"Frontmatter". In: Analysis of Financial Time Series

NONLINEAR MODELS 133the series and obtainr t = 0.043 − 0.022r t−2 + a t ,a t = σ t ɛ tσ 2t = 0.002 + 0.097a 2 t−1 + 0.954σ 2t−1+ (0.056 − 0.051a 2 t−1 − 0.067σ 2t−1 )I (a t−1 > 0),where I (a t−1 ) = 1ifa t−1 > 0 and it is zero otherwise. Because the estimate 0.002of the volatility equation is insignificant at the 5% level, we further refine the modeltor t = 0.043 − 0.022r t−2 + a t ,σ 2t = 0.098a 2 t−1 + 0.954σ 2t−1a t = σ t ɛ t+ (0.060 − 0.052a 2 t−1 − 0.069σ 2t−1 )I (a t−1 > 0), (4.11)where the standard errors of the two parameters in the mean equation are 0.013 and0.010, respectively, and those of the TAR-GARCH(1, 1) model are 0.003, 0.004,0.005, 0.004, and 0.009. All of the estimates are statistically significant at the 5%level. The unconditional mean for r t of model (4.11) is 0.042, which is very closeto the sample mean of r t . Residual analysis based on the Ljung–Box statistics findsno significant serial correlations or conditional heteroscedasticity in the standardizedresiduals. The AR coefficient in the mean equation is small, indicating that, asexpected, the daily log returns of IBM stock are essentially serially uncorrelated.However, the volatility model of the returns shows strong dependence in the innovationalprocess {a t } and evidence of asymmetry in the conditional variance. Rewritingthe TAR-GARCH(1, 1) equation as{σt 2 0.098a2= t−1+ 0.954σt−1 2if a t−1 ≤ 00.060 + 0.046at−1 2 + 0.885σ t−1 2 if a t−1 > 0,(4.12)we obtain some interesting implications. First, if we interpret a t−1 as the deviationof IBM daily log return from its conditional expectation, then volatility followsessentially an IGARCH(1, 1) model without a drift when the deviation is nonpositive.Second, when the deviation is positive, the volatility has a persistent parameter0.046 + 0.885 = 0.931, which is close to, but less than, 1. Therefore, the volatilityfollows a GARCH(1, 1) model when the deviation is positive. Consequently, thevolatility responds differently to positive and negative shocks. Finally, other thresholdvolatility models have also been proposed in the literature (e.g., Rabemananjaraand Zakoian, 1993; Zakoian, 1994).Remark: The RATS program used to estimate the AR(2)-Tar-GARCH(1, 1)model is in the appendix.