is the location (or threshold) parameter. where c In (2), the parameter represents the slope of the transition function G(; ytd; c). Specifically, when = 0 the transition function = 1/2 so that the model nests a G(; ytd; c) LSTAR linear model. Conversely, as | | , the model approaches a threshold autoregressive LSTAR model with two distinct regimes. The transition function is a bounded function of the transition variable and is continuous everywhere in the parameter space for any value of Given that ytd ytd. is odd and monotonically increasing, whenever | | is large and < is G(; ytd; c) ytd c ytd c, yt+1 effectively generated by the linear model: = + = 1, (3) yt+1 wt t+1, t T …, If | | is and < is effectively generated by: ytd c ytd c, yt+1 = ( )wt + = 1, (4) yt+1 + t+1, t T …, (3) (4) Thus, the parameters of the autoregressive process change monotonically as a function of Such a model might be appropriate for real exchange rate behavior if the speed of adjustment ytd. depends on whether the rate is appreciating or depreciating. If is assumed even, it is standard to use the exponential function to the G(; ytd; c) transition between two regimes. The transition function can be written as: ESTAR (5) The parameters , and are defined as Similar to the logistic case, the exponential c, d function is bounded between 0 and 1, cases that correspond to = 0 and | | , respectively.
where c is the location (or threshold) parameter. In (2), the parameter represents the slope of the transition function G(; ytd; c). Specifically, when = 0 the transition function G(; ytd; c) = 1/2 so that the LSTAR model nests a linear model. Conversely, as | | , the LSTAR model approaches a threshold autoregressive model with two distinct regimes. The transition function is a bounded function of the transition variable ytd and is continuous everywhere in the parameter space for any value of ytd. Given that G(; ytd; c) is odd and monotonically increasing, whenever | ytd c | is large and ytd < c, yt+1 is effectively generated by the linear model: yt+1 = wt + t+1, t = 1, …, T (3) If | ytd c | is and ytd < c, yt+1 is effectively generated by: yt+1 = ( + )wt + t+1, t = 1, …, T (4) (3) (4) Thus, the parameters of the autoregressive process change monotonically as a function of ytd. Such a model might be appropriate for real exchange rate behavior if the speed of adjustment depends on whether the rate is appreciating or depreciating. If G(; ytd; c) is assumed even, it is standard to use the exponential function to the transition between two regimes. The ESTAR transition function can be written as: (5) The parameters , c, and d are defined as Similar to the logistic case, the exponential function is bounded between 0 and 1, cases that correspond to = 0 and | | , respectively.