Introduction

1The concept of cointegration, for which Clive W.J. Granger received the highest scientific distinction in 2003, was the result of lengthy reflection on the notion of “long-term equilibrium”. The long-term equilibrium issue concerns all macroeconomics and macroeconometrics [Le Gall, 1993; 2002]. This temporal equilibrium induces systemic dynamics [De Vroey & Malgrange, 2005; 2011] which, if although perfectly formalized in the economic theory framework, generates various problems in econometrics [Artus et al., 1986; Fève, 2005]. Dynamics induce the notion of time in the analysis which requires the adaptation of standard modeling methods.

2Dynamic analysis introduces a possible time dependence, a trend. Series which change under the influence of time (in mean or in variance) are said to be “nonstationary” or “integrated”. The appearance of time conveys a relatively long “memory” phenomenon involving the reaction of the series to its economic environment and the guidance of economic policy. A deterministic trend thus means that the series evolves following the nature of the trend (linear or nonlinear) over an increasing or decreasing gradient, i.e. a series is nonstationary in mean and shocks only have a temporary effect. By contrast, a stochastic trend means that the series is influenced by a random trend, whereby the variance of the process is infinite and shocks have a permanent effect. Nelson and Plosser [1982] showed the consequences of wrong identification of the nature of nonstationarity. However, identification was possible only following the study of Dickey and Fuller [1979, 1981], who worked out the first “unit root tests”. Cointegration arose in this context. Cointegration can be defined as the onset of long-term equilibrium (stationary) between two or more time series which evolve individually under the influence of time (in a deterministic or stochastic way). [2] The advantage of a cointegration relationship is that it enables estimation of a dynamic relation (nonstationary) using standard methods, adding decomposition between the intensity of long-term equilibrium (stable) and short-term dynamic factors (adjustment). This concept made it possible to take systemic information, particularly for economic policy decisions, into better account [Malinvaud, 1991].

3Several people played part in the emergence of this concept. At the beginning, we should mention the work of Alban William Housego Phillips who expressed an interest in macroeconomics. He exploited the mechanisms of equilibrium adjustment and also the control theory [1954; 1957]. Then John Denis Sargan proposed the first “correction to equilibrium” model by putting forward the concept of deviation from the equilibrium path between several time series [1964]. At that time, David Hendry was conducting PhD research under the supervision of Sargan; He was able to ascertain the link between equilibrium correction mechanisms and the spurious regressions presented by Clive Granger and Paul Newbold [1974]. Granger and Newbold determined that nonstationary series led to wrong estimations, i.e. a very high regression coefficient (close to 1) and weak Durbin-Watson statistics [3] could denote a wrong specification of the model because of the autocorrelation between the series–a “spurious regression”. Note that unit root tests did not yet exist at that time. So Hendry often challenged Granger on the relationship between Sargan’s model and spurious regressions. Finally, the cointegration theory emerged when Granger decided to prove to Hendry that there was no link between equilibrium correction mechanisms and spurious regressions. Granger then thought that an integrated series could not move stationary by any linear combination. For him, the individual autocorrelation of the series was an intrinsic characteristic. As he said it [2004], an integrated series was to remain integrated. He refused the idea that a linear combination between integrated series could modify their intrinsic nature. In fact, he did not yet feel an interest in working on the difference between series (linear combination). However, as he discovered later, if the series evolved in the same way then it was highly probable that the difference between these series would be a new stationary series.

4However, what has been relatively overlooked in the emergence of the concept of cointegration is the fundamental role of Hendry who, by his intuition and tenacity, enabled Granger to formalize the concept of cointegration, after many debates. Hendry truly bridged the gap between Sargan and Grangers’ theoretical developments. But let us, however, return to the initial conditions which allowed the econometric development of modern-day dynamics.

