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Economics Committee Newsletter

Cointegration and Antitrust: A Primer

Jonathan L. Rubin, J.D., Ph.D.* American Antitrust Institute

Introduction

On October 8, 2003, Robert F. Engle and It is this last (and most technical) aspect of Clive W . J. Granger were awarded the cointegration which accounts for its Nobel Prize for their research on the influence in the econometric world. statistical analysis of economic . Cointegration methods will inevitably make Both made important contributions on their their way into the statistical analysis of own, but their most influential work by far is antitrust issues and, ultimately, into the contained in a short and elegant paper they courtroom. The purpose of this article is to published together in 1987.1 Their paper introduce the intuition behind cointegration influenced the way perform in the context of antitrust . The almost all . focus will be the multivariate cointegration model pioneered by Johansen.2 Their insight, known as cointegration, has been described as a method of uncovering The Nature of Time Series and the long-run relationships between variables that Problem of Spurious Regression are concealed by the noise of short-term fluctuations. An engineer might look at this Econometric studies relevant to antitrust as disentangling the “signal” from the issues are often concerned with time series, “noise.” An economist could consider it a i.e., a list of n sequential observations, Xt = way of distinguishing between a random {x1, x2, x3, ..., xn} of a particular variable that fluctuation and a correction back to an varies over time. The graph of a typical equilibrium level. A would price series is given in Fig. 1, which shows regard it as a way of doing regression the price of a particular variety and grade of analysis on non-stationary (i.e., lumber over a 22-year period. stochastically trending) variables that gives statistically valid results.

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Figure 1: Real Price of Lumber, 1975-1996

This time series consists of 88 quarterly understating it in the latter part. A strategy observations. The of the sample (i.e., involving waiting a quarter or two for the the average price) is $1,096.75, which is price to revert to the mean would nearly indicated on the graph by the horizontal always fail. Econometricians call this dotted line. What is most obvious about the property “non-stationarity,” and the price in Fig. 1 is its tendency to move from variable in this case is said to be “non- the lower left of the graph to the upper right, stationary.” which is typical in any market in which prices tend to increase over time (the prices An example of a stationary variable would shown are real, that is, they have been be the time series defined as the difference corrected to eliminate the effect of of this price series, telling us to look at the inflation). The significance of this is that time series consisting of the differences of the sample mean summarizes the price quite the prices from one observation to the next. poorly. Except for the periods around 1983 The graph of the lumber prices in or mid-1993, the statement, “The average differences is shown in Fig. 2. price over the sample is $1,096.75” is fairly uninformative, greatly overstating the price in the earlier part of the sample while

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Figure 2: Real Price of Lumber in Differences, 1975-1996

Again, the mean, or average, difference, in “BLUE,” provided that the assumptions this case $6.84, is indicated by the underlying the regression model are horizontal dotted line. While the price in fulfilled. Regression estimates that are not levels in Fig. 1 crossed the mean three times, BLUE are more likely to be excluded under the price in differences crosses the mean 42 the Daubert standard, and regression studies times. Clearly, the difference from any given involving non-stationary data are not BLUE quarter to the next may differ widely from because they do not fulfill the OLS the mean, a quarter or two later the assumptions. Econometric studies that do difference reverts to the average. Such a not take non-stationarity into account are mean-reverting series is said to be flawed, and have little probative value. It “stationary.” has been shown that regressing two non- stationary series leads to false positives, also The distinction between stationary and non- known as “spurious regression.”3 stationary time series is important because Econometric relationships that appear to be these data have dramatically different statistically significant in the presence of statistical properties. Standard regression non-stationarity may not, in fact, have any analysis, also known as “ordinary least meaningful relationship whatsoever. squares,” or “OLS,” is said to give the Best, Linear, Unbiased Estimates, i.e., they are

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Integration and Cointegration Multivariate Cointegration

