Nonlinear cointegration in financial time series 2673 An application to a financial time series
To verify whether there is cointegration amongst financial variables the time series
of the adjusted closing price of the quotations of 15 US stocks were taken from
the S&P 500 basket. For each stock the considered period went from 03.01.2007 to
31.12.2007. Table 1 summarises the 15 stocks considered, the branch of industry they
refer to and the results of the Phillips-Perron test performed on the price series to
assess the presence of unit roots.
Ta b le 1 .p-value for the Phillips–Perron test
p-value
Code Name Industry Stationarity Explosive
AIG American Internat.Group Insurance 0.565 0.435
CR Crane Company Machinery 0.623 0.377
CSCO Cisco Systems Communications equipment 0.716 0.284
F Ford Motor Automobiles 0.546 0.454
GM General Motors Automobiles 0.708 0.292
GS Goldman Sachs Group Capital markets 0.536 0.464
JPM JPMorgan Chase & Co. Diversified financial services 0.124 0.876
MER Merrill Lynch Capital markets 0.585 0.415
MOT Motorola Inc. Communications equip. 0.109 0.891
MS Morgan & Stanley Investment brokerage 0.543 0.457
NVDA NVIDIA Corp. Semiconductor & semiconductor equip. 0.413 0.587
PKI PerkinElmer Health care equipment & supplies 0.655 0.345
TER Teradyne Inc. Semiconductor & semiconductor equip. 0.877 0.123
TWX Time Warner Inc. Media 0.451 0.549
TXN Texas Instruments Semiconductor & semiconductor equip. 0.740 0.260The test was performed both to assess the presence of unit roots vs stationarity
(fourth column) and also the presence of unit roots vs explosiveness (last column).
The null hypothesis of unit roots wasaccepted in all the time series considered. A
further test, the KPSS test [10], was carried out to assess the null hypothesis that the
time series is level or trend stationary. The results have confirmed that all the series are
nonstationary. Considering the results from the nonstationarity tests, we proceeded
to verify the assumption of cointegration in the time series. The acceptation of the
latter assumption is especially interesting: this result can be interpreted in terms of the
mechanisms that affect the quotation of the stocks considered. More specifically, the
shocks that perturb the quotations of a stock imply departure from the system’s steady
state, thus inducing variations that depend on the extent of the shock and the estimable
speed of convergence towards the new equilibrium. From another standpoint, the
presence/absence of cointegration between two stocks may become important when
contemplating the implementation of a trading strategy. Recording the variations in a
variable (quotation of a stock) enables prediction of the “balancing” response provided
by some variables or the purely casual responses of others. Considering the definition
of cointegration introduced in the previous section and taking intoaccount the 15
shares contemplated in this application, there are numerous applicable cointegration
tests, as each possiblem-upla of variables can be considered withm= 2 ,...,15.