where the last inequality is due to Assumption C. Therefore, ∑j=1 E (n' ' 2 ΣLι wi (xi,tΓ}
is bounded. This together with (A8) establishes (A7). B
Proof of Theorem 3: The result under H0 follows easily from Lemma 4 and Lemma
3.1 of Chang et. al. (2001). The result under H1 follows directly from Theorem 3.2
Park and Phillips (2001) and Lemma 3.1 of Chang et. al. (2001). ■
Proof of Theorem 4: (i) Note that [zn(m1),..., zn(mo)] → [z(m1),..., z(m^)] by
Theorem 3.2 of Park and Phillips (2001), for any finite d. In view of Lemma 4(ii)
this ensures that zn converges in distribution to z. Moreover, because supmeM (.)2 is
a continuous mapping from C(M) on R, the result follows.
(ii) The result under H1 follows directly from Theorem 3.2 of Park and Phillips
(2001). ■
Proof of Lemma 6: The result follows easily from Lemma A.B
Proof of Lemma 7: Same as the proof of Theorem 4 of Bierens (1990).■
7 References
Bierens, H.J., 1982. Consistent model specification testing. Journal of Econometrics
20, 3105-134.
Bierens, H.J., 1984. Model specification testing of time series regressions. Journal of
Econometrics 26, 323-353.
Bierens, H.J., 1987. ARMAX model specification testing, with an application to
unemployment in Netherlands. Journal of Econometrics 35, 161-190.
Bierens, H.J., 1988. ARMA memory index modeling of economic time series (with
discussion). Econometric Theory, 4, 35-59.
Bierens, H.J., 1990. A Consistent Conditional Moment Test of Functional Form,
Econometrica, 67, 1341-1383.
Bierens, H.J. and W. Ploberger, 1997. Asymptotic theory of conditional moment
integrated test. Econometrica 65, 1129-1151.
Billinglsey P., 1968. Convergence of Probability Measures, Wiley.
Billinglsey P., 1979. Probability and Measure, Wiley.
Chang, Y., Park J.Y. and P.C.B. Phillips, 2001. Nonlinear econometric models with
cointegrated and deterministically trending regressors. Econometrics Journal 4,
1-36.
de Jong R. M., 1996. The Bierens Test Under Data Dependence, Journal of Econo-
metrics, 72, 1-32.
de Jong R. M., 2004. Addendum to “Asymptotics for nonlinear transformations of
integrated time series”. Econometric Theory 20, 627-635.
de Jong R. M. and Wang, C-H., 2005. Further results on the asymptotics for nonlinear
transformations of integrated time series. Econometric Theory 21, 413-430.
21
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