The name is absent

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 [z_n(m₁),..., z_n(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 z_n converges in distribution to z. Moreover, because sup_meM (.)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

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Bierens, H.J., 1987. ARMAX model specification testing, with an application to
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Bierens, H.J., 1988. ARMA memory index modeling of economic time series (with
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Bierens, H.J., 1990. A Consistent Conditional Moment Test of Functional Form,
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Bierens, H.J. and W. Ploberger, 1997. Asymptotic theory of conditional moment
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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.

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