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
forecast horizon lengthens beyond 7 or 8 periods, the forecasts from the nonlinear models become more similar to those from a linear model. Tables 2 and 3 display the results corresponding to the power simulation exercises. Before proceeding, it is important at this point to describe the differences and similarities between those two types of models. An ESTAR model can be well approximated by an LSTAR model if most of the data generated by the model lie above the value of the threshold parameter c. As such, only the increasing portion of the transition function matters and this can be well mapped by a monotonically increasing function of LSTAR type. Conversely, an LSTAR model may be well approximated by an ESTAR model provided that the move from one regime to the other is not quick. Therefore, the simulations below describe a wide variety of competing testing frameworks to help when one does not know a priori how many terms to include in the Taylor series approximation. Table 2 reveals several findings concerning the power of the test when the true DGPs are LSTAR processes. As increases from 5 to 20, the transition function changes from an approximately linear to a threshold with two states regime. Accordingly, the power increases in for both LSTAR processes. Also, Panels a – f of Figure 3 show that the power of the test also increases with the sample size for both processes. Comparing directly the two DGPs, it seems that the LSTAR2 process displays slightly higher power at all horizons for both samples. Finally, for both DGPs it is evident that the power decreases with the forecasting horizon. This follows as forecasts from nonlinear models should be very close to those from linear models as the forecast horizon gets large. Table 3 shows the results from the power simulation exercises where the DGP is an ESTAR process. Panels a – c of Figure 4 plot the results in Table 3. Note that for ESTAR1, the