KPSS(w) test is stationarity,

so that rejection signals non-stationarity. The KPSS test is preferable becauseof the null of stationarity, but it can be affected by small-sample distortions (see

the Appendix). Hence, adding the ADF test can be useful.Our proposed procedure is then the following. First, we take the two series

of prices and compute the log of the ratio. Second, we apply both the ADFtest, for different values of k, and the KPSS test, for different values of w. If we

can reject stationarity (with the KPSS test) and cannot reject non-stationarity(with the ADF test) we conclude in favor of distinct markets.

By contrast, if we cannot reject stationarity we do not get a definite conclu-

sion. This is a consequence of the arbitrage argument delineated above, whichis inherently asymmetric: while non-stationarity implies different markets, sta-

tionarity could in principle be observed even if markets are distinct. This wouldhappen for instance when prices themselves are stationary, or are affected only

by common sources of non-stationary variation during the sample period. How-ever, if (log) prices are non-stationary in levels and we know that they have been

affected by several sources of variation, a consistent conclusion of stationarity(i.e. difference-stationarity is rejected and stationarity is not) can be interpreted

as an indication in favor of market singleness.It is worth noting that, in the above formulation of the stationarity tests, we

are allowing for the existence of a non-zero mean for the log of the price ratio.This means that we are not necessarily ruling out the possibility of persistent

and large differences in price levels. This is because the mere existence of suchdifferences does not necessarily imply separate markets, unless products are

perfectly homogeneous. For instance, the half-liter brick of fresh milk probably

belongs in the same market as the one-liter brick, but the price of two half litersis higher than the price of one liter (probably because many singles and small

families simply cannot consume one liter in a few days).Perfect product homogeneity is the exception rather than the rule. In most

cases similar products have small but non-negligible differences as for the quality,the package, the location in which they are available, or simply consumers’

perception driven by advertising. Such differences often generate differences inprice levels. The question is not whether such differences exist, but whether they

are stable over time. If, for a given price ratio (different from 1) the elasticityof substitution is very large, a change in one price induced by whatever shock

will be followed, sooner or later, by a proportional change in the other, so thatwe will observe stationarity of the price ratio.

To conclude this section, let us observe that the concept of ‘long run’ plays

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an important role in the arbitrage argument above. But how long is the long runin the present context? The question is relevant for empirical purposes. Clearly

the long run should be long enough to allow for full deployment of competitionforces, which reasonably takes much less than what we consider long run in

Macroeconomics. In the empirical application below we take a sample period oftwo years, with weekly observations. With no data limitations, a sample period

of three or four years would perhaps be preferable.

5 Relations with the literature

Let us begin this section by observing that the DF test (of order zero) perfectlycoincides with one of the tests originally proposed by Horowitz (1981), i.e. the

test described at p.9, with the null γ = 0 in equation (3), the only differencebeing that the test is applied to the difference of prices, rather than the difference

of the logs. The author however does not seem to be aware of this, since he doesnot present the test as a stationarity test and does not mention the difficulties

related to the non-standard distribution of the t-statistic.The arbitrage argument invoked in the previous section is essentially the

same behind the ‘price comovement’ literature, starting from the just mentioned

work by Horowitz (1981) and the paper by Stigler and Sherwin (1985).However, the formulation and the implementation proposed here have some

noticeable advantages with respect to the price comovement tests proposed sofar, while retaining both simplicity and minimal data requirements.

A first problem with correlation is that it ignores lagged relations. Forinstance, assume that (a) data are weekly, (b) the log price of the first good

is a random walk, so that the first difference∆p1t is serially uncorrelated, and(c) the log price of the second good adjusts with a lag of k weeks, i.e. ∆p2t =

∆p1t−k, k > 0. In this case prices are strictly related, but the contemporaneouscorrelation of ∆p1t and ∆p2t is zero. The problem is that we should not limit

ourselves to analyzing contemporaneous correlations, but consider also whathappens at different leads and lags.1 The stationarity analysis proposed here,

being a long-run analysis, does not suffer from this problem. In the exampleabove, the log price ratio would be p1t−p2t = ∆kp1t, which is stationary for any

k.

