3. A Proposed Pretest Teräsvirta (1994) develops a simple test to determine whether the in-sample relationship between and appears to be an or linear process. The test is based on a Taylor yt+1 wt LSTAR, ESTAR, series expansion of the function It is straightforward to extend his methodology to G(; yt-d+h; c). determine the relationship between and yt+h wt. To simplify matters, we temporarily assume = = 0 so that the model becomes: d p (10) Let = (yt and [1 + exp(z)]1. The approach consists in taking a third‐order Taylor z c) series approximation of with respect to evaluated = 0. Although the algebra is tedious, the g z z logistic process can be approximated by a fourth-order polynomial in After using Teräsvirta’s yt. (1994) methodology, it possible to approximate (10) as the following fourth-order polynomial in yt = + 0.5yt + 0.25ytz + (11) yt+1 yt ytz3/48 t+1 so that . In contrast to an process, an series assumes a symmetric smooth LSTAR ESTAR adjustment between regimes. If we continue to assume = = 0, the ESTAR model can be d p written as = + exp((yt + = + exp(z2)) + (12) yt+1 yt yt(1 c)) t+1 yt yt(1 t+1 where . Interestingly, since the ESTAR model is symmetric, the term does not appear in the z3 third-order Taylor series approximation of (12). Consider = + = + yt+1 yt ytz2 yt yt(yt c)2 –
3. A Proposed Pretest Teräsvirta (1994) develops a simple test to determine whether the in-sample relationship between yt+1 and wt appears to be an LSTAR, ESTAR, or linear process. The test is based on a Taylor series expansion of the function G(; yt-d+h; c). It is straightforward to extend his methodology to determine the relationship between yt+h and wt. To simplify matters, we temporarily assume d = p = 0 so that the model becomes: (10) Let z = (yt c) and [1 + exp(z)]1. The approach consists in taking a third‐order Taylor series approximation of g with respect to z evaluated z = 0. Although the algebra is tedious, the logistic process can be approximated by a fourth-order polynomial in yt. After using Teräsvirta’s (1994) methodology, it possible to approximate (10) as the following fourth-order polynomial in yt yt+1 = yt + 0.5yt + 0.25ytz ytz3/48 + t+1 (11) so that . In contrast to an LSTAR process, an ESTAR series assumes a symmetric smooth adjustment between regimes. If we continue to assume d = p = 0, the ESTAR model can be written as yt+1 = yt + yt(1 exp((yt c)) + t+1 = yt + yt(1 exp(z2)) + t+1 (12) where . Interestingly, since the ESTAR model is symmetric, the term z3 does not appear in the third-order Taylor series approximation of (12). Consider yt+1 = yt + ytz2 = yt + yt(yt – c)2
Enders_Pascalau_01082015.2185800.docx