5Since the mid-twentieth century, economists have been interested in the analysis of dynamics and the issue of the structural decomposition of time series between short-term and long-term elements [Persons, 1917; Wold, 1938; 1954; Koopmans-Rubin-Leipnik, 1950; Qin, 1993; Le Gall, 1993; 2002]. This time dichotomy is the basis of systems dynamics, distinguishing between heavy trends and cyclical phenomena. Although the paternity of dynamic analysis in economics goes to Irving Fisher, other economists such as Milton Friedman [1957], then economists from the National Bureau of Economic Research (NBER), such as Ruth Mack or Victor Zarnowitz [1958], were interested in the causes and consequences of time links within series. Note that, at the same time, studies in which spectral analysis was used appeared, coming from the physical science field [Jenkins & Priestley, 1957]. But let us not see there any political feeling: even if the first modern work in time series was the fact of the NBER (thus a place of liberal inspiration), Granger did not have any political inclination in his work. As he said himself, he was simply interested by time series and has been working to develop this econometric field. [4]

6It seems that the most active current of thought on dynamic modeling prevailed at the London School of Economics, particularly with work of John Denis Sargan in 1964. After the Second World War, the London School of Economics adopted a positivist thought process. The School hosted Karl Popper, then Imre Lakatos. It was certainly this philosophical trend, based on experimentation, which underpinned the emergence of a corpus of econometricians. From the end of the fifties, the English economist Lionel Robbins created the “Seminar of Methodology, Measurement and Testing” (M2T) which took place on Wednesday afternoons at the London School of Economics. During one of these seminars, Alban William Housego Phillips presented his “curve” for the first time. Sargan understood the importance of the “Phillips curve” for his own work, i.e. the issue of equilibrium and its dynamic modeling. This was certainly the beginnings of what was identified as the “concept of cointegration” fifteen years later.

7The history of the concept of cointegration is not as clearcut as it seems in the final paper of 1987 published in Econometrica. We drew up this chronological sequence of events—from work of Phillips to the official paper of Engle and Granger in 1987—to put into better perspective the context in which the meeting between Hendry and Granger took place:

Chronology of the discovery of cointegration

Chronology of the discovery of cointegration

8The aim of the paper is to clarify and gain insight into all of the intellectual thought between Granger and Hendry that led to the 1987 paper. The role of Hendry in the emergence of the concept of cointegration is certainly unknown. In the first section, we highlight the influence of Sargan and his works on the notion of “correction to equilibrium”. In the second section, we focus on the debates between Hendry and Granger from 1974 to 1981 which finally led to the theorization of the concept of cointegration after being somewhat antagonistic. Lastly, in the third section, we study, from an historical standpoint, the scientific benefit of Granger’s papers on cointegration, from 1981 to the 1987 paper, which finally acknowledged the cointegration phenomenon.

1 – Beginnings of the concept of cointegration: John Denis Sargan

9John Denis Sargan was first a mathematician; He enhanced his education by studying economics [Hendry, 2003], learned time series with Maurice Stevenson Bartlett at the University of Cambridge. At that time, he read The General Theory of Employment, Interest and Money of John Meynard Keynes [1936] and anticipated the possibility of using mathematics in macroeconomic mechanisms. Over the next ten years (1948 to 1958), he took part in the development of mathematical economics and econometrics. His conception of economics tended towards general equilibrium analysis. He was interested in global macroeconomics solving, following Keynes reasoning. At the same period, the economist Alban William Housego Phillips was teaching at the London School of Economics. He outlined, for the first time in 1958, the “Phillips curve” in the methodological seminars of Lionel Robbins. His interest in macroeconomics led him to analyze equilibrium adjustment mechanisms and the control theory [1954; 1957]. Phillips had a macroeconomic viewpoint; he reasoned in flow and in dynamics. [5] For him, economic policy was the vector of intervention. In his 1957 paper, he outlined the force of convergence (strength of correcting action) which was later at the heart of the cointegration concept. For Phillips, this force of convergence was the stabilizing (multiplying) effect, period after period, of an economic policy on an economic system:

10

It is assumed that the regulating authorities are able to make continuous adjustments in the strength of the correcting action they take but that there is a distributed time lag, L_c, between changes in the strength of the correcting action and the resulting changes in policy demand. The amount by which policy demand would be changed as a direct result of the policy measures if they operated without time lag will be called the potential policy demand, π, the amount by which it is in fact changed as a direct result of policy measures is the actual policy demand E_π. The basic problem in stabilizing production is to relate the actual policy demand to the error in production in such a way that errors caused by un-predicted disturbances are corrected as quickly and smoothly as possible⁽²⁾. For a given correction lag the problem reduces to that of finding the most suitable way of relating the potential policy demand to the error in production.
⁽²⁾ If reliable and frequent measurements of aggregate demand were available the potential policy demand could also be related to the error in demand. This would permit a more rapid correction of errors in production caused by shifts in aggregate demand.

11Here, Phillips was referring to the equilibrium error correction mechanism (the force of convergence) and how it intervened gradually over time: by temporal shifts between the decision of the action and the result. This was the first expression of dynamic economic analysis. This was a description of a mechanism of return to equilibrium with an identified force of convergence relating to the temporal shift between the decision of the measurement of economic policy and its effect. The mathematical writings that he produced were purely algebraic, i.e. ARMA processes were not yet available for econometrics. [6] In 1957, Phillips was already working on impulse response functions (which will have been used by Sims in 1980 as dynamic extension of vector autoregressive models).

12It was in this context that in 1964 Sargan wrote a paper for a conference of the Colston Society on National Economic Planning: Wages and Prices in the United Kingdom: A study in econometric methodology. This paper, which has gone down in history as the “Colston Paper”, revolutionized dynamic modeling. It notably introduced the long-term perspective to understand economic equilibrium and error correction mechanisms in dynamic modeling. Sargan was interested particularly in the Phillips curve, i.e. the relationship between prices and wages (in nominal terms) via inflation rate. As indicated by Hendry [2003, p. 465], “[Sargan’s] analysis highlighted the role of real-wage resistance in wage bargains, interpreting the equilibrium correction‑the deviation of real wages from a productivity trend‑as a ‘catch-up’ mechanism for recouping losses incurred from unanticipated inflation”. He intuitively felt that using first differences of the log of wage and price variables [7] would introduce a distinction between short and long-term aspects in the same structure (model). Sargan [1964] introduced the use of first differences of the log of variables so that the variables were stationary (unit root tests did not yet exist). He understood that working with first differences of the variables amounted to working on stationary series [Hendry, 2003, p. 461] because the deterministic trend (long-term evolution) was removed. The variables could thus evolve independently (variable in level), while the first differences were stationary because one only considered variations between periods without dealing with the trend. The importance of Sargan’s work, for what would later become “cointegration”, was this distinction between short-run oscillations (first differences of the log of variables) which were more unstable than long-run equilibrium (variables in level) which had to converge.

13On the basis of the bivariate wage-price model [Sargan, 1964]:

14In this paper, Sargan provided the basis for a new macroeconometric dynamic modeling. The study integrating the long-run in the analysis produced a new view of macroeconomics. With the error correction concept, Sargan paved the way for real-time dynamic analysis. The error correction model that he developed introduced the idea of a convergent force towards equilibrium. There was thus a distinction made between the short-run dynamics elements and the long-run trend. The framework of the concept of cointegration emerged.

15However, this work had been completely forgotten for several years since Sargan did not seek popularity. It was thanks to David Hendry that Sargan’s work emerged from the darkness, allowing Granger to establish the theory of cointegration, and make the connection with the error correction model a few years later.

2 – The Sargan–Granger connection: David Hendry

16David Hendry became interested in economics after reading Paul Samuelson’s books, particularly Economics: An introductory analysis [1961]. After a short passage at the University of Cambridge, he went to the London School of Economics, where Denis Sargan was teaching, and he entered the econometrics field. In parallel, he attended the seminar of quantitative economics supervised by Sargan and Phillips, among others, and he learned about autoregressive structures and moving averages.