Second, the long-run nature of our approach also render it largely immunefrom another major criticism to the use of correlation and other price comove-

ment tests; namely, that they are related with an economic notion of market,rather than the Antitrust notion, defined in the merger guidelines (Werden and

Froeb, 1993). In fact, non-stationarity provides indirect evidence of potential

1A possible solution is to consider Granger causation as in Slade (1986), Uri and Rifkin (1985),Uri et al. (1985) or focus on long-run dynamic correlation as defined in Croux et al. (2001).

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market power. When the price ratio is non-stationary, there are permanent(and possibly large) changes in one price which are not followed by any reaction

of the other price. This means that a hypothetic monopolist producing one ofthe products involved could have permanently raised its price by a significant

amount, without suffering from the competition of the firms producing the other.In other terms, observing prices that move indefinitely far from each other seems

inconsistent with both the economic and the antitrust market notions.Last, both correlation and Granger causation analysis require that prices are

co-stationary either in levels or in first differences. For instance, if correlationchanges with time, sample correlation is not an estimate of any meaningful

population moment. Similarly, as clarified below, cointegration requires co-stationarity of first differences. By contrast, the test proposed here does not

require that the first differences of the two prices be co-stationary. This is

particularly important when, as often happens in practice, prices are not I(1),so that neither correlation nor Granger causation nor cointegration analysis can

be applied.The stationarity tests proposed here are closely related with cointegration

analysis, which has been proposed for market definition by Ardeni (1989) andhas been applied in several works since then. Precisely, the relation is that, if

prices are co-stationary in first differences of the logs, stationarity of the logof the ratio is equivalent to cointegration of the log prices, with cointegrating

vector ( 1 −1 ).Usually, cointegration studies do not test for the latter restriction, i.e. that

the cointegrating vector must be ( 1 −1 ). Walls (1994) is to my knowledgethe first work in which the importance of testing for the cointegrating vector

( 1 −1 ) is clearly stated. Walls’s procedure is in three steps: First, the ADFtest is applied to each price separately, in order to establish that it is I(1);

second, cointegration is tested and, third, the cointegrating vector restriction is

tested.While being conceptually equivalent, the procedure proposed here has some

advantages with respect to this cointegration test strategy. First, it is muchsimpler. Second, the two conditions in steps two and three are tested jointly,

which is conceptually preferable and more efficient. Third, as observed above, itis more general, in that it does not require that the log prices be both I(1). Fi-

nally, being based on the price ratio, it is insensitive to the use of nominal ratherthan deflated data, whereas cointegration tests can lead to different results.

Finally, I must mention that the use of stationarity tests is proposed byLexecon Ltd.,in a memo dated July 2001 (Lexecon 2001). While the ideas in

this note and the present work are similar, the two works have been producedindependently.

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6 An empirical application to the Italian milk

market

In several merger cases during the nineties, the Italian antitrust Authority ar-

gued that the market for fresh milk has regional geographic dimension. Thisdimension was identified by means of qualitative considerations, and particularly

that the product is highly perishable—a feature partly due to Italian law, which

imposes a tight expiry date (four days). Here we use our procedure to test thishypothesis. Part of the present analysis has been used by the Italian Antitrust

Authority in the merger case Granarolo – Centrale del Latte di Vicenza (seeProvvedimento no. 9557, May 24, 2001, Bollettino no. 19, downloadable at the

web site www.agcm.it).

