Cointegration
Cointegration is a
Introduction
If two or more series are individually integrated (in the time series sense) but some linear combination of them has a lower order of integration, then the series are said to be cointegrated. A common example is where the individual series are first-order integrated () but some (cointegrating) vector of coefficients exists to form a
History
The first to introduce and analyse the concept of spurious—or nonsense—regression was Udny Yule in 1926.[2] Before the 1980s, many economists used
For integrated processes, Granger and Newbold showed that de-trending does not work to eliminate the problem of spurious correlation, and that the superior alternative is to check for co-integration. Two series with trends can be co-integrated only if there is a genuine relationship between the two. Thus the standard current methodology for time series regressions is to check all-time series involved for integration. If there are series on both sides of the regression relationship, then it is possible for regressions to give misleading results.
The possible presence of cointegration must be taken into account when choosing a technique to test hypotheses concerning the relationship between two variables having unit roots (i.e. integrated of at least order one).[3] The usual procedure for testing hypotheses concerning the relationship between non-stationary variables was to run ordinary least squares (OLS) regressions on data which had been differenced. This method is biased if the non-stationary variables are cointegrated.
For example, regressing the consumption series for any country (e.g. Fiji) against the GNP for a randomly selected dissimilar country (e.g. Afghanistan) might give a high
Tests
The six main methods for testing for cointegration are:
Engle–Granger two-step method
If and both have order of integration d=1 and are cointegrated, then a linear combination of them must be stationary for some value of and . In other words:
where is stationary.
If is known, we can test for stationarity with an Augmented Dickey–Fuller test or Phillips–Perron test. If is unknown, we must first estimate it. This is typically done by using ordinary least squares (by regressing on and an intercept). Then, we can run an ADF test on . However, when is estimated, the critical values of this ADF test are non-standard, and increase in absolute value as more regressors are included.[7]
If the variables are found to be cointegrated, a second-stage regression is conducted. This is a regression of on the lagged regressors, and the lagged residuals from the first stage, . The second stage regression is given as:
If the variables are not cointegrated (if we cannot reject the null of no cointegration when testing ), then and we estimate a differences model:
Johansen test
The Johansen test is a test for cointegration that allows for more than one cointegrating relationship, unlike the Engle–Granger method, but this test is subject to asymptotic properties, i.e. large samples. If the sample size is too small then the results will not be reliable and one should use Auto Regressive Distributed Lags (ARDL).[8][9]
Phillips–Ouliaris cointegration test
Multicointegration
In practice, cointegration is often used for two series, but it is more generally applicable and can be used for variables integrated of higher order (to detect correlated accelerations or other second-difference effects). Multicointegration extends the cointegration technique beyond two variables, and occasionally to variables integrated at different orders.
Variable shifts in long time series
Tests for cointegration assume that the cointegrating vector is constant during the period of study. In reality, it is possible that the long-run relationship between the underlying variables change (shifts in the cointegrating vector can occur). The reason for this might be technological progress, economic crises, changes in the people's preferences and behaviour accordingly, policy or regime alteration, and organizational or institutional developments. This is especially likely to be the case if the sample period is long. To take this issue into account, tests have been introduced for cointegration with one unknown structural break,[11] and tests for cointegration with two unknown breaks are also available.[12]
Bayesian inference
Several
See also
References
- .
- S2CID 126346450.
- ^ .
- S2CID 154550363.
- .
- JSTOR 1913236.
- ^ https://www.econ.queensu.ca/sites/econ.queensu.ca/files/wpaper/qed_wp_1227.pdf [bare URL PDF]
- ^ Giles, David (19 June 2013). "ARDL Models - Part II - Bounds Tests". Retrieved 4 August 2014.
- hdl:10983/25617.
- JSTOR 2938339.
- .
- S2CID 153437469.
- ISBN 978-1-4039-4155-8.
Further reading
- Enders, Walter (2004). "Cointegration and Error-Correction Models". Applied Econometrics Time Series (Second ed.). New York: Wiley. pp. 319–386. ISBN 978-0-471-23065-6.
- ISBN 978-0-691-01018-2.
- ISBN 978-0-521-58782-2.
- Murray, Michael P. (1994). "A Drunk and her Dog: An Illustration of Cointegration and Error Correction" (PDF). . An intuitive introduction to cointegration.