Tests for Asymmetry in Possibly Nonstationary Time Series Data

Asymmetry and nonstationarity for a seasonal time series model

“…Note that, by Lemma 2, for each j 2 J 0 , y jt is integrated at frequency y j and, for each jeJ, y jt is stationary. Therefore, combining the argument of Lemma 4 of Shin and Lee (2001) and proof of Theorem 6 of Caner and Hansen (2001), we have that n À2 P y Jt y 0 Jt I kt and n À1 P ðy À Jt 0 I kt , z à kt 0 Þ 0 ðy À Jt 0 I kt , z à kt 0 Þ converge in distribution to positive definite matrices, n À3=2 P y Jt y À Jt 0 I kt ¼ o p ð1Þ for jeJ 0 and n À3=2 P y Jt z à Lemma A.1. If p ðiÞ ¼ 0 and p ðjÞ ¼ 0, iaj, then (i) P t sgn i 1 ðy i 1 ;t Þy j 1 ;t ¼ o p ðn 3=2 Þ, (ii) P t sgn i 1 ðy i 1 ;t Þsgn j 1 ðy j 1 ;t Þ ¼ o p ðnÞ,…”

“…Lett Ãp sj be the same ast p sj except the sign instruments are used instead of the instruments h j ðy jt Þ. By the same argument of Lemma 1 of Shin and Lee (2001),t p sj ¼t Ãp sj þ o p ð1Þ. By the same argument of Theorem 3 of Shin and So (2000), t Ãp sj !…”

“…works of Shin and Lee (2001) and Caner and Hansen (2001) for the model z t ¼ P 2 k¼1 ðr k y tÀd þ P p j¼1 a kj z tÀ1Àj ÞI kt þ e t . The null distribution of the OLSE-based Wald test for the unit root hypothesis r 1 ¼ r 2 ¼ 0 depends on whether threshold effect exists or not, i.e., ða 11 ; .…”

“…Nonstationarity associated with unit root at zero frequency render the unit root test of Shin and Lee (2001) to find some evidence for nonstationarity compounded with presence of unit roots at all seasonal frequencies. However, the evidence captured by the test of Shin and Lee (2001) is not so strong as that captured by the tests developed in this paper, which pinpoint the unit root at frequency zero. We next look into the test statistics for symmetry given in Table 7.…”

Robust Unit Root Tests for a Panel TAR Model

“…Enders and Granger (1998) and Caner and Hansen (2001) provided unit root tests for TAR models. More studies were made by Shin and Lee (2001) for unit root tests and tests for asymmetry, Hansen and Seo (2002) for testing cointegration, Shin and Lee (2008) for panel unit root tests, and others.…”