Abstract
We propose generalized \(C(\alpha )\) tests for testing linear and nonlinear parameter restrictions in models specified by estimating functions. The proposed procedures allow for general forms of serial dependence and heteroskedasticity, and can be implemented using any root-n consistent restricted estimator. The asymptotic distribution of the proposed statistic is established under weak regularity conditions. We show that earlier \(C(\alpha )\)-type statistics are included as special cases. The problem of testing hypotheses fixing a subvector of the complete parameter vector is discussed in detail as another special case. We also show that such tests provide a simple general solution to the problem of accounting for estimated parameters in the context of two-step procedures where a subvector of model parameters is estimated in a first step and then treated as fixed.
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Notes
- 1.
- 2.
The number of observations in the dataset Y could be different from n, say is equal to \(n_{2}, n_{2}\ne n\). If the auxiliary estimate \(\tilde{ \theta }_{2n_{2}}^{0}\) obtained from the second dataset satisfies \(\sqrt{n_{2} }(\tilde{\theta }_{2n_{2}}^{0}-\theta _{20})=O_{p}(1)\), then \(\sqrt{n}(\tilde{ \theta }_{2n_{2}}^{0}-\theta _{20})=\sqrt{n/n_{2}}\sqrt{n_{2}}(\tilde{\theta } _{2n_{2}}^{0}-\theta _{20})=O_{p}(1)\) provided \(n/n_{2}=O(1)\), and the arguments that follow remain valid. When a set of estimating functions \( D_{n_{2}2}(\theta _{2})\) for the second dataset is considered, the argument presented here remains valid provided \(\sqrt{n_{2}}D_{n_{2}2}(\theta _{20})\) obeys a central limit theorem in addition to the previous conditions on the auxiliary estimate and the sample sizes.
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Acknowledgements
The authors thank Marine Carrasco, Jean-Pierre Cotton, Russell Davidson, Abdeljelil Farhat, V. P. Godambe, Christian Genest, Christian Gouriéroux, Stéphane Grégoir, Tianyu He, Frank Kleibergen, Sophocles Mavroeidis, Hervé Mignon, Julien Neves, Denis Pelletier, Mohamed Taamouti, Masaya Takano, Pascale Valéry, two anonymous referees, and the Editor Wai Keung for several useful comments. Earlier versions of this paper were presented at the Canadian Statistical Society 1997 annual meeting and at INSEE (CREST, Paris). This work was supported by the William Dow Chair in Political Economy (McGill University), the Bank of Canada (Research Fellowship), the Toulouse School of Economics (Pierre-de-Fermat Chair of excellence), the Universitad Carlos III de Madrid (Banco Santander de Madrid Chair of excellence), a Guggenheim Fellowship, a Konrad-Adenauer Fellowship (Alexander-von-Humboldt Foundation, Germany), the Canadian Network of Centres of Excellence [program on Mathematics of Information Technology and Complex Systems (MITACS)], the Natural Sciences and Engineering Research Council of Canada, the Social Sciences and Humanities Research Council of Canada, and the Fonds de recherche sur la société et la culture (Québec).
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Appendix
Appendix
Proof of Proposition 1
To simplify notation, we shall assume throughout that \(\omega \in \mathscr {D} _{J}\) (an event with probability 1) and drop the symbol \(\omega \) from the random variables considered. In order to obtain the asymptotic null distribution of the generalized \(C(\alpha )\) statistic defined in (14), we first need to show that \(P(\tilde{\theta }_{n}^{0})\) and \( J_{n}(\tilde{\theta }_{n}^{0})\) converge in probability to \(P(\theta _{0})\) and \(J(\theta _{0})\) respectively. The consistency of \(P(\tilde{\theta } _{n}^{0}),\) i.e.