The technical term for non-stationary time Cointegration theory reaches far beyond series is that they are “integrated.” The explaining, and being able to correct for, cause of such integration can be traced to the spurious regression. It also easily permits a accumulation of random influences on the superior approach to multiple regression variable. The simplest integrated process is modeling which virtually eliminates known as a “.” The random simultaneity bias. Simultaneity bias in walk process is said to be integrated of order regression analysis results when causality one because, like the price series in Fig. 1, if runs not only from the explanatory variable it is differenced once it becomes stationary. to the dependent variable, but “feeds back” More generally, if a variable can be made from the dependent variable to the stationary by differencing it d times, it is explanatory variable, as well. This problem said to be integrated of order d. The concept is discussed in a widely available reference of an integrated time series has not only on scientific evidence, in which Professor been extended to higher orders of d, but to Rubinfeld states, fractional values of d as well. The assumption of no feedback is Ordinarily, the sum of two non-stationary especially important in litigation, because it is possible for the (integrated) time series is also non- defendant (if responsible, for stationary. On occasion, however, a unique example, for price-fixing or combination of two integrated time series discrimination) to affect the values results in a stationary time series, in which of the explanatory variables and case it is said that the data is cointegrated. thus to bias the usual statistical tests that are used in multiple regression.4 Intuitively, two series that are cointegrated may be individually non-stationary, but they The problem is illustrated by supposing that will not move too far apart over time. A the defendant’s expert wants to demonstrate common heuristic example of two that the price of a product, Pt, is determined cointegrated series is that of a drunk dog- by three variables: a demand variable, Dt, a owner walking in a desert. Assuming the cost variable, Ct, and advertising, At. owner is drunk enough to have no sense of Provided that the “no feedback” assumption direction (and does not double-back), his is fulfilled, the researcher might estimate a path might resemble a (non-stationary) multivariate model of the form random walk. His dog’s path might also look like a random walk. At any one time P D C A t = α + β1 t + β 2 t + β 3 t + εt . they may be close together, and at another time further apart, but over the long run they Setting aside for the time being the spurious will move together, and never take off in regression problem, if the explanatory opposite directions. Before any regression variables together account for all but analysis can be considered valid, therefore, residual random variations in the price, and the econometrician must be satisfied either the parameters, ßi, are all statistically that a) the regressors are stationary, or, b) significant, estimations from this model may the regressors are cointegrated. constitute sound statistical evidence. However, if demand also reacts to price, i.e., Pt also causes variations in Dt, then simultaneity bias will invalidate the results.

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To remedy this, Dr. Rubinfeld suggests provide probative evidence, the cointegrated dropping the questionable variable to VAR model represents a superior approach. determine whether its exclusion makes a But because there are numerous contexts in difference, or expanding the model by which the interpretation of a stationary adding one or more equations that explain cointegrating process can be theoretically the feedback effect. meaningful, cointegration analysis can provide a wealth of other kinds of The cointegration approach generally solves information to a fact-finder. Of particular the problem by expanding the model into a interest to the antitrust practitioner is the system of equations in which each variable case in which statistical evidence is needed may influence every other variable. The to determine product or market delineation. of the dependence of each variable on every other variable can The use of statistical correlation between then be tested. Instead of the researcher price series has be justifiably criticized assuming that Pt should be considered the (because of the spurious regression problem, dependent variable and that Dt, Ct, and At inter alia) as a of determining should be the explanatory variables, the whether price realizations from potentially direction of causality as between each substitutable products or from different variable can be tested within the model to geographical areas belong to the same or arrive at a specification that does not suffer separate markets.5 But the cointegration from simultaneity bias. This arrangement paradigm provides a statistically sound basis also permits modeling more complicated on which to perform “price tests” for market dynamics. Thus, a cointegration model can delineation. remain agnostic about which of the variables, Pt, Dt, Ct, At, should be considered Econometric research has begun to explore 6 dependent or explanatory until there is this application of cointegration theory, and statistical evidence to support the in recent years the European Commission specification. has relied on cointegration analysis in antitrust and merger cases.7 Such Research and Applications applications are based on the common-sense notion that non-stationary prices in a single We have already demonstrated the virtue of geographical market or for substitutable the multivariate cointegrated VAR approach goods in the same area will not move too far in the context of litigation as a diagnostic apart over time. In this context the tool to uncover weakness in regression cointegration vector is the analyses presented by an opposing expert. that represents arbitrage and transaction cost The theory offers methods to determine differentials between the two products or whether variables are integrated of order one areas. As Michaels and deVany explain it: or stationary, and thus whether regression results are spurious, whether there is a If two areas are in the same problem of simultaneity bias, and whether a competitive market, their prices will inhabit a band whose width dynamic rather than a static specification is reflects the cost of arbitrage. Those required. costs include transportation, risk exposure, and information about At the same time, the approach has obvious profitable opportunities. If virtues for de novo econometric modeling. competition exists, it will quickly Anywhere a model could bring disparate prices back within their arbitrage limits. If, for Volume 4, Number 1 14 Spring 2004 Economics Committee Newsletter