6.1 The data

Figure 6.1: Average milk prices for Italy and some Italian regions

0 20 40 60 80 1002.05

2.1

2.15

2.2

0 20 40 60 80 1001.8

1.9

2

2.1

2.2

2.3

0 20 40 60 80 100

2.05

2.1

2.15

2.2

2.25

0 20 40 60 80 100

2.1

2.15

2.2

2.25

2.3

a b

c d

Italy

Sicily

Campania

Tuscany

Lazio

Emily

Piedmont

We analyze prices obtained by elaborating on the data base Nielsen Scant-

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Table 6.1: Results for the KPSS test

Cal Cam E.R. Laz Lig Lom Mar Pie Pug Sic Tos Tre Ven

Cal — 0.60* 0.49* 0.55* 0.56* 0.53* 0.56* 0.54* 0.67* 0.57* 0.28 0.59* 0.29Cam 0.99* — 0.54* 0.36o 0.44o 0.47* 0.47* 0.45o 0.52* 0.72* 0.46o 0.34 0.58*E.-R. 0.73* 0.90* — 0.58* 0.51* 0.20 0.61* 0.26 0.21 0.68* 0.10 0.67* 0.28Laz 0.90* 0.56* 0.96* — 0.25 0.38o 0.48* 0.29 0.35o 0.68* 0.35o 0.10 0.67*Lig 0.84* 0.70* 0.77* 0.36o — 0.44o 0.09 0.30 0.31 0.67* 0.36o 0.37o 0.61*Lom 0.72* 0.78* 0.26 0.62* 0.69* — 0.25 0.28 0.12 0.64* 0.20 0.51* 0.33Mar 0.90* 0.76* 1.02* 0.77* 0.11 0.37o — 0.14 0.27 0.70* 0.26 0.53* 0.66*Pie 0.76* 0.73* 0.35o 0.47* 0.47* 0.29 0.21 — 0.16 0.64* 0.35o 0.37o 0.39o

Pug 1.05* 0.84* 0.32 0.56* 0.45o 0.14 0.42o 0.17 — 0.69* 0.17 0.42o 0.25Sic 0.83* 1.24* 1.09* 1.14* 1.10* 1.01* 1.16* 1.03* 1.12* — 0.51* 0.69* 0.63*Tos 0.33 0.76* 0.15 0.57* 0.58* 0.28 0.41o 0.50* 0.21 0.77* — 0.40o 0.09Tre 0.94* 0.51* 1.09* 0.11 0.50* 0.85* 0.75* 0.57* 0.66* 1.13* 0.64* — 0.70*Ven 0.39o 0.96* 0.30 1.12* 0.95* 0.46o 0.98* 0.59* 0.35 1.01* 0.14 1.09* —

KPSS(8) in the lower-left triangle and KPSS(16) in the upper-right triangle*: 5% significance level (0.463); o: 10% significance level (0.347)

rak, referring to the one-liter brick of fresh milk of the kind ‘intero normale’,for all brands sold in supermarkets in the regions Calabria, Campania, Emilia-

Romagna, Lazio, Liguria, Lombardia, Marche, Piemonte, Puglia, Sicilia, Toscana,Trentino-Alto Adige and Veneto. We have 105 weekly data points between

March 13, 1999 and March 10, 2001, obtained by dividing total value (in thou-sands of Italian lire) by total quantity (number of bricks) sold during the week.

Sales promotions are excluded from both values and quantities.Figure 6.1 provides a general idea of the behavior of average milk prices

during this period. In part (a) we have the average price for Italy; in part (b) wehave two southern regions, Sicily and Campania; in part (c) two central regions,

Tuscany and Lazio; in part (d) two northern regions, Emilia and Piedmont.Prices are generally increasing; but they do not seem to increase in the same

way everywhere. In Sicily, for instance, there has been a very large increment,

so that milk is now much more expensive than in Campania, whereas it was lessexpensive in March 1999. In Tuscany and Lazio prices were similar in 1999; in

February 2000 there was a big jump in Tuscany and Lazio followed only partiallyand with a very large delay. Finally, the difference between average prices in

Emilia and Piedmont was about 60 lire during 1999 and the first months of 2000,then reduced to 10-20 lire in May and increased again in July 2000 to stabilize

around 120 lire. Below we show that the percentage deviations between theabove couples of prices are non-stationary.

