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FAQs on Johansen's Cointegration test
Requested and Answered by Vanna on 30-Jan-2011 03:00 (9208 reads)
What is Johansen's cointegration test?
Johansen's testing is
1.A method to method to test if a subset in a set of series has cointegration
2.Can determine cointegrating systems (i.e, details of how the movements occur)
3.Uses VAR (vector auto regression)
What is the original VAR which is modelled?
It is a series of g variables (g>=2) that are I(1) and which are thought to be cointegrated.
Following is setup of the VAR:
What is the vector error correction model considered?
\[
{\Pi} = \left( \sum_{i=1}^k {\beta}_i \right) - I_g
\]
\[
{\Gamma}_i = \left( \sum_{j=1}^i {\beta}_j \right) - I_g
\]
The above output is from
Johansen's testing is
1.A method to method to test if a subset in a set of series has cointegration
2.Can determine cointegrating systems (i.e, details of how the movements occur)
3.Uses VAR (vector auto regression)
What is the original VAR which is modelled?
It is a series of g variables (g>=2) that are I(1) and which are thought to be cointegrated.
Following is setup of the VAR:
\[
y_1 = {\beta}_1 y_{t-1} + {\beta}_2 y_{t-2} + ... + {\beta}_k y_{t-k} + u_t
\]
where
\({\beta}_i\) is g x g matrix, and \({y}_i\) is g x 1 vector
What is the vector error correction model considered?
\[
{\Delta}y_t = {\Pi} y_{t-k} + {\Gamma}_1 {\Delta}y_{t-1} + {\Gamma}_2 {\Delta}y_{t-2} + ...
+ {\Gamma}_{k-1} {\Delta}y_{t-(k-1)} + u_t
\]
where
\[
{\Pi} = \left( \sum_{i=1}^k {\beta}_i \right) - I_g
\]
\[
{\Gamma}_i = \left( \sum_{j=1}^i {\beta}_j \right) - I_g
\]
What is the most important matrix used in Johansen's test?
The \(\Pi\) matrix in the above equation is used for eigen decomposition. The rank of this
matrix is used for finding the no. of cointegration vectors.
How do I interpret Johansen's \(\Pi\) test results?
Let us consider the following example output when a set of 5 series was used
H0 : Rank<=x Test statistic Critical values
90 95 99
0 134.468893 31.2379 33.8777 39.3693
1 99.202023 25.1236 27.5858 32.7172
2 12.03620412 18.8928 21.1314 25.865
3 5.378365285 12.2971 14.2639 18.52
4 0.200411475 2.7055 3.8415 6.6349
The above output is from
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