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Cointegration
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<p>In <a href="/facts/Econometrics/7U52gzHt">econometrics</a>, cointegration is a <a href="/facts/Statistical/DNPsKGYU">statistical</a> property describing a long-term, stable relationship between two or more <a href="/facts/Time_series/fSXPR817">time series</a> variables, even if those variables themselves are individually <a href="/facts/Non-stationary/vozdgfEX">non-stationary</a> (i.e., they have trends). This means that despite their individual fluctuations, the variables move together in the long run, anchored by an underlying equilibrium relationship. </p><p>More formally, if several time series are individually <a href="/facts/Order_of_integration/Y8WBTpXj">integrated of order</a> <i>d</i> (meaning they require <i>d</i> <a href="/facts/Differencing/Ep7xCa8W">differences</a> to become stationary) but a <a href="/facts/Linear_combination/CVjL6kuN">linear combination</a> of them is integrated of a lower order, then those time series are said to be cointegrated. That is, if (<i>X</i>,<i>Y</i>,<i>Z</i>) are each integrated of order <i>d</i>, and there exist coefficients <i>a</i>,<i>b</i>,<i>c</i> such that <i>aX</i> + <i>bY</i> + <i>cZ</i> is integrated of order less than d, then <i>X</i>, <i>Y</i>, and <i>Z</i> are cointegrated. </p><p>Cointegration is a crucial concept in time series analysis, particularly when dealing with variables that exhibit trends, such as <a href="/facts/Macroeconomics/4dcQAdCr">macroeconomic</a> data. In an influential paper, Charles Nelson and <a href="/facts/Charles_Plosser/eJuoCPAc">Charles Plosser</a> (1982) provided statistical evidence that many US macroeconomic time series (like GNP, wages, employment, etc.) have stochastic trends. </p>