6.2 The tests

Let us come to the tests. For each couple of regions, we computed the difference

of the logs of prices. Then we computed the KPSS test with two Bartlett windowsizes, 8 months and 16 months, and the ADF test of orders 4 and 8. Results are

reported in Tables 6.1 and 6.2.As shown in Table 6.1, stationarity is very often rejected at least at the 10%

significance level. Some results are notable. First, Sicily, which, being an island,

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Table 6.2: Results for the ADF test[10pt] Cal Cam E.R. Laz Lig Lom Mar Pie Pug Sic Tos Tre Ven

Cal — 1.09 2.34 1.97 2.19 2.38 2.01 2.24 1.71 0.76 2.89* 2.27 2.51(6.0)

Cam 1.50 — 0.87 1.42 1.08 1.13 1.17 1.24 1.22 0.38 1.21 1.69 0.85

E.R. 2.34 1.46 — 1.66 1.84 1.67 1.55 1.54 0.76 1.03 1.98 2.27 3.43*(8.0)

Laz 1.97 1.74 1.86 — 2.06 1.80 1.86 1.62 1.70 1.30 1.52 2.39 2.47

Lig 2.17 1.53 2.08 2.00 — 1.54 2.55 1.10 1.94 1.17 1.60 2.54 2.49

Lom 2.56 1.51 2.06 1.66 1.48 — 1.85 2.69o 2.65o 0.86 1.71 2.21 2.38(5.9) (5.2)

Mar 2.09 1.60 1.98 1.85 2.47 1.97 — 2.12 1.52 1.27 1.72 2.48 2.59o

(7.9)

Pie 2.24 1.58 2.25 1.73 1.04 3.26* 2.07 — 2.97* 0.65 1.69 2.30 1.75(4.7) (9.9)

Pug 1.54 1.88 1.20 1.97 2.13 3.01* 1.98 3.01* — 0.27 2.23 2.20 2.04(3.9) (5.5)

Sic 1.07 0.74 1.39 1.52 1.47 1.19 1.45 0.76 0.44 — 1.48 1.34 1.02

Tos 3.22* 1.12 2.15 1.26 1.72 2.15 1.55 2.02 2.53 2.20 — 1.67 2.03(4.4)

Tre 2.11 2.11 2.04 2.84o 2.35 1.95 2.73o 2.22 2.35 1.56 1.49 — 3.07*(4.3) (5.9) (15.6)

Ven 2.29 1.35 3.28* 2.72o 2.31 1.80 2.73o 1.30 1.72 0.96 2.07 2.79o —(4.4) (19.8) (11.4) (17.0)

– ADF(4) in the lower-left triangle and – ADF(8) in the upper-right triangle*: 5% significance level (-2.89); o: 10% significance level (-2.58)

has no neighboring regions, is the only region for which stationarity is stronglyrejected in relation with all other regions for both window sizes.

Second, for several couples of regions we cannot reject stationarity. Obvi-ously, in some cases this happens by mere chance, simply because of the stochas-

tic nature of the experiment and the fact that the number of couples involvedis high. But in some cases the low value of the test is probably due to a relative

openness of the market. For instance, regarding Emilia, we can reject station-

arity for all regions except the neighboring Tuscany, Veneto, and Lombardy.As for Veneto, we cannot reject stationarity with respect to Emilia. Finally,

stationarity cannot be rejected for the neighboring Piedmont and Lombardy.Overall, these results seem quite reasonable.

Table 6.2 reports the ADF test statistics. The table provides essentially aconfirmation of the above results. The null of non-stationarity

is rejected only in a small number of cases even at the 10% level. Manyrejections are specular to the rejections lacking in the previous table.

