Error correction model: Difference between revisions

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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|doi=10.2307/2341482 |jstor=2341482 }}</ref><ref>{{cite journal |lastlast1=Granger |firstfirst1=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 |doi=10.1016/0304-4076(74)90034-7 |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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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 |lastlast1=Davidson |firstfirst1=J. E. H. |first2=D. F. |last2=Hendry |author-link2=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 |doi=10.2307/2231972 |jstor=2231972 }}</ref>

==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 |lastlast1=Engle |firstfirst1=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 |doi=10.2307/1913236 |jstor=1913236 |url=http://pe.cemi.rssi.ru/pe_2015_3_106-135.pdf }}</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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: <math> A(L) \, \Delta y_t = \gamma + B(L) \, \Delta x_t + \alpha (y_{t-1} -\beta_0 - \beta_1 x_{t-1} ) + \nu_t. </math>

[define A and B]

''If'' both variables are integrated and this ECM exists, they are cointegrated by the Engle–Granger representation theorem.

The second step is then to estimate the model using [[ordinary least squares]]: <math> y_t = \beta_0 + \beta_1 x_t + \varepsilon_t </math>

If the regression is not spurious as determined by test criteria described above, [[Ordinary least squares]] will not only be valid, but in fact superalso [[consistent estimator|consistent]] (Stock, 1987).

Then the predicted residuals <math>\hat{\varepsilon_t}= y_t -\beta_0 - \beta_1 x_t </math> from this regression are saved and used in a regression of differenced variables plus a lagged error term

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One can then test for cointegration using a standard [[t-statistic]] on <math>\alpha</math>.

While this approach is easy to apply, there are, however numerous problems:

* The univariate unit root tests used in the first stage have low [[statistical power]]

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==Further reading==

* {{cite book |lastlast1=Dolado |firstfirst1=Juan J. |last2=Gonzalo |first2=Jesús |last3=Marmol |first3=Francesc |chapter=Cointegration |pages=[https://archive.org/details/companiontotheor00balt/page/n646 634]–654 |title=A Companion to Theoretical Econometrics |url=https://archive.org/details/companiontotheor00balt |url-access=limited |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 |author-link=Helmut Lütkepohl |title=New Introduction to Multiple Time Series Analysis |url=https://archive.org/details/newintroductiont00ltke |url-access=limited |___location=Berlin |publisher=Springer |year=2006 |isbn=978-3-540-26239-8 |pages=[https://archive.org/details/newintroductiont00ltke/page/n251 237]–352 }}

* {{cite book |lastlast1=Martin |firstfirst1=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]]