17At the same time, Hendry was interested in economics and psychology, and after having read George Katona (who was interested in the relationship between economic science and psychology and sought to understand economic phenomena from the standpoint of human behavior), his project was to use the psychological factors of human behavior in intertemporal utility maximization models. The role of expectations was then unsophisticated and the models had to be reappraised. Sargan was thus interested in Hendry’s work and introduced him to vector autoregressive dynamic modeling systems. The vector structure enabled him to overcome problems generated by autocorrelation (using factorization and orthogonalization of the polynomial matrix resulting from the vector structure). [8] Throughout these years, Hendry promoted Sargan’s ideas.

18When Hendry took a sabbatical year to go to Yale and Berkeley in 1975, [9] he realized that Sargan’s works were unknown outside of Great Britain, particularly those concerning error correction models. Later, Hendry diligently endeavored to improve dynamic modeling, thus contributing to the promotion of Sargan’s work about dynamic modeling, the genesis they constituted for the future theory of cointegration. In November 1975, Hendry took part in a conference [10] organized by Christopher Sims, for which he wrote a paper [11] in which he stated his agreement with Granger and Newbold [1974] [12] on the existence of bad practices in econometric modeling, in particular that a high coefficient of regression associated with weak Durbin-Watson statistics indicated a mis-specification of the model‒distortions generated by the treatment in statics of time variables. [13] Hendry’s paper suggested an equilibrium correction model based on consumer expenditures. [14] Hendry met Granger for the first time at this conference.

19Hendry was very interested in econometric modeling data processing tools. Back at the London School of Economics, he was influenced by the innovative work of Phillips on the control theory, and he remained enthusiastic about the equilibrium correction mechanisms of Sargan. However, although in agreement with Granger on the possibility of regressions without any sense (“spurious regressions”), they were at odds on the way of identifying them even though they understood that the indication was a residual autocorrelation [Qin, 1993]. Hendry therefore used the equilibrium correction mechanisms proposed by Sargan that were, at that time, not coherent for Granger: he was skeptical about adding first differences to series in level could stationarize the system, the corrections (first differences) had a short-term effect while the long-term relationship remained nonstationary. For Granger, the introduction of first differences into a nonstationary system could not correct the long-term nonstationarity between variables. The discussion between Granger and Hendry was strictly mathematical. Granger was a famous theorist and his work primarily concerned mathematical developments applied to economics. Hendry understood that if a long-run equilibrium relationship existed between two series, they thus evolved in a similar way, and if this phenomenon appeared, it had been “caused” by a common economic conditioning. However, Granger had the conviction that a nonstationary system could not be stationarized by the addition of differentiations. In other words, he refused a priori the possibility of a stationarization of the variables by a simple correction of equilibrium deviations because the long-run trend of the series was its trajectory–so a nonstationary series had a nonstationary trajectory. Granger rejected the idea that this trajectory could be modified by differentiation [Hendry, 2004, p. 19]. The response of Granger to Hendry at that time was that if the variables were integrated (nonstationary), then they had to remain integrated, i.e. the transformation did not have to change the nature (in the long term) of the variables of a system, otherwise this showed a sort of instability that Granger claimed was due to the data, not to the economic theory. This phenomenon was due to mis-specification of the model, or ad hoc models. Even if Granger was right as for the order of integration of the series considered individually, he did not perceive, at that time, the possibility of a reduction of the order of integration between two series or more by just a linear combination. Several factors underlie Granger’s standpoint, all of a mathematical nature: the ARMA model was not appropriate for nonstationary processes (they were designed to satisfy the matrix inversion rule); the operation of first difference (i.e. Y_t – Y_t-1 = ΔY_t) no longer concerned relations between levels of variables but instead only focused on variations between successive periods. The ARMA model thus could not analyze a long-run relationship between nonstationary variables [Lardic & Mignon, 2002].