Where non-stationarity is rejected at least at the 10% level, we report themean lag of the response to the shock implied by the estimated coefficients of

the ADF equations (the number in brackets). The impulse-response functionswere taken in absolute values. The mean lag gives an idea of the time taken by

the price deviation to go back to its average level. This can be useful because,in presence of a stationarity result, it seems reasonable to require that it not

take too much time to go back to equilibrium.

For some couples the mean lag is very large. Lazio-Veneto (about twenty

13

weeks) and Trentino-Veneto (16-17 weeks) are the most striking examples. Verylarge values of the mean lag should induce some caution in interpreting the

stationarity result as evidence in favor of a single market. Our suspicion isconfirmed by the fact that for both the couples above stationarity is rejected by

the KPSS test.

6.3 Summary results

Table 6.3 summarizes results. We say that stationarity (non-stationarity) is

strongly rejected when it is rejected at the 5% level with both the KPSS window

sizes (both orders of the ADF test). Stationarity (non-stationarity) is weaklyrejected when it is rejected at the 10% level with at least one of the KPSS

window sizes (ADF orders), but strong rejection does not hold. In all othercases we say that stationarity (non-stationarity) is not rejected.

Table 6.3: Summary results

H0: Non-stationarity (ADF test)H0:Stationarity strongly weakly not(KPSS test) rejected rejected rejected total

strongly rejected 0 4 33 37weakly rejected 0 0 26 26not rejected 2 4 9 15total 2 8 68 78

As shown in the Table, stationarity is strongly rejected for almost half of thecouples and weakly rejected for another third. It is worth noting that in a few

cases (4) we do not get a definite conclusion in favor of non-stationarity becausenon-stationarity is also rejected (even if in the weak form). Such rejection of both

stationarity and non-stationarity has already been obtained in the literature(see e.g. Kwiatkowski et al. 1992). It shows that the ADF test can add useful

information with respect to the KPSS test, not only by reinforcing, but also by

contradicting the conclusion of non-stationarity.For only 15 cases out of 78 (less than 20%) stationarity is not rejected. In

most of these cases, also non-stationarity cannot be rejected, so that we cannotconclude in favor of stationarity. A consistent conclusion of stationarity is ob-

tained only in 6 cases, for which stationarity is not rejected and non-stationarityis rejected, at least weakly. The percentage of such cases is not far from the fig-

ure that we would have obtained because of the stochastic nature of the series,the null of non-stationarity being true for all the log price ratios.

14

Figure 6.2: Results for each couple of regions

Calabria

Campania

Emilia

Liguria

Lombardia

Veneto

Trentino

Toscana

Sicilia

Puglia

Piemonte

Marche

Cal Cam Emi Li g Lom Mar Pie Pu g Sic Tos Tre Ven

Lazio

Laz

White: Stationarity is strongly rejected, non-stationarity is not rejected; Light gray: Stationarity is weakly rejected, non-stationarity is not rejected; Dark gray: Stationarity is not rejected, non-stationarity is weakly rejected; Black: Stationarity isnot rejected, non-stationarity is strongly rejected; Medium gray: All other cases.

Figure 6.2 reports the summary results for each couple of regions. ClearlySicily forms a separate market, since stationarity is strongly rejected with respect

to all regions. For the other regions we conclude in favor of a separate marketif we can reject stationarity without ambiguity for all the neighboring regions,

at least in the weak sense (white and light-gray cells).With this criterion Calabria, Campania, Puglia, Lazio, Liguria and Marche

represent distinct markets. For the other regions the picture is less clear. How-ever, Lombardy cannot be in the same market with Veneto or Trentino; Tuscany

is separate with respect to Lazio, Marche and Liguria, Piedmont cannot be inthe same market with Liguria; Emilia-Romagna cannot be in the same market

with Liguria and Marche. On the other hand, Piedmont could be linked to Lom-bardy and Veneto could be linked to Emilia. Overall, results seem consistent

with the hypothesis of separate regional markets.

