Handbook of Exchange Rates

7.7 Appendix: TAR modeling

Let the deviations from the LOP for a sector i (i.e., the sectoral real exchange rate for sector i) be q_t(i). An econometric model that captures the predictions of a discrete switch in q_t(i) is the following symmetric three-regime TAR model:¹⁸

7.9

where Δ denotes the first difference operator; θ_i is the threshold parameter for sector i; and q_t−d(i) represents the threshold variable for sector i, with d denoting an integer chosen from the set Ψ ∈ [1, d]. The indicator function, given by l( · ), takes a value of unity when the bracketed expression is true, and is zero otherwise. The error term ε _it is assumed independently and identically distributed (iid)Gaussian. The autoregressive order of q_t is . Note that essentially the TAR specified has two regimes given that the behavior of the outer regimes is identical because of the symmetry assumption.

The integer d represents the delay parameter and reflects the possibility that market participants react to deviations of the LOP from equilibrium with a lag. As long as |q_t−d| ≤ θ, the time series q_t follows a unit root process. Thus, in this regime (or region), there is no tendency for the series q_t to move back toward equilibrium. Once |q_t−d| > θ, however, q_t becomes a stationary process and has a tendency to ...