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==History==
[[Udny Yule|Yule]] (1926) and [[Clive Granger|Granger]] and [[Paul Newbold|Newbold]] (1974) were the first to draw attention to the problem of [[spurious correlation]] and find solutions on how to address it in time series analysis.<ref>{{cite journal|last1=Yule|first1=Georges Udny|title=Why do we sometimes get nonsense correlations between time series? – A study in sampling and the nature of time-series|journal=Journal of the Royal Statistical Society|date=1926|volume=89|issue=1|pages=1–63|jstor=2341482 }}</ref><ref>{{cite journal |last=Granger |first=C.W.J. |first2=P.|last2=Newbold |year=1978 |title=Spurious regressions in Econometrics | volume=2| issue=2| journal=[[Journal of Econometrics]] |pages=111–120 |jstor=2231972 }}</ref> Given two completely unrelated but integrated (non-stationary) time series, the [[regression analysis]] of one on the other will tend to produce an apparently statistically significant relationship and thus a researcher might falsely believe to have found evidence of a true relationship between these variables. [[Ordinary least squares]] will no longer be consistent and commonly used test-statistics will be non-valid. In particular, [[Monte Carlo method|Monte Carlo simulations]] show that one will get a very high [[coefficient of determination|R squared]], very high individual [[t-statistic]] and a low [[Durbin–Watson statistic]]. Technically speaking, Phillips (1986) proved that parameter estimates will not [[Convergence in probability|converge in probability]], the [[Y-intercept|intercept]] will diverge and the slope will have a non-degenerate distribution as the sample size increases.<ref>{{cite journal|last1=Phillips|first1=Peter C.B.|title=Understanding Spurious Regressions in Econometrics|journal=Cowles Foundation Discussion Papers 757|date=1985|url=http://cowles.yale.edu/sites/default/files/files/pub/d07/d0757.pdf|publisher=Cowles Foundation for Research in Economics, Yale University}}</ref> However, there might be a common [[cointegration|stochastic trend]] to both series that a researcher is genuinely interested in because it reflects a long-run relationship between these variables.
Because of the stochastic nature of the trend it is not possible to break up integrated series into a deterministic (predictable) [[trend-stationary process|trend]] and a stationary series containing deviations from trend. Even in deterministically detrended [[random walk]]s spurious correlations will eventually emerge. Thus detrending does not solve the estimation problem.
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==Estimation==
Several methods are known in the literature for estimating a refined dynamic model as described above. Among these are the [[Robert F. Engle|Engle]] and Granger 2-step approach, estimating their ECM in one step and the vector-based VECM using [[Johansen test|Johansen's method]].<ref>{{cite journal |last=Engle |first=Robert F. |last2=Granger |first2=Clive W. J. |year=1987 |title=Co-integration and error correction: Representation, estimation and testing |journal=[[Econometrica]] |volume=55 |issue=2 |pages=251–276 |jstor=1913236 }}</ref>
===Engle and Granger 2-step approach===
The first step of this method is to pretest the individual time series one uses in order to confirm that they are [[Stationary process|non-stationary]] in the first place. This can be done by standard [[unit root]] [[Dickey–Fuller test|DF]] testing and [[ADF test]] (to resolve the problem of serially correlated errors).
Take the case of two different series <math>x_t</math> and <math>y_t</math>. If both are I(0), standard regression analysis will be valid. If they are integrated of a different order, e.g. one being I(1) and the other being I(0), one has to transform the model.
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