Johansen, Engle-Granger and Hansen-Seo cointegration analyses of futures contracts
| Period (Johansen) | Futures | Lag order | Rank | Trace test | p-value | λ max | p-value |
|---|---|---|---|---|---|---|---|
| Mar 21, 1995–May 30, 2017 | CL2-CL3-CL6-CL12 | 1 | 0 | 244.52 | 0.00 | 148.62 | 0.00 |
| 1 | 95.90 | 0.00 | 69.88 | 0.00 | |||
| 2 | 26.02 | 0.00 | 23.48 | 0.00 | |||
| 3 | 2.54 | 0.11* | 2.54 | 0.11* |
| Period (Johansen) | Futures | Lag order | Rank | Trace test | |||
|---|---|---|---|---|---|---|---|
| Mar 21, 1995–May 30, 2017 | CL2-CL3-CL6-CL12 | 1 | 0 | 244.52 | 0.00 | 148.62 | 0.00 |
| 1 | 95.90 | 0.00 | 69.88 | 0.00 | |||
| 2 | 26.02 | 0.00 | 23.48 | 0.00 | |||
| 3 | 2.54 | 0.11* | 2.54 | 0.11* |
| Period (Engle-Granger) | Futures | Lag order | ADF | p-value | |||
|---|---|---|---|---|---|---|---|
| Mar 21, 1995–May 30, 2017 | CL2 | 1 | −1.69 | 0.76 | |||
| CL3 | 1 | −1.61 | 0.79 | ||||
| CL6 | 1 | −1.46 | 0.84 | ||||
| CL12 | 1 | −1.25 | 0.90 | ||||
| Residuals | 1 | −10.22 | 0.00* |
| Period (Engle-Granger) | Futures | Lag order | ADF | ||||
|---|---|---|---|---|---|---|---|
| Mar 21, 1995–May 30, 2017 | CL2 | 1 | −1.69 | 0.76 | |||
| CL3 | 1 | −1.61 | 0.79 | ||||
| CL6 | 1 | −1.46 | 0.84 | ||||
| CL12 | 1 | −1.25 | 0.90 | ||||
| Residuals | 1 | −10.22 | 0.00* |
| Period (Hansen and Seo Sup LM Test) | Futures | Lag order | Test statistics | p-values | Nboot | Fixed regressor | Intercept |
|---|---|---|---|---|---|---|---|
| Mar 21, 1995–May 30, 2017 | CL2 vs CL3 | 1 | 21.77 | 0.036* | 1,000 | yes | Yes |
| CL3 vs CL6 | 1 | 19.26 | 0.121 | 1,000 | yes | Yes | |
| CL6 vs CL12 | 1 | 13.55 | 0.587 | 1,000 | yes | Yes | |
| CL3 vs CL12 | 1 | 18.10 | 0.172 | 1,000 | yes | Yes | |
| CL2 vs CL12 | 1 | 18.18 | 0.157 | 1,000 | yes | Yes | |
| CL2 vs CL6 | 1 | 19.52 | 0.082 | 1,000 | yes | Yes |
| Period (Hansen and Seo Sup LM Test) | Futures | Lag order | Test statistics | Nboot | Fixed regressor | Intercept | |
|---|---|---|---|---|---|---|---|
| Mar 21, 1995–May 30, 2017 | CL2 vs CL3 | 1 | 21.77 | 0.036* | 1,000 | yes | Yes |
| CL3 vs CL6 | 1 | 19.26 | 0.121 | 1,000 | yes | Yes | |
| CL6 vs CL12 | 1 | 13.55 | 0.587 | 1,000 | yes | Yes | |
| CL3 vs CL12 | 1 | 18.10 | 0.172 | 1,000 | yes | Yes | |
| CL2 vs CL12 | 1 | 18.18 | 0.157 | 1,000 | yes | Yes | |
| CL2 vs CL6 | 1 | 19.52 | 0.082 | 1,000 | yes | Yes |
Note(s): * Indicates cointegration at 5% level
Lag order is defined with BIC criterion after first differencing values to achieve stationarity (Johansen test)
Source(s): Personal elaborations on Quandl (2021)