Linearity tests and stochastic trend under the STAR framework
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Linearity tests and stochastic trend under the STAR framework Lingxiang Zhang1 Received: 18 December 2017 © Springer-Verlag GmbH Germany, part of Springer Nature 2018
Abstract This study investigates the linearity test of smooth transition autoregressive models when the true data generating process is a stochastic trend process. Results show that, under the null hypothesis of linearity, the asymptotic distribution of the W statistic proposed by Teräsvirta (J Am Stat Assoc 89:208–218, 1994) follows the χ2 distribution, whereas the finite sample distribution does not. A maximized Monte Carlo simulation-based test is used to perform the linearity test, and the results show good performance. Keywords Linearity · STAR · Stochastic trend · Maximized Monte Carlo simulation-based test JEL Classification C12 · C32 · C52
1 Introduction The smooth transition autoregressive (STAR) model is one of the most important nonlinear models used to model the dynamics of financial and economic data (van Dijk et al. 2002; Teräsvirta et al. 2010). Initiative linearity tests against the STAR models proposed by Teräsvirta (1994) assume that the time series is stationary and that the LM- or Wald-type statistic has an asymptotic χ 2 distribution. However, the stationarity assumption may be invalid because substantial economic and financial data are highly persistent or present trend characteristics. Kiliç (2004), Harvey and Leybourne (2007) and Kruse (2011) emphasized that the asymptotic distribution of Wald-type linearity tests under a random walk process is non-standard. Zhang (2012) emphasized that the Wald-type statistic degenerates at the speed of T when the data generation process (DGP) includes a deterministic trend. The present study extends
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Lingxiang Zhang [email protected] School of Management and Economics, Beijing Institute of Technology, Beijing, China
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L. Zhang
these abovementioned studies into the stochastic trend case. The results show that, if the true DGP is a stochastic trend process, then the asymptotic distribution of the linearity test statistic does follow χ 2 distribution, but the finite sample distribution does not, especially in the case of small samples. Therefore, we cannot use the χ 2 distribution to test for linearity in this case. To overcome this problem, this study uses a maximized Monte Carlo (MC) simulation-based finite sample linearity test against STAR models, and the results show good performance. As we know that the linearity tests are related to the unit root test because certain economic and financial time series present high persistence. An issue that emerged from the widespread application of nonlinear STAR models to highly persistent time series is how traditional tests of the unit root can benefit the process of distinguishing unit root processes from stationary but nonlinear processes. In fact, several studies show that standard linear unit root tests display poor performance against nonlinear processes (Pippenger and Goering 1993; Balke and Fomby 1997; Taylor et al. 2001). This p
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