20Hendry nevertheless remained persuaded that there could be linear combinations of nonstationary variables (in a model) which were stationary, and which would evolve in the same way because they were driven by the same economic phenomenon. [15] Let us note that this definition is really that of the concept of cointegration. He wondered, however, about modifications induced by unit roots within the estimator distributions and within the tests. He decided, along with Grayham Mizon, to verify this issue. By a Monte Carlo simulation, they generated a series of processes and they reached the conclusion that the presence of a unit root did not modify the estimator distribution; they found this distribution almost normal. Hendry thus concluded that, as intuited by Granger, integrated variables inside a system remained integrated irrespective of the imposed linear transformation. Otherwise, in a symmetrical way, if the system was moved by long-term equilibrium, then the variables could not be integrated. This conclusion should have led to the downfall of the concept of cointegration. But Hendry, prompted by his first intuition, realized that the Monte Carlo data simulation processes that he carried out with Mizon had produced variables with a very strong growth rate (series with an order of integration higher than one). A transformation by first differences could therefore not stationarize the variables.

21During his stay at the University of California (San Diego) in 1976, Hendry once again discussed with Granger about the existence of relationships without any direction, (spurious regressions) and on the validity of the equilibrium correction model promoted by Hendry at the conference of the previous year, i.e. the possibility of stationarization of a system of nonstationary variables. For Hendry, the existence of a convergence to a long run equilibrium ensured by the model of Sargan (the residual process is thus stationary since the joint evolution of the variables is convergent over time) made it possible to specifically avoid estimation of spurious regressions since, if there was no convergence in the error-correction mechanism it was because the variables evolved without any economic link. Hendry did not, however, see the mathematical condition of the rank-reduction which closed the demonstration.

22Granger was steadfast, maintaining that the structure of the model with error correction could not solve the initial nonstationarity of the variables–the equilibrium correction model associated a differentiated variable on the left side of the equation with variables in level and in differences on the right side. He stated that if the variables were integrated of order one, then the model built could not represent any equilibrium because the right side contained integrated as well as stationary variables. He remained skeptical about any specification in level. Using his theorizing skills, he decided to mathematically prove that the reasoning of Hendry did not hold, and that his intuition was wrong. However, trying to refute Hendry’s reasoning, Granger actually demonstrated the opposite: a linear combination of nonstationary variables in level could indeed be stationary. Surprisingly, Granger succeeded in connecting his intuition about spurious regressions with the long-term equilibrium or “cointegration” via the error correction model of Sargan. This economic situation corresponded to quite real particular cases and not only ad hoc cases as Granger supposed would be the case. These cases expressed common exogenous conditionings of two or more variables, conferring the same long-run evolution (a kind of equilibrium path, or common trend) on these variables. Cointegration represents therefore the existence of a stationary evolution of the difference between nonstationary variables. The equilibrium correction of Sargan provides the theoretical possibility of convergence towards long-run equilibrium–indeed, one cannot measure any convergence between variables if they are stationary. Granger talked about his astonishment in his speech at Nobel Prize Awards:

23

I am often asked how the idea of cointegration came about; was it the result of logical deduction or a flash of inspiration? In fact, it was rather more prosaic. A colleague, David Hendry, stated that the difference between a pair of integrated series could be stationary. My response was that it could be proved that he was wrong, but in attempting to do so, I showed that he was correct, and generalized it to cointegration, and proved the consequences such as the error-correction representation.

24From that time, Hendry and Granger shared the same view of time series–improving the empirical economics [Hendry, 2010, p. 4].

3 – The concept of cointegration by Granger

25The obstinacy of two men propelled econometrics towards one of its most tremendous advances, i.e. cointegration. This word was coined by Granger: first called “co-integration”, in the literature this particular relation became “cointegration”. The importance of the concept led to advances in econometric methodology and in economic knowledge. This is why Granger received the Nobel Prize in 2003.

