Mathematical and Statistical Methods for Actuarial Sciences and Finance

268 Claudio Pizzi

In this paper the test of hypothesis of cointegration has been restricted to analysing
the relationships between all the different pairs of time series. The Phillips Ouliaris
test [15] was performed on each pair, the results of which are summarised in Table 2.
The table highlights in bold thep-values lower than 0.05 that show the pairs of stocks
for which the hypothesis of no cointegration was rejected. The test was performed
resorting to R software. Note that the software outputs 0.15 for any realp-value
greater than 0.15. Hence the values 0.15 in the Table 2 mean that the null hypothesis
of no cointegration is accepted with ap-value≥ 0 .15.

Ta b le 2 .p-value of the Phillips Ouliaris test for the nullhypothesis that the time series are not
cointegrated

AIG CR CSCO F GM GS JPM MER MOT MS NVDA PKI TER TWX

CR 0.15

CSCO 0.15 0.15

F 0.15 0.15 0.15

GM 0.15 0.15 0.15 0.15

GS 0.15 0.15 0.15 0.15 0.15

JPM 0.15 0.15 0.15 0.15 0.15 0.15

MER 0.15 0.15 0.15 0.15 0.15 0.15 0.15

MOT 0.06 0.15 0.15 0.15 0.15 0.15 0.15 0.07

MS 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15

NVDA 0.01 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.05

PKI 0.15 0.15 0.15 0.15 0.15 0.15 0.01 0.06 0.15 0.15 0.15

TER 0.15 0.04 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15

TWX 0.05 0.15 0.04 0.08 0.15 0.15 0.15 0.15 0.15 0.14 0.07 0.15 0.15

TXN 0.11 0.15 0.15 0.15 0.15 0.15 0.15 0.04 0.01 0.11 0.10 0.10 0.15 0.15

Ta b le 3 .p-value for the nonlinear cointegration test for the null hypothesis that the time series
are not cointegrated

AIG CR CSCO F GM GS JPM MER MOT MS NVDA PKI TER TWX

CR 0.19

CSCO 0.65 0.00

F 0.93 0.97 0.99

GM 0.95 0.97 0.81 0.59

GS 0.03 0.31 0.06 0.00 0.20

JPM 0.27 0.13 0.98 0.00 0.99 0.00

MER 0.23 0.14 1.00 0.99 0.99 0.98 0.28

MOT 0.08 0.53 0.14 0.00 0.99 0.01 0.02 0.01

MS 0.00 0.00 0.16 0.94 1.00 0.94 0.59 0.86 0.46

NVDA 0.13 0.51 1.00 0.00 1.00 0.14 0.09 0.77 0.89 0.00

PKI 0.02 0.01 0.00 0.00 1.00 0.00 0.00 0.00 0.13 0.89 0.01

TER 0.45 0.01 0.00 0.99 0.99 0.99 0.00 0.05 0.00 0.99 0.00 0.00

TWX 0.42 0.06 1.00 1.00 1.00 0.99 0.00 0.42 0.01 1.00 1.00 0.02 0.00

TXN 0.41 0.00 0.99 1.00 1.00 0.99 0.00 0.79 0.02 1.00 0.01 0.00 0.00 0.01

The figures in bold show that only 7 pairs, out of the total 105 combinations, of
time series are linearly cointegrated, highlighting for the majority of cases the lack of a
long-run relationship and adjustment mechanism. To assess the presence of nonlinear