15

6.4 Against a smaller geographic dimension

What about the possibility of a smaller geographic dimension?

Unfortunately, we do not have data for provinces, so that we cannot perform

a more disaggregated analysis. Nonetheless, we have data for the province ofMilan and the rest of Lombardy. Moreover, we can get useful indications by

looking at prices of different brands operating in the same region. For Lom-bardy we have prices for twelve important brands, which are present for the

whole sample period: Lactis, Latte Magenta, Latte Milano, Pavilat, Polenghi,Prealpina, Granarolo, Granarolo S.Giorgio, Centrale di Brescia, Centrale di Mi-

lano, Centrale di Monza, Marche Private. If the market is truly regional, weshould not observe frequent rejection of stationarity between different brands.

On the other hand, since many brands are sold mainly in a single province ora couple of provinces, a frequent rejection of stationarity could be read as an

indication in favor of a smaller geographical dimension.

Figure 6.3: Two examples

0 20 40 60 80 100−0.05

0

0.05

0.1

0.15

0.2

0 20 40 60 80 100−0.1

−0.05

0

0.05

0.1

0 20 40 60 80 1000.55

0.6

0.65

0.7

0.75

0.8

0.85

0 20 40 60 80 1000.65

0.7

0.75

0.8

0.85Sicily and Trentino Milan and the rest of Lombardy

logs

difference of logs difference of logs

logs

16

Let us consider first the log-price ratio of Milan and the rest of Lombardy.Stationarity is not rejected (KPSS(8)=0.118, KPSS(12)=0.093) and non-stationarity

is weakly rejected (ADF(4)=-3.32, ADF(8)=-2.32). The mean lag is 2.8 weekswith the ADF(4) and 4.6 with the ADF(8). These results suggest that Milan

and the rest of Lombardy lie in the same market.Figure 6.3 compares the case of Milan and the rest of Lombardy (right-hand

side) and the case of Sicily and Trentino (left-hand side). These cases can betaken as paradigms for the two opposite outcomes of the stationarity tests and

can then help the reader to get a more concrete idea of the method. In thelatter case it is seen that the two prices move far apart during the sample pe-

riod. The percentage deviation shown in the lower part of the figure follows apositive trend. With reference to the simulated series of Figure 4.1, its behav-

ior resembles the non-stationary series of part (b). By contrast, Milan and the

rest of Lombardy follow similar paths. The graph of the percentage deviationexhibits important differences for some periods, but such differences are tem-

porary. With reference to Figure 4.1, the graph is much more similar to thestationary series of part (a).

Next we test for each one of the above twelve Lombard brands with respect tothe reference brands Latte Milano, Centrale di Milano and Centrale di Monza,

which are diffused in the geographical core of the region. Results are shownin Table 6.4. With respect to Centrale di Monza, we have a weak rejection of

stationarity for Granarolo. With respect to Latte Milano, we get a weak rejec-tion for Lactis, Granarolo and Marche Private; however, also non-stationarity

can be weakly rejected for all these brands, so that rejection is not consistent.With respect to Centrale di Milano, we get a weak, but consistent rejection for

Lactis, Granarolo and Marche Private. These rejections are too few, weak andinconsistent with respect to different reference points and tests to contradict the

hypothesis that Lombardy makes up a single market.

Obviously, in most cases administrative boundaries can at best be an approx-

imation for the economic boundaries which are relevant for antitrust purposes.Probably the Emilian province of Piacenza, which is close to Milan and con-

nected to Milan by a fast motorway, is in the same market with Milan, whilethe Lombard mountains of Sondrio are not. With our data we cannot verify

this conjecture. Moreover, in principle the relevant area should be centered onthe place where the firms involved are located. Hence if the firms are located

in Milan, Lombardy can be the geographic market, but if the firms are locatede.g. in Mantova, the relevant area should include part of Lombardy, Veneto

and Emilia. With these caveats, we interpret our results as suggesting that theregional market size is a reasonable description of reality.

