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Asymptotic Expansions for Several GEL-Based Test Statistics and Hybrid Bartlett-Type Correction with Bootstrap

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Research Papers in Statistical Inference for Time Series and Related Models

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Abstract

This paper mainly discusses two issues about asymptotic expansions for the distributions of \(\chi ^2\)-type test statistics. First, it is shown that the generalized empirical likelihood ratio, Wald-type, and score-type test statistics for a subvector hypothesis in the possibly over-identified moment restrictions are, in general, not Bartlett-correctable, except for the empirical likelihood ratio test statistic. Second, starting with the classical likelihood or the modern generalized empirical likelihood, the Bartlett-type corrected test statistics, with the bootstrap procedure, are proposed to achieve a higher-order accurate testing inference for the nonparametric setup as well as the parametric setup.

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Notes

  1. 1.

    The idea of using “estimating equations” has a long history in the literature, at least since Karl Pearson’s introduction of the so-called method of moments. It seems that a motivation behind Qin and Lawless [62] is Godambe’s optimal estimating equations theory. In the econometric literature, Hansen’s [32] generalized method of moments is a basic inferential technique. We do not mention these two big methodologies anymore, to save space.

  2. 2.

    The finding was first announced by the author at The Mathematical Society of Japan (Spring Meeting 2012).

  3. 3.

    The bootstrap method was introduced by Efron [25], inspired by an earlier work on the jackknife. The methodology of the bootstrap is a computer-intensive technique that provides a basis for every field from modern statistical science. It is well recognized that its theoretical validity, as well as a deep understanding of the bootstrap procedure, is closely related to higher-order asymptotic statistical theory (e.g., Hall [30]); the details are omitted here.

  4. 4.

    A hybrid method of “the bootstrap-based Bartlett-type adjustment” from the ordinary parametric likelihood/GEL testing inference was reported by the author at Nara-Symposium (September 2014) that Professor Taniguchi organized, and also at the Japanese Joint Statistical Meeting (September 2014).

  5. 5.

    The following properties hold:

    • If \(Y^{(N)}=o_F^{(N)}(q_1,q_2)\), then,  (i) \(Y^{(N)}=o_F^{(N)}(q_1',q_2')\) for any \(q_1 \ge q_1'\)(\(\ge 0\)) and \(q_2' \ge q_2\); (ii) \(N^{-\tau } Y^{(N)}=o_F^{(N)}(q_1,q_2')\) for any \(\tau >0\) and \(q_2' \ge 0\).

    • If \(Y_\textrm{I}^{(N)}=o_F^{(N)}(q,q_2)\) and \(Y_\textrm{II}^{(N)}=o_F^{(N)}(q,q_2')\), then,

      $$Y_\textrm{I}^{(N)}+Y_\textrm{II}^{(N)}=o_F^{(N)}(q,\max (q_2,q_2')) \quad \text{ and } \quad Y_\textrm{I}^{(N)} Y_\textrm{II}^{(N)}=o_F^{(N)}(q,q_2+q_2')\,. $$
  6. 6.

    The GELR is a GEL counterpart of Wilks [76] from the parametric likelihood. As a special case of \(\rho (\cdot )=\rho _\textrm{EL}(\cdot )\), the ELR by Qin and Lawless [62] is most popular in the literature.

  7. 7.

    The Wald-type is a GEL counterpart of Wald [75] from the parametric likelihood.

  8. 8.

    The score-type (or Lagrange multiplier) is a GEL counterpart of [65] (see also [69]) from the parametric likelihood. For a history about Rao’s score test, we refer the readers to [5]. See also [8] for the score-type test statistic from the estimating equation models.

  9. 9.

    The gradient-type is a GEL counterpart of [73] from the parametric likelihood.

  10. 10.

    Newey and Smith [56] established that

    $$N^{1/2}(\widehat{\boldsymbol{{\eta }}}^{(N)\rho }-\boldsymbol{{\eta }}_0) {\mathop {\longrightarrow }\limits ^{d}}\textrm{N}\left( \textbf{0}_{p+M}\,,\, \left( \begin{array}{@{}c@{~}c@{}} ( \boldsymbol{{\nu }}_{\theta \lambda } \boldsymbol{{\nu }}_{\lambda ,\lambda }^{-1} \boldsymbol{{\nu }}_{\lambda \theta } )^{-1} &{} \textbf{O}_{p,M} \\ \textbf{O}_{M,p} &{} \textbf{P} \\ \end{array} \right) \right) \,, $$

    where \(\boldsymbol{{\nu }}_{\lambda ,\lambda }=-\boldsymbol{{\nu }}_{\lambda \lambda }\) and

    $$\textbf{P} =\boldsymbol{{\nu }}_{\lambda ,\lambda }^{-1} -\boldsymbol{{\nu }}_{\lambda ,\lambda }^{-1} \boldsymbol{{\nu }}_{\lambda \theta } ( \boldsymbol{{\nu }}_{\theta \lambda } \boldsymbol{{\nu }}_{\lambda ,\lambda }^{-1} \boldsymbol{{\nu }}_{\lambda \theta } )^{-1} \boldsymbol{{\nu }}_{\theta \lambda } \boldsymbol{{\nu }}_{\lambda ,\lambda }^{-1}\,. $$

