"Frontmatter". In: Analysis of Financial Time Series

THRESHOLD CO-INTEGRATION 333ously buying (short-selling) the security index and selling (buying) the index futureswhenever the log prices diverge by more than the cost of carrying the index over timeuntil maturity of the futures contract. Under the weak stationarity of zt ∗ , for arbitrageto be profitable, zt ∗ must exceed a certain value in modulus determined by transactioncosts and other economic and risk factors.It is commonly believed that the f t,l and s t series of the S&P 500 index containa unit root, but Eq. (8.32) indicates that they are co-integrated after adjusting forthe effect of interest rate and dividend yield. The co-integrating vector is (1, −1)after the adjustment, and the co-integrated series is zt ∗ . Therefore, one should usean error-correction form to model the return series r t = (∇ f t , ∇s t ) ′ ,where∇ f t =f t,l − f t−1,l and ∇s t = s t − s t−1 , where for ease in notation we drop the maturitytime l from the subscript of ∇ f t .8.6.1 Multivariate Threshold ModelIn practice, arbitrage tradings affect the dynamic of the market, and hence the modelfor r t may vary over time depending on the presence or absence of arbitrage tradings.Consequently, the prior discussions lead naturally to the model⎧⎪⎨r t =⎪⎩c 1 + ∑ pi=1 Φ(1) ir t−i + β 1 z t−1 + a (1)t if z t−1 ≤ γ 1c 2 + ∑ pi=1 Φ(2) ir t−i + β 2 z t−1 + a (2)t if γ 1 < z t−1 ≤ γ 2c 3 + ∑ pi=1 Φ(3) ir t−i + β 3 z t−1 + a (3)t if γ 2 < z t−1 ,(8.33)where z t = 100zt ∗, γ 1 < 0