3.2 – The concept of cointegration: a merely brilliant idea

27When in 1980 Granger [17] formalized cointegration, he simultaneously achieved the unification of dynamic modeling by the connection with Sargan’s equilibrium correction model [1964], which has been called the “error correction model” since 1983. The concept of cointegration was the cornerstone, harmonizing problems generated by the nonstationarity of time series, and thus of economic phenomena in their all complexity. However, it seemed that the international community of econometricians had some hesitations about cointegration. The theory was too simple (just a linear combination between variables), so it did not seem to be scientific enough. Granger twice submitted a paper to Econometrica, and both were rejected.

28

I wrote up a version of the theory, including a simple version of the ʽrepresentation theoremʼ and submitted it to Econometrica. I also mentioned cointegration in a few other places as it was attracting a lot of interest. Econometrica rejected the paper for various reasons, such as wanting a deeper theory and some discussion of the testing question and an application. As I knew little about testing I was very happy to accept Rob’s [Engle] offer of help with the revision. I re-did the representation theorem and he produced the test and application, giving a paper by Granger and Engle which evolved into a paper by Engle and Granger, whilst I was away for six months leave in Oxford and Canberra. This new paper was submitted to Econometrica but it was also rejected for not being sufficiently original. I was anxious to submit it to David Hendry’s new econometrics journal [18] but Rob wanted to explore other possibilities. These were being considered when the editor of Econometrica asked us to re-submit because they were getting so many submissions on this topic that they needed our paper as some kind of base reference. A few citations and twenty years later and here we are, although I still believe that the paper would have been successful wherever it was published!
…Once the first major paper was written it appeared as a UCSD working paper, [19] which immediately generated a great deal of interest and was quite widely cited in many papers. The idea of cointegration was so easy to understand and its possible implications were immediately clear that a whole industry was started. The error correction model was rejuvenated by this literature and its relevance and success of economic forecasting established.
…The number of applied papers using the concept of cointegration is enormous and far too large to attempt to summarize.

29The very first version of Granger’s cointegration was a paper published in 1981 in the Journal of Econometrics, entitled “Some properties of time series data and their use in econometric model specificationˮ. In this paper, Granger was extremely careful in his formalizations. He systematically referred to spectral analysis and went by the variable distributions. Finally, only the last two of the nine pages of the paper were about cointegration. However, before these last two pages, Granger very carefully explained the few possible cases of cointegration just as to support his digression with economic examples. Thus one could read on the sixth page: [20]

30

As an extreme case of model (3.2) inconsistency, suppose that d_x<1/2, so variance of x_t is finite, but 1>d_y>1/2, so variance of y_t is infinite. Using just finite polynomials in the filters, clearly y_t cannot be explained by the model, if variance ε_t is finite, which is generally taken to be true. Similarly if d_y<1/2 but 1>d_x>1/2, then one is attempting to explain a finite variance series by a infinite variance one. This same problem occurs when the d’s can take integer values, of course. Suppose that one knows that change in employment has d = 0, and that level of production has d = 1, then one would not expect to build a model of the form Change in employment = α + β (level of production) + f(B) ε_t.

31Granger later developed the idea of cointegration but again using spectral analysis. He expounded the possibilities to find this kind of phenomena on the economic reality:

32

Co-integrated pairs of series may arise in a number of ways, for example:
(i) If x_t is the input and z_t the output of a black box of limited capacity, or of finite memory, the x_t, z_t will be co-integrated, for instance the series might be births and deaths in an area with no immigration or emigration, cars entering and leaving the Lincoln Tunnel, patients entering and leaving a maternity hospital, or houses started and houses completed in some region. For these examples to hold, it is necessary to have d > 0.
(ii) Series for which a market ensures that they cannot drift too far apart, for example interest rates in different parts of a country or gold prices in London and New York.

33Granger thus enumerated some possible cases of cointegration as if he felt the need to reassure himself. On the next to last page of the paper, he finally referred to Sargan and Hendry, as if he relied on their former analyses, as if he refused the responsibility of the concept of cointegration [Granger, 1981, p. 129].