    We can see that

    which yields \(N^{1/2}(\widehat{\boldsymbol{{\theta }}}_{(1)}^{(N)\rho }-\boldsymbol{{\theta }}_{(1)0}) {\mathop {\longrightarrow }\limits ^{d}}\textrm{N}(\textbf{0}_{p_1},\boldsymbol{{\nu }}_{(11 \cdot 2)}^{-1}) \), since

    $$(\textbf{I}_{p_1}~ \textbf{O}_{p_1,p_2}) ( \boldsymbol{{\nu }}_{\theta \lambda } \boldsymbol{{\nu }}_{\lambda \lambda }^{-1} \boldsymbol{{\nu }}_{\lambda \theta } )^{-1} \left( \begin{array}{@{}c@{}} \textbf{I}_{p_1} \\ \textbf{O}_{p_2,p_1} \\ \end{array} \right) =-( \boldsymbol{{\nu }}_{\theta _{(1)}\lambda } \textbf{M}_{\lambda \lambda } \boldsymbol{{\nu }}_{\lambda \theta _{(1)}} )^{-1} =\boldsymbol{{\nu }}_{(11 \cdot 2)}^{-1}\,. $$

    It follows that \(\textrm{W}^{(N)\rho } {\mathop {\longrightarrow }\limits ^{d}}\chi ^2_{p_1}\) and \(\textrm{W}_\dagger ^{(N)\rho } {\mathop {\longrightarrow }\limits ^{d}}\chi ^2_{p_1}\).

  11. 11.

    By Proposition 10.2 of Sect. 10.6.3, we have, in principle,

    $$E_F^{(N)}[\textrm{ELR}^{(N)}] =p_1 +\frac{1}{N}\, (\kappa _{b_1,b_2}^{-2,-6,1/3,0,1/4,-1/12}+\kappa _{b_1}^{-2,1/3}\kappa _{b_2}^{-2,1/3}) \nu _{(11\cdot 2)}^{b_1b_2} +o(N^{-1})\,. $$
  12. 12.

    Some authors may use the terminology of the Bartlett correction in the sense that the expectation of the test statistic is closer to that of the original (uncorrected) test statistic.

  13. 13.

    We stress that the classical likelihood-based parametric case is also allowed here. See Kakizawa [45,46,47,48,49].

  14. 14.

    There were, at least for me, confusing expressions in Cordeiro and Ferrari [20, (1) and (2)]; indeed, using the relation \(2g_{\nu +2}(x)=G_\nu (x)-G_{\nu +2}(x)\), one can rearrange Harris’s [34, (3.2)] asymptotic expansion for the distribution of the Rao test statistic \(S_\textrm{R}\), as follows:

    $$\begin{aligned}{} & {} \Pr (S_\textrm{R} \le x) \\= & {} G_m(x) +\frac{1}{24n}\,[ A_3 G_{m+6}(x) +(A_2-3A_3) G_{m+4}(x) +(A_1-2A_2+3A_3) G_{m+2}(x) \\{} & {} \qquad \qquad \qquad +(-A_1+A_2-A_3) G_m(x) ] +o(n^{-1}) \\= & {} G_m(x) -\frac{1}{12n} \Bigl [ \frac{A_1-A_2+A_3}{m} +\frac{(A_2-2A_3)x}{m(m+2)} +\frac{A_3\,x^2}{m(m+2)(m+4)} \Bigr ] x g_m(x) +o(n^{-1})\,. \end{aligned}$$
  15. 15.

    A seminal work in this area is an unpublished working paper of Rothenberg [68], cited by Anderson’s book (see also Fujikoshi [27]); it would, however, not available almost anywhere from the world.

  16. 16.

    Similar conclusion holds for the modern GEL framework; indeed, Bravo [10] explicitly derived \(N^{-1/2}\)-asymptotic expansion for the distribution (under a sequence of contiguous alternatives) of a class of ECR test statistics for testing a simple full vector parameter hypothesis in the just-identified moment restrictions. To the best of our knowledge, the literature is, however, little.

  17. 17.

    Note that

    $$1-P_F^{(N)}(\mathcal{X}_0^{(N)}) \le P_F^{(N)} \Bigl [ \max _{i \in \{ 1,\ldots ,N \}} \sup _{\theta \in \Theta } ||\textbf{g}(\textbf{X}_i,\boldsymbol{{\theta }})|| > \frac{1}{2}\,N^{1/2-\xi _0} \Bigr ] =o(N^{-1}) $$

    (see Lemma 10.2).