example, bad weather increases acquisition by Gencor Ltd. of South Africa price in area i while price at area j (a holding company) and Lonrho plc of the and transmission cost are unchanged, transactions in a U.K. of the Gencor platinum mining and competitive market will restore an refining assets. One of the issues in the case equilibrium at which the two prices was whether platinum, gold, silver, rhodium again differ by no more than the and palladium should be considered separate arbitrage limits. If the cost of antitrust product markets. Noting that the arbitrage varies little over time, two areas are in the same market if the prices of these commodities are “highly difference between their prices is correlated,” the EC recognized that “a high relatively constant. The statistical correlation does not in itself imply a causal technique known as cointegration relationship. Indeed economic price-series provides a criterion under which to data are often non-stationary (i.e. trended) determine the relative constancy of 11 such a difference. If the prices are and therefore automatically correlated.” not cointegrated, there are no well- To reach beyond the limitations of cross- defined bounds on the difference correlations, the Commission undertook a between them. If prices in two cointegration analysis, which they dubbed areas are cointegrated, the areas are “an econometric method which can test in the same economic market. Although the difference between whether there is a systematic equilibrium (or the prices varies with some long-run) relationship between two or more 12 , there is a high time-series of data.” The Commission probability that it will remain concluded: within arbitrage bounds.8 The results of the analysis show In their study of the BP/Arco merger, Hayes, that the data do not suggest any et al. (2001, note 7, supra, at 7) cite equilibrium (or long-run) cointegration studies of the price series of relationship between the respective various types of crude oil because, price levels of platinum, rhodium, palladium, gold and silver, nor of “evidence that prices are cointegrated has any subset of these metals. This been interpreted as an indication that the econometric analysis of metal products in question trade in the same prices indicates that [prices of these antitrust market. In particular, the fact that metals] tend to vary, over the long the price of [Alaskan North Slope] is run, independently of each other, thus confirming the view that [they cointegrated with world crude prices constitute] separate relevant indicates that ANS trades in a world market product markets.13 for crude oil.”9 Cointegration analysis also played a role in Unlike the U.S. antitrust enforcement the EC’s decision rejecting the acquisition of agencies and U.S. courts, who may as yet Lenzing AG, a synthetic fiber manufacturer, not been called upon to expressly rely on by CVC Capital Partners Group Ltd., owner cointegraton analysis as part of their of Acordis, also a manufacturer of synthetic decision-making, the EC has recognized the fibers.14 The relevant market issue depended utility of cointegration methods in the in part on whether various varieties of fibers antitrust context. One notable case is the belonged to the same product market. While Gencor/Lonrho merger case,10 in which the the correlation coefficients were found to be advantage of cointegration compared to quite high, a statistical test based on the cross-correlation analysis was emphasized. distribution theory that underlies The case involved the proposed joint cointegration analysis revealed that the price

Volume 4, Number 1 15 Spring 2004 Economics Committee Newsletter gap between “commodity VSF (viscose Coast, while the plaintiff asserts that the staple fibers)” and “spun-dyed VSF” was relevant geographic market is narrower, and non-stationarity, leading the EC to conclude that Florida and Georgia constitute a that these two varieties of fiber do not separate market. To keep the example belong to the same product market. simple, we will consider real wholesale prices from suppliers in Boston, Atlanta, and An Illustration Miami. The data are shown in Fig. 3.

