1. In the case of two regimes, Potter [1990], [1993] proposed to call this class of non-linear, non-normal models the single index generalized multivariate autoregressive (SIGMA) model.
2. In threshold autoregressive (TAR) processes, the indicator function is defined in a switching variable z
t-d
, d ≥ 0. In addition, indicator variables can be introduced and treated with error-in-variables techniques. Refer for example to Cosslett & Lee [1985] and Kaminsky [1993].
3. If F(·) is even, e.g. F(y
t-d
– r) = 1 – exp {-(y
t-d
– r)2}, a generalized exponential autoregressive model as proposed by Ozaki [1980] and Haggan & Ozaki [1981] ensues.
4. The reader is referred to Hamilton [1994a] for an excellent introduction into the major concepts of Markov chains and to Titterington, Smith & Makov [1985] for the statistical properties of mixtures of normals.
5. Models where the regime is switching between deterministic and stochastic trends are considered by McCulloch & Tsay [1994a].