  18. 18.

    There exists an integer \(N_{0,\rho }\) such that \(\pm N^{-\xi _0} \log N \in \mathcal{N}_\rho \)(\(\subset \mathcal{V}_\rho \)) for all \(N \ge N_{0,\rho }\).

  19. 19.

    Apply the fact (10.1) to get

    $$\begin{aligned} \rho _{M+1}^\uparrow (\boldsymbol{{\nu }}+\boldsymbol{{\Psi }}^{(N)\rho }(\boldsymbol{{\eta }}))\ge & {} \rho _{M+1}^\uparrow (\boldsymbol{{\nu }})-||\boldsymbol{{\Psi }}^{(N)\rho }(\boldsymbol{{\eta }})|| \ge \frac{1}{ ||\boldsymbol{{\nu }}^{-1}|| }-||\boldsymbol{{\Psi }}^{(N)\rho }(\boldsymbol{{\eta }})|| >\frac{1}{ 2||\boldsymbol{{\nu }}^{-1}|| }\,, \\ \rho _M^\uparrow (\boldsymbol{{\nu }}+\boldsymbol{{\Psi }}^{(N)\rho }(\boldsymbol{{\eta }}))\le & {} \rho _M^\uparrow (\boldsymbol{{\nu }})+||\boldsymbol{{\Psi }}^{(N)\rho }(\boldsymbol{{\eta }})|| \le -\frac{1}{ ||\boldsymbol{{\nu }}^{-1}|| }+||\boldsymbol{{\Psi }}^{(N)\rho }(\boldsymbol{{\eta }})|| <-\frac{1}{ 2||\boldsymbol{{\nu }}^{-1}|| }\,, \end{aligned}$$

    since \(\min \{ \rho _{M+1}^\uparrow (\boldsymbol{{\nu }}),-\rho _M^\uparrow (\boldsymbol{{\nu }}) \} =1/||\boldsymbol{{\nu }}^{-1}|| \) (see (10.6)).

  20. 20.

    Some straightforward but tedious calculations show that

    $$\begin{aligned} \kappa _{a_1}^{\rho _3,\tau _1}= & {} -\nu ^\mathcal{G}_{a_1r,r'} \nu _{(22)}^{rr'} +\frac{1}{2}\,{\nu ^\rho }^\mathcal{G}_{a_1r_1r_2} \nu _{(22)}^{r_1r_1'} \nu _{r_1',r_2'} \nu _{(22)}^{r_2r_2'} \\{} & {} +\frac{1}{2} \Bigl ( \tau _1 {\nu ^\rho }^\mathcal{G\,\,G\,\,G}_{a_1b\,b'} -\nu ^\mathcal{G\,\,G\,\,G}_{a_1b,b'} -\sum _{\beta '=1}^M \mathcal{G}_{[\beta ']b'} {\nu ^\rho }^\mathcal{G\,\,G}_{a_1b[\beta ']} \Bigr ) \nu _{(11 \cdot 2)}^{bb'}\,, \end{aligned}$$

    and that \(\kappa _{a_1,a_2,a_3}^{\rho _3,\tau _1}\) and \(\kappa _{a_1,a_2,a_3,a_4}^{-2,\rho _4,1/3,\tau _2,\tau _3,\tau _4}\) are explicitly given in the proof of Theorem 10.1; the lengthy general expression for \(\kappa _{a_1,a_2,a_3,a_4}^{\rho _3,\rho _4,\tau _1,\tau _2,\tau _3,\tau _4}\) is, however, omitted here. Although, in principle, we can write down \(\kappa _{a_1,a_2}^{\rho _3,\rho _4,\tau _1,\tau _2,\tau _3,\tau _4}\), we have not rearranged it, due to the rather lengthy algebra (the task would be no practical importance, in many cases).

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  1. Hokkaido University, Kita-ku, Sapporo, Hokkaido, 060-0809, Japan

    Yoshihide Kakizawa

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  1. Faculty of Science and Engineering, Waseda University, Shinjuku-ku, Tokyo, Japan

    Yan Liu

  2. Faculty of Science, Niigata University, Nishi-ku, Niigata, Japan

    Junichi Hirukawa

  3. Faculty of Economics and Business, Hokkaido University, Kita-ku, Sapporo, Hokkaido, Japan

    Yoshihide Kakizawa

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Kakizawa, Y. (2023). Asymptotic Expansions for Several GEL-Based Test Statistics and Hybrid Bartlett-Type Correction with Bootstrap. In: Liu, Y., Hirukawa, J., Kakizawa, Y. (eds) Research Papers in Statistical Inference for Time Series and Related Models. Springer, Singapore. https://doi.org/10.1007/978-981-99-0803-5_10

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