Error correction model: Difference between revisions

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[[Udny Yule|Yule]] (1926) 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 process|trend]] and a stationary series containing deviations from trend. Even in deterministically detrended [[random walk]]s spurious correlations will eventually emerge. Thus detrending doesn'tdoes not 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.

This led [[John Denis Sargan|Sargan]] (1964) to develop the ECM methodology, which retains the level information.<ref>Sargan, J. D. (1964). "Wages and Prices in the United Kingdom: A Study in Econometric Methodology", 16, 25–54. in ''Econometric Analysis for National Economic Planning'', ed. by P. E. Hart, G. Mills, and J. N. Whittaker. London: Butterworths</ref><ref>{{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 }}</ref>