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 “longterm equilibrium”. The longterm 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 longterm 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 longterm equilibrium (stable) and shortterm 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 DurbinWatson 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 modernday dynamics.
5Since the midtwentieth century, economists have been interested in the analysis of dynamics and the issue of the structural decomposition of time series between shortterm and longterm elements [Persons, 1917; Wold, 1938; 1954; KoopmansRubinLeipnik, 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:
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 unpredicted 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 longterm 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 realwage resistance in wage bargains, interpreting the equilibrium correction‑the deviation of real wages from a productivity trend‑as a ‘catchup’ 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 longterm 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 (longterm 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 shortrun oscillations (first differences of the log of variables) which were more unstable than longrun equilibrium (variables in level) which had to converge.
13On the basis of the bivariate wageprice model [Sargan, 1964]:
14In this paper, Sargan provided the basis for a new macroeconometric dynamic modeling. The study integrating the longrun in the analysis produced a new view of macroeconomics. With the error correction concept, Sargan paved the way for realtime 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 shortrun dynamics elements and the longrun 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 DurbinWatson statistics indicated a misspecification 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 shortterm effect while the longterm relationship remained nonstationary. For Granger, the introduction of first differences into a nonstationary system could not correct the longterm 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 longrun 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 longrun 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 misspecification 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_{t1} = 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 longrun 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 longterm 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 errorcorrection mechanism it was because the variables evolved without any economic link. Hendry did not, however, see the mathematical condition of the rankreduction 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 longterm 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 longrun 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 longrun 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:
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 errorcorrection 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 “cointegration”, 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.
3.1 – When reason exceeds convictions…
26On March 20^{th} 2008, Granger was invited as Emeritus Professor to the Institute of Econometrics of Rotterdam (Netherlands) where he occupied the Henri Theil chair for ten days. On this occasion, he wrote a paper to celebrate the twenty year anniversary of cointegration, which appeared posthumously in a special issue of the Journal of Econometrics [2010, 158, p. 36] on the initiative of Boswijk, Franses and Van Dijk. [16] In this paper, Granger reconsidered the history of this concept. After having refuted the stability of the equilibrium correction model beyond an ad hoc experiment, Granger wondered about the possibility of stationarizing using only a linear combination (subtraction) of two integrated variables of order one, the famous assertion of Hendry. It was while trying to definitively and negatively reply the speculation of Hendry that all the pieces of the puzzle finally came together as if by magic: two nonstationary series in level could have a stationary linear combination if their respective evolution paths follow the same trajectory. Indeed, two variables driven by the same economic conditioning had a high probability to evolve in a similar way towards a common longterm equilibrium. Sargan’s equilibrium correction model formalized in 1964 was rejuvenated thanks to Hendry’s work. Without the determination of Hendry, cointegration would certainly not have emerged, at least not in 1981. Even though Hendry was convinced at that time that a linear combination of nonstationary variables could be stationary, he did not perceive the rank condition involved in the model [Hendry 2004]. Moreover, Granger did not realize that the outcome of cointegration depended on this reduction in the rank of the matrix coefficients of the model–it was necessary to wait a few years and Johansen’s work in 1988. At that moment, cointegration appeared only in a bivariate form.
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.
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 redid 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 resubmit 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]
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:
Cointegrated 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 cointegrated, 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, “CoIntegrated Variables and Models ErrorCorrectingˮ, 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_{t1}, he divided the dynamics of the system into a longrun component (trend) and a shortrun component (cyclical variations). The longrun component (target) was given by the lagged cointegration variable (z_{t1}), and the shortrun 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. 12]. 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 “Cointegration” 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 longterm. 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 errorcorrecting form of vector autoregressive models has thus been derived from cointegratedness, 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 multicointegrated” [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 (thirtythree 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:
45In 1983, Granger developed the error correction model, including cointegration as a longterm equilibrium relationship:
47➀ : Granger introduced the cointegration relationship as a longterm 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:
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 timeseries analysts and a more closely specified class of models than now used by the more classical econometricians.
3.3 – Completion of the cointegration theory: Robert Engle