Let us conclude with a brief illustrative Cross-correlation reveal that the application of the cointegration approach to for the Boston-Miami delineating the relevant geographical market price pair is 0.41, the Boston-Atlanta pair is for a fictional product manufactured by 0.55, and the Miami-Atlanta pair is 0.72. several competitors, each with several While these correlation coefficients tend to facilities throughout the U.S. Assume that suggest that Boston is in a separate market, the defendant finds it advantageous to claim the results can only be ad hoc, and in fact that the relevant market is the U.S. East give no reliable guidance to the decision- maker.

Figure 3: W holesale Price of W idgets Boston, Atlanta and Miami, in Logs

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Figure 4: The Cointegrating Relation

Suppose, however, that a cointegration they do not enter into the cointegrating system is estimated, and that the results relation, but they are also strongly indicate that there is a single, unique exogenous because the variable is not combination of the three time series that is affected by the stationary equilibrium stationary. We can restrict the model to represented by the price gap between prices conform with these finding, estimate the in Atlanta and Miami, i.e., the cointegrating parameters of the cointegrating relation, and relation. determine which variables react to it. Graphs of the cointegrating relation are Finding a single, stationary cointegrating shown in Fig. 4. relation in a three-dimensional system is the same as finding that all three time series are In the upper panel, the cointegrating relation driven by two non-stationary components, or is shown without restrictions on the short- random walks (in the multivariate context, run process. The lower panel shows the called “common stochastic trends”), because cointegrating relation “cleaned” of the the number of cointegrating relations is the influence of the short-run process by complement of the number of common regression. Further testing indicates that the stochastic trends. The model, therefore, Boston prices do not belong in the provides strong evidence that the same cointegrating relation, nor do they react to it. common trend drives both the Atlanta and Being unable to reject these restrictions Miami price series, while a separate establishes the strict exogeneity of the stochastic trend drives prices in Boston. Boston price series. Not only are the Boston Moreover, since the Boston prices do not prices weakly exogenous, in the sense that enter into the equilibrium represented by the

