"Frontmatter". In: Analysis of Financial Time Series

NONLINEAR MODELS 135els to the monthly returns of 3M stock, we obtain the modelR t = 0.017 + a t , a t = σ t ɛ t ,σ 2t = (0.002 + 0.256a 2 t−1 + 0.141a2 t−2 ) + 0.002 − 0.314a2 t−11 + exp(−1000a t−1 ) , (4.15)where the standard error of the constant term in the mean equation is 0.002 andthose of the estimates in the volatility equation are 0.0003, 0.092, 0.056, 0.001, and0.102, respectively. The scale parameter 1000 of the logistic transition function isfixed a priori to simplify the estimation. This STAR model provides some supportfor asymmetric responses to positive and negative prior shocks. For a large negativea t−1 , the volatility model approaches the ARCH(2) modelσ 2t = 0.002 + 0.256a 2 t−1 + 0.141a2 t−2 .Yet for a large positive a t−1 , the volatility process behaves like the ARCH(2) modelσ 2t = 0.005 − 0.058a 2 t−1 + 0.141a2 t−2 .The negative coefficient of at−1 2 in the prior model is counterintuitive, but the magnitudeis small. As a matter of fact, for a large positive shock a t−1 , the ARCH effectsappear to be weak even though the parameter estimates remain statistically significant.The RATS program used is given in the appendix.4.1.4 Markov Switching ModelThe idea of using probability switching in nonlinear time series analysis is discussedin Tong (1983). Using a similar idea, but emphasizing aperiodic transition betweenvarious states of an economy, Hamilton (1989) considers the Markov switchingautoregressive (MSA) model. Here the transition is driven by a hidden two-stateMarkov chain. A time series x t follows an MSA model if it satisfiesx t ={c1 + ∑ pi=1 φ 1,i x t−i + a 1t if s t = 1,c 2 + ∑ pi=1 φ 2,i x t−i + a 2t if s t = 2,(4.16)where s t assumes values in {1, 2} and is a first-order Markov chain with transitionprobabilitiesP(s t = 2 | s t−1 = 1) = w 1 , P(s t = 1 | s t−1 = 2) = w 2 .The innovational series {a 1t } and {a 2t } are sequences of iid random variables withmean zero and finite variance and are independent of each other. A small w i meansthat the model tends to stay longer in state i. In fact, 1/w i is the expected durationof the process to stay in State i. From the definition, an MSA model uses a hidden