In the previous post https://statcompute.wordpress.com/2018/07/29/co-integration-and-pairs-trading, it was shown how to identify two co-integrated stocks in the pair trade. In the example below, I will show how to form a mean reverting portfolio with three or more stocks, e.g. stocks with co-integration, and also how to find the linear combination that is stationary for these stocks.
First of all, we downloaded series of three stock prices from finance.yahoo.com.
For the residual-based co-integration test, we can utilize the Pu statistic in the Phillips-Ouliaris test to identify the co-integration among three stocks. As shown below, the null hypothesis of no co-integration is rejected, indicating that these three stocks are co-integrated and therefore form a mean reverting portfolio. Also, the test regression to derive the residual for the statistical test is also given.
Based on the test regression output, a linear combination can be derived by [FITB + 1.097465 – 0.152637 * MTB – 0.140457 * BAC]. The ADF test result confirms that the linear combination of these three stocks are indeed stationary.
Alternatively, we can also utilize the Johansen test that is based upon the likelihood ratio to identify the co-integration. While the null hypothesis of no co-integration (r = 0) is rejected, the null hypothesis of r <= 1 suggests that there exists a co-integration equation at the 5% significance level.
Similarly, based on the above Eigenvectors, a linear combination can be derived by [FITB + 0.6216917 – 0.1398349 * MTB – 0.1916826 * BAC]. The ADF test result also confirms that the linear combination of these three stocks are stationary.
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