50On October 5^{th}, 1984, Granger and Engle presented their respective results on cointegration at a conference on time series at Davis (University of California) organized by NBER and NSF. They presented two distinct lectures: Granger expounded his unpublished paper of 1983, while Engle proposed a test for cointegration (at that time, cointegration was only in a bivariate form). Engle then stayed at San Diego (University of California): his time series work led him to contact Granger. Thus, at the hour of cointegration, Granger and Engle were collaborating. Engle [2004] was very interested in Granger’s cointegration. He perceived that the unit root tests of Dickey and Fuller [1979, 1981] could be useful to test this relationship between two time series, as Nelson and Plosser had done earlier [1982]. So Engle developed the test procedure at that time, the CRDW test–Cointegration Regression Durbin Watson. This was the first test of bivariate cointegration, inspired from the work of Bhargava in 1984 then 1986. It was not until 1987 that the Econometrica journal published the paper, although it was mentioned several times in reference, e.g. in Engle & Granger [1985].
51In 1986, Granger took the opportunity of a stay in England at Nuffield College and the Institute of Economics and Statistics–where Hendry was teaching–to write a new version of his paper on cointegration. That was “Developments in the study of cointegrated variables”, published in the Oxford Bulletin of Economics and Statistics. In this paper, Granger again asserted real cases of cointegration that could be encountered in economics, but this time right from the beginning of the paper. In this introduction, he outlined the concept in a literary form:
The idea underlying cointegration allows specification of models that capture part of such beliefs, at least for a particular type of variable that is frequently found to occur in macroeconomics. Since a concept such as the longrun is a dynamic one, the natural area for these ideas is that of timeseries theory and analysis. It is thus necessary to start by introducing some relevant time series models.
53Granger again explained integrated series as compared to stationary series. As usual, the demonstrations were simple and understandable. In the second section of the paper, Granger introduced the formalization of a cointegration relationship: z_{t} = x_{t}  Ay_{t}. At that time, Granger described the relation only between two series, x_{t} and y_{t}, integrated of order one, with variable z_{t} being stationary or integrated of order zero. The concept of cointegration meant that the linear combination of integrated variables was stationary. The simplicity of the concept was obvious, i.e. simply the result of a subtraction operation between two variables, which was Hendry’s initial intuition. [23] This relation conveyed de facto the longterm relationship between series, which was called “target”. We understand therefore what could have repulsed Granger for such a long time, i.e. how two series which evolved in a nonstationary way could have a concomitant stationary evolution? In addition, the formulation was so simple that it could be perplexing–the Econometrica journal twice refused cointegration papers by Granger, considering them too simplistic in their theoretical formalization. In the 1986 version, Granger further developed the connection with equilibrium correction models [1986, p. 216], which he this time attributed to Phillips [1957] and Sargan [1964]. He developed a test procedure on the 1985 publication signed Granger & Engle and indicated as ‘forthcoming’ in Econometrica. [24] However, he preferred to go back to digressions of spectral analysis, “a testing procedure would involve estimating the equilibrium regression (5.2) using some TVP [25] techniques, such as a Kalman filter procedure, probably assuming that the components of α r [26] are stochastic but slowly changing.” [Granger, 1986, p. 225]
54He then endeavored to show the connection between cointegration and the error correction model, but he did not speak here about “Representation Theorem”. It seemed that Granger was bound by the various already written papers. Let us recall that the 1983 paper outlined the representation theorem, and that the 1985 paper (in collaboration with Engle) presented a cointegration test procedure. Indeed, Granger appeared in a hurry to publish on the subject, but this 1986 article seemed hesitant: Granger dwelt on the mathematical developments without giving the representation theorem; in the same way he outlined the test procedure while still relying on spectral analysis. Thus, the conclusion of the paper could leave readers in a quandary, being both modest and persuasive about the benefits of the concept of cointegration for the development in time series:
This paper has attempted to expand the discussion about differencing macroeconomic series when model building by emphasizing the use of a further factor, the’equilibrium error’, that arises from the concept of cointegration. This factor allows the introduction of the impact of long run or’equilibrium’ economic theories into the models used by the timeseries analysts to explain the shortrun dynamics of economic data. The resulting errorcorrection models should produce better shortrun forecasts and will certainly produce longrun forecasts that hold together in economically meaningful ways.
…Whilst the paper has not attempted to link errorcorrection models with optimizing economic theory, through control variables for example, there is doubtless much useful work to be done in this area. Testing for cointegration in general situations is still in an early stage of development. Whether or not cointegration occurs is an empirical question but the beliefs of economists do appear to support its existence and the usefulness of the concept appears to be rapidly gaining acceptance.
56Was it because Granger had decided to publish alone and quickly that the feeling with regard to this article remained ambivalent? When one seemed to understand something, all was reappraised in the following paragraph, with references in this paper of 1985 in collaboration with Engle. Two papers seemed to contain the true substance of cointegration, i.e. the ‘unpublished Discussion Paper’ of 1983 and the 1985 paper in collaboration with Engle.
57Finally, in 1987 the concept of cointegration was validated and recognized by the whole econometric community. Although all was written between 1983 and 1985, it was not until 1987 that Econometrica published the paper and recognized the concept. Granger and Engle had already both written on the subject: mathematical developments, test procedure and implications for economic theory. They had already presented most of their results at the time of the 1984 conference at Davis. However, Engle [2010] felt that, although the concept was interesting, the formalization seemed to be too obvious for the scientific community. Hence, although the theory of cointegration was written between 1980 and 1983, it was only officially published in 1987.
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 longterm 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 “cointegration” 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 longterm trend between two variables even though they are nonstationary individually. The fouryear 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.
Notes