17

Table 6.4: Results for some Lombard brands

stationarity non-stationarityBrands Latte Centrale Centrale Latte Centrale Centrale

Milano Milano Monza Milano Milano MonzaLactis w w n w n nLatte Magenta n n n n n nLatte Milano - n n - s wPavilat n n n w n nPolenghi n n n w n nPrealpina n n n w w nGranarolo w w w w n nGranarolo S.G. n n n s s nCentrale Brescia n n n w n nCentrale Milano n - n s - nCentrale Monza n n - w n -Marche Private w w n w n nn: not rejected; w: weakly rejected; s: strongly rejected.

7 Summary and conclusions

We have argued that if two products or geographic areas belong to the same

market, their relative prices should behave like a stationary time series ratherthan a difference-stationary one. Hence stationarity tests can provide useful

indications for the definition of antitrust markets.This is particularly important when other empirical methods cannot be used,

either because of the ‘cellophane fallacy’ or because the data needed for estima-tion of demand elasticities are not available.

The proposed procedure is in the spirit of the ‘price comovement’ litera-ture following Horowitz(1981) and Stigler and Sherwin (1985), but avoids some

important shortcomings of existing methods. Moreover, our test is related to

cointegration analysis, but has more general validity.To illustrate the procedure, we have studied the Italian market for fresh

milk. Our results indicate that the market size is regional, thus providing anempirical confirmation for the thesis of the Italian Antitrust Authority.

18

Appendix: The Dickey-Fuller and the KPSS tests

The DF test consists in estimating by OLS the equation

∆xt = α + βxt−1 + εt, t = 1, . . . , T, (1)

where xt is the variable of interest and εt is a serially uncorrelated process. Inthe ‘augmented’ version (the ADF test) the lagged differences ∆xt−1, ∆xt−2, . . . , ∆xk

appear as additional explanatory variables on the right-hand side. If the maxi-mum lag is k, we have the ADF test of order k, or ADF(k), so that the DF test

is the ADF test of order zero.The null hypothesis H0 : β = 0 implies that the first difference of the series

is stationary, but the level is not, i.e. xt is DS. The alternative is H1 : β < 0,

which implies that the level xt is already stationary, without differencing. Toperform the test we have to compute the t-statistic for the coefficient β. The only

difference with respect to the usual t-test is that the distribution of the t-statisticis non-standard under the null; the critical values, tabulated by Dickey and

Fuller (1979), are reported in all time-series textbooks. The common practiceis to try the ADF test for different values of k and conclude in favor of non-

stationarity if the null cannot be rejected.The KPSS test is as follows. Let xt be the series of interest, expressed in

deviation from the mean, and St be the cumulated sum x1 + x2 + · · · + xt.Moreover, set

s2(l) = T−1T∑

t=1

x2t + 2T−1

l∑

s=1

w(s, l)T∑

t=1

xtxt−s, (2)

where w(s, l) is an optional weighting function corresponding to the choice

of a spectral window. In the empirical application below we follow Kwiatkowski

et al. (1992) and use a Bartlett window.The KPSS test statistic is

η = T−2∑

S2t /s

2(l). (3)

The asymptotic distribution of this statistic under the null is known and can

be used to compute the critical values.Here the null and the alternative hypotheses are reversed with respect to the

ADF test: the null is stationarity, while the alternative is difference-stationarity.From this point of view, the KPSS test is preferable in our context, because what

we really want to know is whether we can or cannot reject stationarity at a pre-specified significance level. On the other hand, the ADF test can give useful

additional information, particularly because here we have only an asymptoticdistribution and small sample size distortions can be large (see Kwiatkowski et

al. 1992, Section 5).

19

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21