Cointegration is a statistical property of a collection (X_{1}, X_{2}, ..., X_{k}) of time series variables. First, all of the series must be integrated of order d (see Order of integration). Next, if a linear combination of this collection is integrated of order less than d, then the collection is said to be cointegrated. Formally, if (X,Y,Z) are each integrated of order d, and there exist coefficients a,b,c such that aX + bY + cZ is integrated of order less than d, then X, Y, and Z are cointegrated. Cointegration has become an important property in contemporary time series analysis. Time series often have trends—either deterministic or stochastic. In an influential paper, Charles Nelson and Charles Plosser (1982) provided statistical evidence that many US macroeconomic time series (like GNP, wages, employment, etc.) have stochastic trends—these are also called unit root processes, or processes integrated of order .^{[1]} They also showed that unit root processes have nonstandard statistical properties, so that conventional econometric theory methods do not apply to them.
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Transcription
Contents
Introduction
If two or more series are individually integrated (in the time series sense) but some linear combination of them has a lower order of integration, then the series are said to be cointegrated. A common example is where the individual series are firstorder integrated () but some (cointegrating) vector of coefficients exists to form a stationary linear combination of them. For instance, a stock market index and the price of its associated futures contract move through time, each roughly following a random walk. Testing the hypothesis that there is a statistically significant connection between the futures price and the spot price could now be done by testing for the existence of a cointegrated combination of the two series.
History
The first to introduce and analyse the concept of spurious—or nonsense—regression was Udny Yule in 1926.^{[2]} Before the 1980s, many economists used linear regressions on nonstationary time series data, which Nobel laureate Clive Granger and Paul Newbold showed to be a dangerous approach that could produce spurious correlation,^{[3]}^{[4]} since standard detrending techniques can result in data that are still nonstationary.^{[5]} Granger's 1987 paper with Robert Engle formalized the cointegrating vector approach, and coined the term.^{[6]}
For integrated processes, Granger and Newbold showed that detrending does not work to eliminate the problem of spurious correlation, and that the superior alternative is to check for cointegration. Two series with trends can be cointegrated only if there is a genuine relationship between the two. Thus the standard current methodology for time series regressions is to check alltime series involved for integration. If there are series on both sides of the regression relationship, then it's possible for regressions to give misleading results.
The possible presence of cointegration must be taken into account when choosing a technique to test hypotheses concerning the relationship between two variables having unit roots (i.e. integrated of at least order one).^{[3]} The usual procedure for testing hypotheses concerning the relationship between nonstationary variables was to run ordinary least squares (OLS) regressions on data which had been differenced. This method is biased if the nonstationary variables are cointegrated.
For example, regressing the consumption series for any country (e.g. Fiji) against the GNP for a randomly selected dissimilar country (e.g. Afghanistan) might give a high Rsquared relationship (suggesting high explanatory power on Fiji's consumption from Afghanistan's GNP). This is called spurious regression: two integrated series which are not directly causally related may nonetheless show a significant correlation; this phenomenon is called spurious correlation.
Tests
The three main methods for testing for cointegration are:
Engle–Granger twostep method
If and are nonstationary and Order_of_integration d=1, then a linear combination of them must be stationary for some value of and . In other words:
where is stationary.
If we knew , we could just test it for stationarity with something like a Dickey–Fuller test, Phillips–Perron test and be done. But because we don't know , we must estimate this first, generally by using ordinary least squares, and then run our stationarity test on the estimated series, often denoted .
A second regression is then run on the first differenced variables from the first regression, and the lagged residuals is included as a regressor.
Johansen test
The Johansen test is a test for cointegration that allows for more than one cointegrating relationship, unlike the Engle–Granger method, but this test is subject to asymptotic properties, i.e. large samples. If the sample size is too small then the results will not be reliable and one should use Auto Regressive Distributed Lags (ARDL).^{[7]}^{[8]}
Phillips–Ouliaris cointegration test
Peter C. B. Phillips and Sam Ouliaris (1990) show that residualbased unit root tests applied to the estimated cointegrating residuals do not have the usual Dickey–Fuller distributions under the null hypothesis of nocointegration.^{[9]} Because of the spurious regression phenomenon under the null hypothesis, the distribution of these tests have asymptotic distributions that depend on (1) the number of deterministic trend terms and (2) the number of variables with which cointegration is being tested. These distributions are known as Phillips–Ouliaris distributions and critical values have been tabulated. In finite samples, a superior alternative to the use of these asymptotic critical value is to generate critical values from simulations.
