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MODULE OBJECTIVE
This module attempts to explore the long run association among the time series variable. It is very important for studying causality in the time series setting. In this section, we highlight the followings:
**1. What is cointegration test?
- Various Tests for cointegration
- Choice of lag length in the cointegration test**
WHAT IS COINTEGRATION TEST?
Cointegration means despite of being individually non stationary, a linear combination of two or more time series can be stationary. Cointegration of two time series suggests that there is a long run or equilibrium or relationship between them. It is quite possible trend that the two time series share the same common trend so that the regression of one on the other will not be necessarily spurious. Regression of one time series variable on one or more time series variables often can give nonsensical or spurious results. This phenomenon is known as spurious regression. One way to guard against it is to find out if the time series are cointegrated. However, the minimum requirement for cointegration test is to know the order of integration (unit root) of time series variables. This is already explained in the earlier lecture.
For better understanding of cointegration, we can cite the following example. Let us now consider PCE and PDI that is the personal consumption expenditure and the disposable personal income respectively
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Suppose we now subject u t to unit root analysis such that it is stationary that is it is I (0).Thus it indicated that both PCE and PDI are individually I (1) but their linear combination is I (0). In the above case we can say that the two variables are cointegrated. We can say that the first equation in the example is the cointegrating regression and the slope parameter β 2 is the cointegrating parameter. The concept of cointegration can be extended to regression model containing k regressors where we will have k cointegrating parameters.
TEST FOR COINTEGRATION
There two popular tests of cointegration:
- ENGLE AND GRANGER TEST
- JOHANSEN AND JUSELIUS TEST
ENGLE AND GRANGER TEST
To examine the test, let us take X and Y are two variables. Now, two time series variables (say X and Y) are said to be cointegrated, if the following three conditions are satisfied:
- Both X and Y are integrated of same order, which represents that number of times each variable has to be differenced in order to turn the series stationary.
- There exists a linear relationship between the two, i.e., Yt = α +βXt + εt , where β should be statistically significant.
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ti t
p
X t A Xt p i Ai X
1 (^01)
Where, is impact matrix and contains information about long run relationships between
variables in the data vector. If the rank of (say r ) is equal to zero, the impact matrix is a null
vector. If has full rank, n , then the vector process x t is stationary. If 0 < r < n, then there
exists r cointegrating vectors. Here, the impact matrix is
Where, both and are ( n x r ) matrices. The cointegrating vectors have the property that
X^ t is stationary [ I (0)] even though Xt is non-stationary [ I (1) ].
The cointegrating rank, r , can be formally tested with two statistics. The test statistic for the null hypothesis that there are at most r cointegrating vectors is the trace test and is computed as:
Trace Ti ^ nr 1 Log 1 ˆ i
Where ˆ^ r 1 , ….. ˆ n^ are (n-r) smallest estimated eigenvalues. The test statistic for the null
hypothesis of r cointegrating vectors against the alternative of r + 1 cointegrating vectors is the maximum eigenvalue test and is given by
max TLog^ 1 ˆ r 1 (10)
Here, the null hypothesis of r cointegrating vectors is tested against an alternative hypothesis of r +1 cointegrating vectors. Hence, the null hypothesis r = 0 is tested against r = 1 and r =1 is tested against r = 2 and so on. It is well known that the cointegration tests are very to the choice
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of lag length. The Schwarz Bayesian Criterion (SBC) is used to select the number of lags required in the cointegration test. The SBC test is explained in detail in the previous section.
COINTEGRATION AND ERROR CORRELATION MECHANISM
The error correction mechanism (ECM) developed by Engle and Granger is a means of reconciling the short-run behavior of an economic variable with its long-run behavior. While the concept of cointegartion is clearly an important theoretical underpinning of the error correction model there are still a number of problems surrounding its practical application.