follows simply from the consistency of \(\tilde{\theta }_{n}^{0}\) [Assumption 9] and the continuity of \(P(\theta )\) at \(\theta _{0}\) [Assumption 7]. Further, by Assumption 8, since \( P(\theta )\) is continuous in open neighborhood of \(\theta _{0},\) we also have
Consider now \(J_{n}(\tilde{\theta }_{n}^{0}).\) By Assumption 5, for any \(\varepsilon >0\) and \(\varepsilon _{1}>0,\) we can choose \(\delta _{1} :=\delta (\varepsilon _{1}, \varepsilon )>0\) and a positive integer \(n_{1}(\varepsilon , \delta _{1})\) such that: (i) \(U_{J}(\delta _{1}, \varepsilon , \theta _{0})\le \varepsilon _{1}/2,\) and (ii) \(n>n_{1}(\varepsilon , \delta _{1})\) entails
Further, by the consistency of \(\tilde{\theta }_{n}^{0}\) [Assumption 9], we can choose \(n_{2}(\varepsilon _{1}, \delta _{1})\) such that \(n>n_{2}(\varepsilon _{1}, \delta _{1})\) entails \(\mathsf {P}[\Vert \tilde{\theta }_{n}^{0}-\theta _{0}\Vert \le \delta _{1}]\ge 1-(\varepsilon _{1}/2).\) Then, using the Boole-Bonferroni inequality, we have for \(n>\max \{n_{1}(\varepsilon , \delta _{1}), n_{2}(\varepsilon _{1}, \delta _{1})\}\):
Thus,
hence
or, equivalently,
By Assumption 4, we can write [setting \( 0/0=0]:\)
where
and \(\mathop {\lim \sup }\nolimits _{n\rightarrow \infty }\,\mathsf {P}\big [ r_{n}(\delta , \theta _{0})>\varepsilon \big ] <U_{D}(\delta , \varepsilon , \theta _{0}).\) Thus, for any \(\varepsilon >0\) and \(\delta >0,\) we have:
hence, using the consistency of \(\tilde{\theta }_{n}^{0},\)
Since \(\mathop {\lim }\nolimits _{\delta \downarrow 0}\,U_{D}(\delta , \varepsilon , \theta _{0})=0,\) it follows that \(\mathop {\lim }\nolimits _{n\rightarrow \infty }\, \mathsf {P}[\Vert R_{n}(\tilde{\theta }_{n}^{0}, \theta _{0})\Vert /\Vert \tilde{\theta }_{n}^{0}-\theta _{0}\Vert \le \varepsilon ]=1\) for any \( \varepsilon >0,\) or equivalently,
Since \(\sqrt{n}\,(\tilde{\theta }_{n}^{0}-\theta _{0})\) is asymptotically bounded in probability (by Assumption 9), this entails:
and
By Taylor’s theorem and Assumptions 7 and 8, we also have the expansion:
for \(\theta \in N\subseteq N_{0}\cap V_{0},\) where N is a non-empty open neighborhood of \(\theta _{0}\) and
i.e., \(R_{2}(\theta , \theta _{0})=o(\big \Vert \theta -\,\theta _{0}\big \Vert )\), so that, using Assumption 10,
for \(\tilde{\theta }_{n}^{0}\in N,\) and
By (74) and (76) jointly with the Assumptions 3, 6, 7, 8, 11 and 12, we have:
so the matrices \(\tilde{J}_{n},\;\tilde{I}_{n},\) and \(W_{n}\) all have full column rank. Since \(\mathop {\text {plim}}\limits _{n\rightarrow \infty }\, \tilde{P}_{n}=P(\theta _{0})\) and \(\mathop {\text {plim}}\limits _{n\rightarrow \infty } \tilde{J}_{n}=J(\theta _{0})\), we can then write:
where \(\tilde{Q}_{n} :=\tilde{Q}[W_{n}]=\tilde{P}_{n}[\tilde{J} _{n}^{\,\prime } W_{n} \tilde{J}_{n}]^{-1}\tilde{J} _{n}^{\,\prime } W_{n}.\) Then, using (78) and (81), it follows that:
We conclude that the asymptotic distribution of \(\sqrt{n}\,\tilde{Q}_{n}D_{n}(\tilde{\theta }_{n}^{0})\) is the same as the one of \( Q(\theta _{0})\sqrt{n}\,D_{n}(\theta _{0}),\) namely (by Assumption 2) a \(\mathrm {N}[0,V_{\psi }(\theta _{0})] \) distribution where