34It was extremely unusual to read such a cautious paper of Granger. All the skepticism reported by Hendry seemed to show through in this first 1981 paper. Contrary to what emerged thereafter, Granger did not develop “his” error correction model in this paper. He just reproduced the formulation of Sargan’s model without introducing the cointegration relation. The beginnings were there, but Granger had not yet formulated the error correction model which included the cointegration relation.

The unpublished paper: 1983

35In 1983, Granger again wrote about the concept of cointegration. This document, “Co-Integrated Variables and Models Error-Correctingˮ, remained indexed as an Unpublished Discussion Paper. It was, however, in this document that Granger formally introduced the concept of cointegration into the error correction model–and the first version of the representation theorem–and he interpreted the implications of cointegration: by addition of the cointegration variable, z_t-1, he divided the dynamics of the system into a long-run component (trend) and a short-run component (cyclical variations). The long-run component (target) was given by the lagged cointegration variable (z_t-1), and the short-run component by the variations in the variables in first difference lagged Δx_t,y_t; the dynamic evolution was represented by the lag operators:

36This formalization of the error correction model, including the cointegration relation, was published in 1986, but not the representation theorem [21] which was only included in the 1987 paper.

37In this document, Granger formulated the error correction model in the introduction [Granger, 1983, p. 1-2]. Granger’s care and the progression in his reasoning–which had to be his following discussions with Hendry–could be readily perceived. On the other hand, the introduction ended on the certainty that cointegration existed in relation with the error correction model. Then Granger clearly announced the content of the following debate: this “Co-integration” and its identification and its formalization. After exposing the mathematical writing of the integrated variables and the (bivariate) cointegrated systems, Granger departed from the formalization and described these series as even smoother forms since the order of integration was higher. He seemed amazed by the fact that two series of this nature could evolve in a similar way in the long-term. This enthusiasm led him to wonder about the economic significance of systems of integrated variables of different order, whereas mathematically the demonstration was valid.

38Granger was interested in the multivariate cointegration case with different orders of integration for variables within a system: in this demonstration, he achieved the rank condition of the coefficient matrix of the variables, and he used “r” in reference to the cointegration rank. Johansen resumed this a few years later, in 1988. This rank condition allowed the identification of systems of cointegrated variables. Thus, if the rank was at least one unit lower than that of the size of the system, then there was at least one linear combination between these variables. If there was one linear combination between integrated variables, then there was one cointegration relationship. He then put forward two theorems, which were obviously the first draft of what became the “Granger Representation Theorem” in the official 1987 paper. He completed the ‘cointegration/error correction model’ connection using this corollary: “The error-correcting form of vector autoregressive models has thus been derived from co-integratedness, or equilibrium, constraints between variables. Equally, the reverse will be true in that if data is generated by an error correcting model, then it will be cointegrated or multi-cointegrated” [Granger, 1983, p. 16]. Granger then attached this demonstration of a relationship between variables to his concept of causality, which validated the connection with economic theory and thus closed on the existence of “spurious regressions”. [22] A last short section was devoted to estimating cointegration relationships. As he admitted thereafter, he was not very at ease with the testing procedures. The first sentence of this section clearly stated that, although the subject was interesting, it would be discussed later in another paper. Then Granger continued the reasoning from a theoretical standpoint: he literally distinguished a first procedure necessary to identify the order of integration of time series and of course he suggested unit root tests.

39In the last paragraph (only two sentences), he wondered about the estimate of the cointegration vector and the identification of its slow changes. No procedure was suggested here. It would have been necessary to await Engle’s contribution, suggesting the estimate and an inventory of tests for cointegration the following year.

40The conclusion of the paper was short. Granger referred to the interest of the concept of cointegration for the economic theory, particularly the equilibrium theory, via the error correction model. Granger had obviously achieved an important breakthrough in the analysis of time series, but he was perhaps overwhelmed by the impact of what he had just found.

