Damodar Gujarati
Econometrics by Example

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There is a unique case where a regression of a nonstationary series on another nonstationary series does not result in spurious regression.
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This is the situation of cointegration .
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If two time series have stochastic trends (i.e., they are nonstationary), a regression of one on the other may cancel out the stochastic trends, which may suggest that there is a long-run, or equilibrium, relationship between them even though individually the two series are nonstationary.
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Keep in mind that unit root and nonstationarity are not synonymous. A stochastic process with a deterministic trend is nonstationary but not unit root.
Damodar Gujarati
Econometrics by Example

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In the context of testing for cointegration, the Dickey-Fuller
(DF) and augmented Dickey-Fuller (ADF) tests are known as
Engle-Granger (EG) and augmented Engle-Granger (AEG) tests, which are now incorporated in several software packages.
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If you have more than three variables, there might be more than one cointegrating relationship.
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Once we go beyond two time series, we will have to use
Johansen methodology to test for cointegrating relationships among multiple variables.
Damodar Gujarati
Econometrics by Example

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Tests for unit roots are performed on single time series, whereas cointegration deals with the relationship among a group of variables, each having a unit root.
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It is better to test each series for unit roots, as some of the series in a group may have more than one unit root, in which case they will have to be differenced more than once to make them stationary.
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If two time series Y and X are integrated of different orders then the error term in the regression of Y and X is not stationary and this regression equation is said to be unbalanced .
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On the other hand, if the two variables are integrated of the same order, then the regression equation is said to be balanced .
Damodar Gujarati
Econometrics by Example

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Granger Representation Theorem : If two variables Y and X are cointegrated, the relationship between the two can be expressed as an error correction mechanism (ECM).
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The ECM postulates that changes in the dependent variable depend on changes in the independent variable and the lagged equilibrium error term, u t-1
.
 If this error term is zero, there will not be any disequilibrium between the two variables and in that case the long-run relationship will be given by the cointegrating relationship.
 But if the equilibrium error term is nonzero, the relationship between the two variables will be out of equilibrium.
 For multiple time series, we need to use the vector error correction model (VECM).
Damodar Gujarati
Econometrics by Example