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==History of ECM==
Yule (1936) and Granger and 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|trend]] and a stationary series containing deviations from trend. Even in deterministically detrended [[random walk]]s spurious correlations will eventually emerge. Thus detrending doesn't solve the estimation problem.
In order to still use the [[Box–Jenkins| Box–Jenkins approach]], one could difference the series and then estimate models such as [[ARIMA]], given that many commonly used time series (e.g. in economics) appear to be stationary in first differences. Forecasts from such a model will still reflect cycles and seasonality that are present in the data. However, any information about long-run adjustments that the data in levels may contain is omitted and longer term forecasts will be unreliable. ▼
▲In order to still use the [[Box–Jenkins| Box–Jenkins approach]], one could difference the series and then estimate models such as [[ARIMA]], given that many commonly used time series (e.g. in economics) appear to be stationary in first differences. Forecasts from such a model will still reflect cycles and seasonality that are present in the data. However, any information about long-run adjustments that the data in levels may contain is omitted and longer term forecasts will be unreliable.
==Estimation==
Several methods are known in the literature for estimating a refined dynamic model as described above. Among these are the 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===
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This structure is common to all ECM models. In practice, econometricians often first estimate the cointegration relationship (equation in levels), and then insert it into the main model (equation in differences).
==References==
{{Reflist}}
==Further reading==
▲* {{cite journal |last=Davidson |first=J. E. H. |first2=D. F. |last2=Hendry |authorlink2=David Forbes Hendry |first3=F. |last3=Srba |first4=J. S. |last4=Yeo |year=1978 |title=Econometric modelling of the aggregate time-series relationship between consumers' expenditure and income in the United Kingdom |journal=[[Economic Journal]] |volume=88 |issue=352 |pages=661–692 |jstor=2231972 }}
* {{cite book |last=Dolado |first=Juan J. |last2=Gonzalo |first2=Jesús |last3=Marmol |first3=Francesc |chapter=Cointegration |pages=634–654 |title=A Companion to Theoretical Econometrics |editor-first=Badi H. |editor-last=Baltagi |___location=Oxford |publisher=Blackwell |year=2001 |isbn=0-631-21254-X |doi=10.1002/9780470996249.ch31 }}
* {{cite book |first=Walter |last=Enders |title=Applied Econometric Time Series |edition=Third |___location=New York |publisher=John Wiley & Sons |year=2010 |isbn=978-0-470-50539-7 |pages=272–355 }}
* {{cite book |last=Lütkepohl |first=Helmut |authorlink=Helmut Lütkepohl |title=New Introduction to Multiple Time Series Analysis |___location=Berlin |publisher=Springer |edition= |year=2006 |isbn=978-3-540-26239-8 |pages=237–352 }}
* {{cite book |last=Martin |first=Vance |last2=Hurn |first2=Stan |last3=Harris |first3=David |title=Econometric Modelling with Time Series |___location=New York |publisher=Cambridge University Press |year=2013 |isbn=978-0-521-13981-6 |pages=662–711 }}
[[Category:Error detection and correction]]
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