41Granger had formulated the whole cointegration theory within six pages of a document dated 1983, which will never be published: definition of cointegrated variables, formalization of cointegrated systems related with error correction models, and mathematical law of identification of cointegrated systems. It was, however, only the second paper of Granger on the subject. As Granger indicated himself, he was “surprised” to verify the existence of this phenomenon whereas he did not believe in it. This paper was really a working paper: it was thick (thirty-three pages) but priority was given to the theorization and formalization. Although all the elements were there, we were far from the esthetics of the paper of 1987, written in collaboration with Engle and revised several times for Econometrica.

Comparison of the models of Sargan [1964] and Granger [1983]

42Sargan’s main paper on error correction mechanisms goes back to 1964. Note that Box and Jenkins’ work on the ARMA model appeared later (1970). In addition, Sargan’s work remained unknown until Hendry went to the United States in 1975. This was a crucial point in the discussion between Hendry and Granger. We compare the model of Sargan [1964] with the “theorem of representation” of Granger [1983] in order to fully understand how and why the concept of cointegration was so revolutionary. We uniformized the notations starting from the model of Sargan.

43In 1964, Sargan described error correction mechanisms as:

44

45In 1983, Granger developed the error correction model, including cointegration as a long-term equilibrium relationship:

46

47➀ : Granger introduced the cointegration relationship as a long-term equilibrium relationship between series. This relationship exists only if variables are: (i) integrated and of the same order, (ii) they obey a significant causality relationship, and (iii) the linear combination is stationary over time (to enable the development of the first cointegration test: CRDW of Engle and Granger [1987]). Conditions (i) and (iii) were testable only via the unit root tests of Dickey and Fuller [1979, 1981]. Conditions (ii) and (iii) referred to the spurious regressions described by Granger and Newbold [1974].

48In the 1983 unpublished paper, Granger wrote in the conclusion:

49

Thus, if it is desired to specify models involving equilibrium theories, [error correction] models are shown to be relevant. At the very least, this theory suggests a richer class of models than now used by time-series analysts and a more closely specified class of models than now used by the more classical econometricians.

Conclusion

58The meeting between Hendry and Granger in November 1975 was instrumental in the advent of the concept of cointegration. First of all, Hendry knew about Sargan’s work at the London School of Economics, including the reasoning underlying equilibrium correction: Sargan understood that deviations from long-term equilibrium embodied the intrinsic dynamics of the system. At the University of California in San Diego, Granger perceived “irregularities” in the regressions, particularly that a very high regression coefficient could be the sign of a residual autocorrelation, which led to interest being focused on Durbin and Watson statistics. Granger published a paper on these spurious regressions in collaboration with Newbold in 1974. When Hendry met Granger at the conference at Davis in 1975, he was the only one able to understand that there was complementarity between the ideas of Sargan and Granger, more so as he realized that Sargan’s work was unknown in the United States. Both Sargan and Granger identified estimate problems due to residual autocorrelation derived from nonstationary variables. Sargan [1964] proposed to solve this problem using first difference variables and their correction to the equilibrium path, while Granger warned against the danger of estimating, in the same model, variables not having the same level of integration. It was thanks to Hendry, who struggled for several years to convince Granger that Sargan’s and Granger’s work could be fused. The way cointegration took form is a peculiar feature of the story: it was while wanting to show that Hendry was mistaken that Granger found the cointegration relation. The collaboration with Engle produced the first test procedure required for both scientific and academic validation of the concept of cointegration.

59The study of these various steps made it possible to understand the development of Granger‘s ideas. In his first writings, Granger was still dealing with spectral analysis and he interpreted this famous “co-integration” relationship starting from a frequency approach. In 1983, Granger had all the elements required to formalize what was going to become a concept: there could be a common long-term trend between two variables even though they are nonstationary individually. The four-year gap until the 1987 publication was used to enhance the ideas and to engage a very profitable collaboration with Engle, which led them to be awarded the Nobel Prize of Economy in 2003.