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Atlanta-Miami price gap, and, in turn, the 1 Engle, Robert F. and C.W.J. Granger (1987), Atlanta-Miami equilibrium relation has no “Cointegration and error correction: Representation, estimation and testing,” 55, 251–76. influence on prices in Boston, the model 2 The “Johansen method” is explored in detail in provides strong evidence that suppliers in Johansen, S. (1995), Likelihood-Based Inference in Atlanta and Miami serve a separate Cointegrated Vector-Autoregressive Models, Oxford geographical market from the market served University Press, but most econometrics textbooks by suppliers in Boston. published since 1997 contain a description of Johansen’s approach. 3 The spurious regression phenomenon was first Conclusion identified by Yule, G.U. (1926), “Why Do We Sometimes Get Nonsense Correlations Between The levels of a variable contain its history, Time-series?” Journal of the Royal Statistical while its differences reflect short-term Society, 89:1–69, and established rigorously by changes. The multivariate cointegrated simulation studies by Granger, C.W.J. and P. VAR model is able to examine both types of Newbold (1974), “Spurious Regression in Econometrics,” , 2:111–120. effects by splitting up the movement of each Not until more than ten years later, however, was the variable into a long-run and short-term asymptotic distribution theory applicable to the effect, and permitting the researcher to Granger-Newbold worked out by perform separate on Phillips, P.C.B. (1986), “Understanding Spurious each. For variables integrated of order one, Regression in Econometrics,” Journal of Econometrics, 33:311-340. this is equivalent to splitting the system into 4 Rubinfeld, D.L. (2000), “Reference Guide on non-stationary and stationary components. Multiple Regression,” in Reference Manual on This approach represents a vast Scientific Evidence, 2nd Ed., Federal Judicial Center, improvement over traditional linear at 196, available at: http://www.fjc.gov/public/ regression analysis, which is invalid in the pdf.nsf/lookup/sciman00.pdf/$file/sciman00.pdf 5 See, e.g., Werden G. and L. Froeb (1993) presence of non-stationarity. “Correlation, Causality, and All That Jazz: The Inherent Shortcomings of Price Tests for Antitrust Because regression analysis is such an Market Delineation,” Review of Industrial important econometric tool, and because so Organization, 8: 329–353. 6 many variables are non-stationary, See, e.g., Møllgaard, P. and C.K. Nielsen (2004), cointegration is likely to find its way into the “The Competition Law & Economics of Electricity Market Regulation”, European Competition Law courtroom soon. Cointegration offers a way Review, 25(1): 37-43; la Cour, L.F. and P. of explaining why the regression results Møllgaard, (2002), "Market Domination: Tests presented by the opposing expert are not Applied to the Danish Cement Industry,” European statistically valid, and a way for your expert Journal of Law and Economics 14(2), 99-127; to present statistical relationships that can Wårell, L. (2002), “Market Integration in the International Coal Industry: An Error Correction withstand methodological attack. Decision- Model,” Luleå University of Technology Department makers are likely to rely on such statistical of Business Administration and Social Science, methods for help in determining a wide Division of Economics, available at: of antitrust-related issues. http://www.kkv.se/ forskare student/pdf/proj122 Congratulations to Professors Engle and 2001.pdf; Hayes, J., C. Shapiro, and R.Town (2001), “Market Definition in Crude Oil: Estimating the Granger for their tremendous contribution, Effects of the BP/ARCO Merger,” Working Paper, and for a Nobel Prize well-deserved. available at: http://www.ftc.gov/bc/gasconf/comments2/oilpaperjo * The views expressed herein are solely those of the hnhayesetal.pdf; Dhar, T.P. and S. Ray (2000), author and not of the American Antitrust Institute. “Understanding Dynamic Retail Competition For further information, the author may be contacted Through the Analysis of Strategic Price Response at JRubin@ antitrustinstitute.org or at (202) 415- Using Time Series Techniques,” presented at the 0616. American Agricultural Economics Association

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Meeting, Tampa, Florida, 2000, available at: http://www.aae.wisc.edu/fsrg/publications/wp2002 7.pdf; Michaels, R.J. and A.S. deVany, (1995) “Market-based rates for interstate gas pipelines: The relevant market and the real market,” 16 EnergyL.J. 299–345; and, deVany, A.S., and W.D. Walls, (1993). “Pipeline Access and Market Integration in the Natural Gas Industry: Evidence from Cointegration Tests”, The Energy Journal, Vol. 14, No. 4, pp. 1-19. 7 See, e.g., Commission Decision of 17 October 2001 declaring a concentration to be incompatible with the common market and the functioning of the EEA Agreement (Case No COMP/M.2187 - CVC/Lenzing) (slip opinion available at: http://europa.eu.int/comm/competition/mergers/cases /decisions/m2187_en.pdf); Commission Decision of 24 April 1996 declaring a concentration to be incompatible with the common market and the functioning of the EEA Agreement (Case No IV/M.619 - Gencor/Lonrho) Official Journal L 011 , 14/01/1997 p. 0030 - 0072; and Commission Decision of 31 January 1994 declaring a concentration to be compatible with the common market (Case No IV/M.315- Mannesmann/Vallourec/Ilva) Official Journal L 102 , 21/04/1994 p. 0015 - 0037. 8 Michaels, R.J. and A.S. deVany (1995), note 6, supra, at 327. 9 Citing Rodriguez, A.E. and M.D. Williams (1993), “Is the World Oil Market ‘One Great Pool’? A Test,” Energy Studies Journal, 121–130, for the proposition that cointegration tests “show that a relevant antitrust product market is no narrower than crude oil and the appropriate geographic market is the world.” 10 Note 7, supra. 11 Id., at 52. 12 Id., at 53. 13 Id. 14 Note 7, supra, at 109–10.

Volume 4, Number 1 19 Spring 2004