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{{Short description|Type of time series model}}
An '''error correction model''' ('''ECM''') belongs to a category of multiple [[time series]] models most commonly used for data where the underlying variables have a long-run common stochastic trend, also known as
==History
[[Udny Yule|Yule]] (
Because of the stochastic nature of the trend it is not possible to break up integrated series into a deterministic (predictable) [[trend
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|
==Estimation==
Several methods are known in the literature for estimating a refined dynamic model as described above. Among these are the
===
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
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.
If they are both integrated to the same order (commonly I(1)), we can estimate an ECM model of the form
: <math> A(L) \, \Delta y_t = \gamma + B(L) \, \Delta x_t + \alpha (
[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
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
* The univariate unit root tests used in the first stage have low [[statistical power]]
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* The cointegration test on <math>\alpha </math> does not follow a standard distribution
* The validity of the long-run parameters in the first regression stage where one obtains the residuals cannot be verified because the distribution of the OLS estimator of the cointegrating vector is highly complicated and non-normal
* At most one cointegrating relationship can be examined.{{Citation needed|date=March 2019}}
===VECM===
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* Step 1: estimate an unrestricted VAR involving potentially non-stationary variables
* Step 2: Test for cointegration using [[Johansen test]]
* Step 3: Form and analyse the VECM.
===An example of ECM===
The idea of cointegration may be demonstrated in a simple macroeconomic setting. Suppose, consumption <math>C_t</math> and disposable income <math>Y_t</math> are macroeconomic time series that are related in the long run (see [[Permanent income hypothesis]]). Specifically, let [[average propensity to consume]] be 90%, that is, in the long run <math>C_t = 0.9 Y_t</math>. From the econometrician's point of view, this long run relationship (aka cointegration) exists if errors from the regression <math>C_t = \beta Y_t+\varepsilon_t</math> are a [[Stationary process|stationary]] series, although <math>Y_t</math> and <math>C_t</math> are non-stationary. Suppose also that if <math>Y_t</math> suddenly changes by <math>\Delta Y_t</math>, then <math>C_t</math> changes by <math>\Delta C_t = 0.5 \, \Delta Y_t</math>, that is, [[marginal propensity to consume]] equals 50%. Our
In this setting a change <math>\Delta C_t = C_t - C_{t-1}</math> in consumption level can be modelled as <math>\Delta C_t = 0.5 \, \Delta Y_t - 0.2 (C_{t-1}-0.9 Y_{t-1}) +\varepsilon_t</math>. The first term in the RHS describes short-run impact of change in <math>Y_t</math> on <math>C_t</math>, the second term explains long-run gravitation towards the equilibrium relationship between the variables, and the third term reflects random shocks that the system receives (e.g. shocks of consumer confidence that affect consumption). To see how the model works, consider two kinds of shocks: permanent and transitory (temporary). For simplicity, let <math>\varepsilon_t</math> be zero for all t. Suppose in period ''t'' − 1 the system is in equilibrium, i.e. <math>C_{t-1} = 0.9 Y_{t-1}</math>. Suppose that in the period t, disposable income <math>Y_t</math> increases by 10 and then returns to its previous level. Then <math>C_t</math> first (in period t) increases by 5 (half of 10), but after the second period <math>C_t</math> begins to decrease and converges to its initial level. In contrast, if the shock to <math>Y_t</math> is permanent, then <math>C_t</math> slowly converges to a value that exceeds the initial <math>C_{t-1}</math> by 9.
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 book |
▲* {{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 |
* {{cite
▲* {{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 }}
[[Category:Error detection and correction]]
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