Likelihood-Ratio-Based Confidence Intervals for Multiple Threshold Parameters

Appendix: Discussion of assumptions

In ensuring the asymptotical equivalence between the limiting distribution of the likelihood ratio statistic for the construction of confidence intervals in multiple threshold and multiple break point settings, the assumptions in Eo and Morley (2015) are followed closely and adapted to the threshold regression environment of equation (1). Let the true values of the parameters be denoted with a superscript 0 and, for any i = 1, 2, …, m + 1, define the moment functionals M ( γ ) = E x t − 1 x t − 1 ′ 1 i , t γ i , d , D ( γ ) = E x t − 1 x t − 1 ′ | q t − d = γ and V ( γ ) = E x t − 1 x t − 1 ′ e t 2 | q t − d = γ . Then, the assumptions are given as follows:

To verify that equations (7) and (8) satisfy these assumptions, note that Assumption 1 restricts the threshold γ _i so that each regime contains a minimum number of observations, where t _i is the number of sample of observations in the ith regime. This ensures the threshold parameters to be asymptotically different and it is enforced during the estimation process as discussed in the main text, setting τ = 0.1. Assumption 2 is important given the dynamics of time series and is trivially satisfied for independent observations. Threshold processes of the types in equations (7) and (8) exhibit a strong form of geometric ergodicity, which implies Assumption 2 (Chan 1989). Assumption 3 imposes that the error term is zero-meaned with finite variance, so that the model in equation (1) correctly specifies the conditional mean, and ordinary least squares (OLS) estimation is appropriate. The error terms associated with equations (7) and (8) follow a normal distribution with mean zero and variance σ², which satisfy this assumption.

Assumptions 4 and 5 determine the structure of the x _t e _t and e _t processes and imply conditional and unconditional bounded fourth moments. Since the relevant characteristic polynomials of equations (7) and (8) have roots that lie outside the unit circle and the errors distribute normally and are i.i.d. (that is, they have a bounded and continuous probability density function), Assumptions 4 and 5 are satisfied (Chan 1993; Gonzalo and Pitarakis 2002; Hansen 2000). Assumption 6 requires the threshold variable to have a continuous distribution, effectively requiring the conditional variance E e t 2 | q t − d = γ to be continuous at γ i 0 for i = 1, …, m, which excludes the possibility of regime-dependent heteroskedasticity. Since the error terms follow the same distributions across regimes, Assumption 6 is satisfied.

Assumption 7 implies that the magnitude of the threshold effect Δ β _i → 0 as T → ∞, which guarantees that the limiting distribution is free of nuisance parameters and, therefore, allow for the construction of confidence intervals. Of course, the threshold effect for any give sample size T is of fixed magnitude. However, the asymptotic distribution of the likelihood ratio statistic for a shrinking threshold effect provides an upper bound on the distribution under a fixed threshold effect under normal errors (Hansen 2000). Therefore, any confidence intervals will overcover for any given experiment.

Finally, Assumption 8 is a full-rank condition which excludes multicollinearity and is necessary to ensure non-degenerate asymptotic distributions. Since γ ∈ Γ _i , and all threshold parameters are required to lie in the bounded set Γ = γ ̲ , γ ̄ , Assumption 8 is then satisfied.