and \(V_{\psi }(\theta _{0})\) has rank \(p_{1}=\,\mathrm {rank}\big [ Q(\theta _{0})\big ] =\,\mathrm {rank}[P(\theta _{0})].\) Consequently, the estimator
converges to \(V_{\psi }(\theta _{0})\) in probability and, by (82),
Thus the test criterion
has an asymptotic \(\chi ^{2}(p_{1})\) distribution.\(\square \)
Proof of Proposition 2
Consider the (non-empty) open neighborhood \(N=N_{1}\cap N_{2}\) of \(\theta _{0}.\) For any \(\theta \in N\) and \(\omega \in \mathscr {Z},\) we can write
By Assumption 14b, we have
and we can find a subsequence \(\left\{ J_{n_{t}}(\theta ,\omega ):t{=}1,2,\dots \right\} \) of \(\left\{ J_{n}(\theta ,\omega ):n=1,2,\dots \right\} \) such that
Let
and \(\varepsilon >0.\) By definition, \(\mathsf {P}\big [ \omega \in CS\big ] =1.\) For \(\,\omega \in CS,\) we can choose \(t_{0}(\varepsilon , \omega )\) such that
Further, since \(J_{n}(\theta , \omega )\) is continuous in \(\theta \) at \( \theta _{0},\) we can find \(\delta (n, \omega )>0\) such that
Thus, taking \(t_{0}=t_{0}(\varepsilon , \omega )\) and \(n=n_{t_{0}},\) we find that \(\big \Vert \theta -\theta _{0}\big \Vert <\delta (n_{t_{0}},\omega )\) implies
In other words, for any \(\varepsilon >0,\) we can choose \(\delta =\delta (n_{t_{0}},\varepsilon )>0\) such that
and the function \(J(\theta )\) must be continuous at \(\theta _{0}.\) Part (a) of the Proposition is established.
Set \(\overline{\varDelta }_{n}(N_{2}, \omega ) :=\sup \left\{ \Vert J_{n}(\theta , \omega )-J(\theta )\Vert :\theta \in N_{2}\right\} .\) To get Assumption 5, we note that
for any \(\delta >0,\) hence, by Assumption 14b,
for any function \(U_{J}(\delta , \varepsilon , \theta _{0})\) that satisfies the conditions of Assumption 5. The latter thus holds with \(V_{0}\) any non-empty open neighborhood of \(\theta _{0}\) such that \(V_{0}\subseteq N_{2}\). To obtain Assumption 4, we note that Assumption 14 entails \(D_{n}(\theta , \omega )\) is continuously differentiable in an open neighborhood of \(\theta _{0}\) for all \(\omega \in \mathscr {D}_{J},\) so that we can apply Taylor’s formula for a function of several variables (see Edwards [26, Section II.7]) to each component of \(D_{n}(\theta , \omega ):\) for all \(\theta \) in an open neighborhood U of \(\theta _{0}\) (with \(U\subseteq N_{2}\)), we can write
where \(J_{n}(\theta , \omega )_{i\cdot }\) and \(J(\theta )_{i\cdot }\) are the i-th rows of \(J_{n}(\theta , \omega )\) and \(J(\theta )\) respectively,
and \(\bar{\theta }_{n}^{\,i}(\omega )\) belongs to the line joining \(\theta \) and \(\theta _{0}.\) Further, for \(\theta \in U,\)
hence, on defining \(N_{0}=U,\)
we see that
and
Thus \(r_{n}(\delta , \theta _{0}, \omega )\underset{n\rightarrow \infty }{ \overset{\mathsf {p}}{\longrightarrow }}0\) and
must hold for any function that satisfies the conditions of Assumption 4. This completes the proof.\(\square \)
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Dufour, JM., Trognon, A., Tuvaandorj, P. (2016). Generalized \(C(\alpha )\) Tests for Estimating Functions with Serial Dependence. In: Li, W., Stanford, D., Yu, H. (eds) Advances in Time Series Methods and Applications . Fields Institute Communications, vol 78. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-6568-7_7
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