Cointegration

Cointegration is a statistical property of a collection (X1, X2, ..., Xk) 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 co-integrated. 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 : I ( 1 ) {displaystyle 1:I(1)} . They also showed that unit root processes have non-standard statistical properties, so that conventional econometric theory methods do not apply to them. Cointegration is a statistical property of a collection (X1, X2, ..., Xk) 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 co-integrated. 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 : I ( 1 ) {displaystyle 1:I(1)} . They also showed that unit root processes have non-standard statistical properties, so that conventional econometric theory methods do not apply to them. 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 first-order integrated ( I ( 1 ) {displaystyle I(1)} ) 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. The first to introduce and analyse the concept of spurious—or nonsense—regression was Udny Yule in 1926.Before the 1980s, many economists used linear regressions on non-stationary time series data, which Nobel laureate Clive Granger and Paul Newbold showed to be a dangerous approach that could produce spurious correlation, since standard detrending techniques can result in data that are still non-stationary. Granger's 1987 paper with Robert Engle formalized the cointegrating vector approach, and coined the term. For integrated I ( 1 ) {displaystyle I(1)} processes, Granger and Newbold showed that de-trending does not work to eliminate the problem of spurious correlation, and that the superior alternative is to check for co-integration. Two series with I ( 1 ) {displaystyle I(1)} trends can be co-integrated only if there is a genuine relationship between the two. Thus the standard current methodology for time series regressions is to check all-time series involved for integration. If there are I ( 1 ) {displaystyle I(1)} 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). The usual procedure for testing hypotheses concerning the relationship between non-stationary variables was to run ordinary least squares (OLS) regressions on data which had been differenced. This method is biased if the non-stationary 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 R-squared relationship (suggesting high explanatory power on Fiji's consumption from Afghanistan's GNP). This is called spurious regression: two integrated I ( 1 ) {displaystyle I(1)} series which are not directly causally related may nonetheless show a significant correlation; this phenomenon is called spurious correlation.