A test due to Johansen[1] allows us to determine if three or more time series are cointegrated. This would be useful analysis to translate into a mean reversion trading strategy.
The following three times series data sets will be used for the example:
## [1] "GOOG"| GOOG.Open | GOOG.High | GOOG.Low | GOOG.Close | GOOG.Volume | GOOG.Adjusted |
|---|---|---|---|---|---|
| 1102.09 | 1122.310 | 1096.010 | 1114.22 | 1072400 | 1114.22 |
| 1120.00 | 1128.227 | 1115.000 | 1115.65 | 1315100 | 1115.65 |
| 1099.00 | 1128.000 | 1093.800 | 1127.46 | 1217300 | 1127.46 |
| 1135.82 | 1135.820 | 1100.020 | 1102.89 | 679000 | 1102.89 |
| 1110.53 | 1127.500 | 1108.480 | 1124.27 | 1066700 | 1124.27 |
| 1123.58 | 1140.930 | 1120.737 | 1140.17 | 996100 | 1140.17 |
## [1] "AAPL"| AAPL.Open | AAPL.High | AAPL.Low | AAPL.Close | AAPL.Volume | AAPL.Adjusted |
|---|---|---|---|---|---|
| 184.10 | 186.21 | 183.80 | 185.50 | 17365200 | 181.4822 |
| 186.29 | 187.19 | 182.91 | 185.11 | 22737700 | 181.1006 |
| 183.82 | 187.30 | 183.42 | 187.18 | 17731300 | 183.1258 |
| 187.79 | 187.95 | 183.54 | 183.92 | 13954800 | 179.9364 |
| 185.26 | 186.41 | 184.28 | 185.40 | 16604200 | 181.3844 |
| 185.42 | 188.43 | 185.20 | 187.97 | 17485200 | 183.8987 |
## [1] "IBM"| IBM.Open | IBM.High | IBM.Low | IBM.Close | IBM.Volume | IBM.Adjusted |
|---|---|---|---|---|---|
| 138.10 | 140.54 | 137.95 | 140.04 | 4523700 | 130.5398 |
| 140.32 | 141.36 | 139.61 | 139.70 | 3710100 | 130.2229 |
| 138.28 | 140.22 | 138.20 | 139.86 | 3405400 | 130.3720 |
| 140.65 | 140.94 | 139.37 | 139.57 | 1963200 | 130.1017 |
| 140.48 | 141.43 | 139.93 | 141.43 | 3744700 | 131.8355 |
| 141.53 | 142.94 | 141.17 | 142.48 | 2849000 | 132.8143 |
The following variables will be used to hold backward: (Close Price)
GOOGAdj = unclass(GOOG$GOOG.Close)
AAPLAdj = unclass(AAPL$AAPL.Close)
IBMAdj = unclass(IBM$IBM.Close)##The Johansen Test produces the following results:
##
## ######################
## # Johansen-Procedure #
## ######################
##
## Test type: trace statistic , with linear trend in cointegration
##
## Eigenvalues (lambda):
## [1] 4.449165e-03 3.125569e-03 1.804695e-03 6.938894e-18
##
## Values of teststatistic and critical values of test:
##
## test 10pct 5pct 1pct
## r <= 2 | 5.54 10.49 12.25 16.26
## r <= 1 | 15.15 22.76 25.32 30.45
## r = 0 | 28.84 39.06 42.44 48.45
##
## Eigenvectors, normalised to first column:
## (These are the cointegration relations)
##
## GOOG.Close.l2 AAPL.Close.l2 IBM.Close.l2 trend.l2
## GOOG.Close.l2 1.000000 1.0000000 1.000000 1.0000000
## AAPL.Close.l2 -1.439142 -17.5234126 -15.570925 0.1619705
## IBM.Close.l2 3.571236 -1.8561071 2.470901 -3.7774703
## trend.l2 -0.293524 0.5046487 0.748573 -0.1262161
##
## Weights W:
## (This is the loading matrix)
##
## GOOG.Close.l2 AAPL.Close.l2 IBM.Close.l2 trend.l2
## GOOG.Close.d -0.0051307496 -7.289428e-04 2.125545e-04 3.210602e-18
## AAPL.Close.d -0.0004784436 6.136602e-05 1.478564e-04 -2.436872e-20
## IBM.Close.d -0.0011838080 2.052026e-04 -6.299388e-05 1.538684e-18##
## #####################################################
## # Johansen-Procedure Unit Root / Cointegration Test #
## #####################################################
##
## The value of the test statistic is: 5.5436 15.151 28.836| 10pct | 5pct | 1pct | |
|---|---|---|---|
| r <= 2 | | 10.49 | 12.25 | 16.26 |
| r <= 1 | | 22.76 | 25.32 | 30.45 |
| r = 0 | | 39.06 | 42.44 | 48.45 |
##Reference:
[1] Johansen, S. (1991) Estimation and Hypothesis Testing of Cointegration Vectors in Gaussian Vector Autoregressive Models, Econometrica 59 (6): 1551-1580