[1]
CIRAD, UMR 5281 ARTDev – TA C113/15  34398 Montpellier – France. Email: veronique.meuriot@cirad.fr

[2]
As an illustration, consider the income/consumption system. Both series increase over time, they are nonstationary (or integrated). However, because of the economic theory between income and consumption, we could expect that they would evolve in the same way. If so, the difference between income and consumption over time must be a stable relationship (stationary).

[3]
DurbinWatson statistics identifies residual autocorrelations in time series from a OLS regression.

[4]
In a correspondence with the author, Granger indicated “I was never connected to Cowles or to the NBER as neither paid much attention to time series. In practice we have built up our own school developing causality, cointegration, ARCH, longmemory, etc.”

[5]
The expressions “time responses”, “lag forms” and “lagged responses” were very often used in Phillips’ 1957 paper.

[6]
AutoRegressive Moving Average: from the work of Wold (1954), time series can be decomposed into the past values (AR process) and past and present values of a random process of zero mean (MA process). In 1970, Box and Jenkins combined the results of Wold, and built a coherent theoretical corpus for ARMA processes and time series analysis. This concept has been the basis of unit root tests from 1979.

[7]
First differences concern the evolution of a variable from one period (t) to the previous period (t1); it refers to the short run. The level of a variable refers to the long run. So, the first log difference can be interpreted as growth rates of variables in level and thus refers to the short run.

[8]
In 1970, Hendry defended his thesis under Sargan’s supervision.

[9]
He ended this sabbatical year with a visit to the Australian National University of Canberra where he met, among others, Ted Hannan and Adrian Pagan.

[10]
New Methods in Business Cycle Research: papers from this conference are pooled in the eponymous book of Christopher Sims (Federal Reserve Bank of Minneapolis, 1977).

[11]
Hendry David F., [1977], “On the time series approach to econometric model building”. In A.C. Sims (ed.), New Methods in Business Cycle Research: p. 183202. Minneapolis: Federal Reserve Bank of Minneapolis.

[12]
The Englishman Paul Newbold obtained a PhD in Statistics under the supervision of George Box at the University of Wisconsin [1970]. He answered a call for a postdoc position initiated by Clive Granger. Their collaboration notably led to the 1974 article on “spurious regressions”.

[13]
Nelson and Plosser highlighted the problems of nonstationarity of series for macroeconomic analysis in 1982: “Trends and Random Walks in Macroeconomic Time Series: some Evidence and Implications”, Journal of Monetary Economics, 10(2), p. 139162.

[14]
This article became famous as DHSY [1978]: Davidson, Hendry, Srba, Yeo “Econometric modeling of the aggregate timeseries relationship between consumers’ expenditure and income in the United Kingdom”. Economic Newspaper, 88, p. 661692.

[15]
At that time, Hendry was always at the London School Economics and evolved towards dynamic macroeconomics.

[16]
Peter Boswijk is Professor of Econometrics at the University of Amsterdam; Philip Hans Franses (Dean of the Institute of Econometrics of Rotterdam) and Dick Van Dijk are both Professors of Econometrics at the Institute.

[17]
We consider here the story of Hendry who met Granger in 1981 and for whom cointegration was already formulated. Of course, the basic article was published in 1981 in the Journal of Econometrics [16, p. 121130].

[18]
The Econometrics Journal.

[19]
Granger Clive W.J., [1983], “CoIntegrated Variables and ErrorCorrecting Models”, Unpublished UCSD Discussion Paper 8313.

[20]
“d” indicates the degree of integration. If d = 1, the series is said to be “integrated of order 1” and it is nonstationary because of an infinite variance: x_{t}≈I 1. If d = 0, the series is stationary and its variance is finite: X_{t}≈I 0. When 0<d<1, the series follows an autoregressive process: X_{t} ≈ I d<1. The difficulty is to determine if the variance of the series is stationary (d < ½) or not (d > 1/2). α is the constant, β is the coefficient of shape (trend) and f(B) is a lag operator which describes the evolution of ε_{t} (residuals) over time.

[21]
The representation theorem specifies that, if two series are cointegrated (i.e. a linear combination of the series is stationary when individual series are nonstationary), then we can cover all of the dynamics in an equation system as follows :z_{t1} denotes the longrun equilibrium relationship (target), and denotes the shortrun deviations of the system. The corollary is that data generated by an error correction model are de facto cointegrated.

[22]
Granger Clive W.J. and Paul Newbold, [1974], “Spurious Regressions in Econometric Model Specification”, Journal of Econometrics, 2, p. 111120.Online

[23]
“A colleague, David Hendry, stated that the difference between a pair of integrated series could be stationary”, in Granger [2003], speech for the Nobel Prize.

[24]
Granger refers here to the famous article Granger, C.W.J. and RF. Engle [1985]: “Dynamic model specification with equilibrium constraints”, Mimeo. (University of California, San Diego, CA), submitted to Econometrica, but which was rejected twice and which was finally published only in 1987. At this period, Granger referred to it as: “Granger, C. W. J. and Engle, R. F. [1985]. Dynamic Specification with Equilibrium Constraints: Cointegration and ErrorCorrection. (forthcoming, Econometrica)”.

[25]
Timevarying parameter.

[26]
α r are the “r” cointegrating vectors, “r” indicates the number of cointegrating relationships in the model.