Multicointegration
In practice, cointegration is often used for two series, but it is more generally applicable and can be used for variables integrated of higher order (to detect correlated accelerations or other seconddifference effects). Multicointegration extends the cointegration technique beyond two variables, and occasionally to variables integrated at different orders.
Variable shifts in long time series
Tests for cointegration assume that the cointegrating vector is constant during the period of study. In reality, it is possible that the longrun relationship between the underlying variables change (shifts in the cointegrating vector can occur). The reason for this might be technological progress, economic crises, changes in the people's preferences and behaviour accordingly, policy or regime alteration, and organizational or institutional developments. This is especially likely to be the case if the sample period is long. To take this issue into account, tests have been introduced for cointegration with one unknown structural break,^{[10]} and tests for cointegration with two unknown breaks are also available.^{[11]}
See also
References
 ^ Nelson, C. R.; Plosser, C. R. (1982). "Trends and random walks in macroeconmic time series". Journal of Monetary Economics. 10 (2): 139. doi:10.1016/03043932(82)900125.
 ^ Yule, U. (1926). "Why do we sometimes get nonsensecorrelations between time series?  A study in sampling and the nature of time series". Journal of the Royal Statistical Society. 89 (1): 11–63. doi:10.2307/2341482. JSTOR 2341482.
 ^ ^{a} ^{b} Granger, C.; Newbold, P. (1974). "Spurious Regressions in Econometrics". Journal of Econometrics. 2 (2): 111–120. CiteSeerX 10.1.1.353.2946. doi:10.1016/03044076(74)900347.
 ^ Mahdavi Damghani, Babak; et al. (2012). "The Misleading Value of Measured Correlation". Wilmott. 2012 (1): 64–73. doi:10.1002/wilm.10167.
 ^ Granger, Clive (1981). "Some Properties of Time Series Data and Their Use in Econometric Model Specification". Journal of Econometrics. 16 (1): 121–130. doi:10.1016/03044076(81)900798.
 ^ Engle, Robert F.; Granger, Clive W. J. (1987). "Cointegration and error correction: Representation, estimation and testing". Econometrica. 55 (2): 251–276. doi:10.2307/1913236. JSTOR 1913236.
 ^ Giles, David. "ARDL Models  Part II  Bounds Tests". Retrieved 4 August 2014.
 ^ Pesaran, M.H.; Shin, Y.; Smith, R.J. (2001). "Bounds testing approaches to the analysis of level relationships". Journal of Applied Econometrics. 16 (3): 289–326. doi:10.1002/jae.616.
 ^ Phillips, P. C. B.; Ouliaris, S. (1990). "Asymptotic Properties of Residual Based Tests for Cointegration". Econometrica. 58 (1): 165–193. doi:10.2307/2938339. JSTOR 2938339.
 ^ Gregory, Allan W.; Hansen, Bruce E. (1996). "Residualbased tests for cointegration in models with regime shifts". Journal of Econometrics. 70 (1): 99–126. doi:10.1016/03044076(69)416857.
 ^ HatemiJ, A. (2008). "Tests for cointegration with two unknown regime shifts with an application to financial market integration". Empirical Economics. 35 (3): 497–505. doi:10.1007/s0018100701759.
Further reading
 Enders, Walter (2004). "Cointegration and ErrorCorrection Models". Applied Econometrics Time Series (Second ed.). New York: Wiley. pp. 319–386. ISBN 9780471230656.
 Hayashi, Fumio (2000). Econometrics. Princeton University Press. pp. 623–669. ISBN 9780691010182.
 Maddala, G. S.; Kim, InMoo (1998). Unit Roots, Cointegration, and Structural Change. Cambridge University Press. pp. 155–248. ISBN 9780521587822.
 Murray, Michael P. (1994). "A Drunk and her Dog: An Illustration of Cointegration and Error Correction" (PDF). The American Statistician. 48 (1): 37–39. doi:10.1080/00031305.1994.10476017. An intuitive introduction to cointegration.