Testing for serial independence in vector autoregressive models

Abstract

We consider tests for serial independence of arbitrary finite order for the innovations in vector autoregressive models. The tests are expressed as L2-type criteria involving the difference of the joint empirical characteristic function and the product of corresponding marginals. Asymptotic as well as Monte-Carlo results are presented.

Appendix: Proofs of the results

Recall that ${\mathcal {C}}= C(\mathbb {R}^M \times \mathbb {R}^M, {\mathbb {C}})$, and define on ${\mathcal {C}}$ the metric

$$\begin{aligned} \rho (x,y)= \sum _{j=1}^{\infty } 2^{-j}{\rho _j(x,y) \over 1+\rho _j(x,y) }, \quad \forall j \ge 1, \quad \rho _j(x,y) = \sup _{||w|| \le j} \left| x(w)-y(w) \right| . \end{aligned}$$

It is well known that endowed with $\rho$, ${\mathcal {C}}$ is a separable Fréchet space, and that convergence in this metric corresponds to the uniform convergence on all compact sets. That is for all $x,y \in {\mathcal {C}}$, $\rho (x,y)=0 \Longleftrightarrow \forall j \ge 1, \ \rho _j(x,y)=0$. For random elements $x_n$ and $y_n$ of ${\mathcal {C}}$, $\rho (x_n,y_n) {\mathop {\longrightarrow }\limits ^{P}}0 \Longleftrightarrow \forall j \ge 1, \ \rho _j(x_n,y_n) {\mathop {\longrightarrow }\limits ^{P}}0$.

1.1 Proof of Theorem 4.1

For the proof of Theorem 4.1, from Proposition 16.6, Lemma 16.2 and Theorem 16.3 of Kallenberg (2001), it suffices to show that $\sqrt{T}\varDelta _{T,\varepsilon ,h}^*$ is tight and that its finite-dimensional distributions converges to those of any complex Gaussian process with covariance $\varLambda _h$ and pseudo-covariance ${\widetilde{\varLambda }}_h$.

The convergence of the finite-dimensional distributions is a simple consequence of the Central Limit Theorem (CLT) for h dependent complex-valued random variables.

For the tightness, one can write, for all $u,v \in \mathbb {R}^M$,

$$\begin{aligned} \sqrt{T}\varDelta _{T,\varepsilon ,h}^*(u,v)= & {} {1 \over \sqrt{T}} \sum _{t=1}^T \left[ e^{i(u' \varepsilon _t+v' \varepsilon _{t+h})} -\varphi (u)\varphi (v) \right] \\&+\,\varphi (v){1 \over \sqrt{T}} \sum _{t=1}^T [e^{iu' \varepsilon _t} -\varphi (u)]\\&+\, \varphi (u){1 \over \sqrt{T}} \sum _{t=1}^T [e^{iv' \varepsilon _{t+h}}-\varphi (v)], \end{aligned}$$

which shows that under the null hypothesis ${\mathbb {H}}_0$, $\sqrt{T}\varDelta _{T,\varepsilon ,h}^*$ is a sum of empirical characteristic function processes which are shown to be tight on compact sets by Csörgő (1985).

Now, denoting by ${\mathcal {D}}_h$ any zero-mean complex-valued Gaussian process with covariance and pseudo-covariance kernels $\varLambda _h$ and ${\widetilde{\varLambda }}_h$ respectively, one can see easily that for all $j \ge 1$, $\rho _j(\sqrt{T} \varDelta _{T, \varepsilon ,h}^*, {\mathcal {D}}_h)$ tends in probability to 0 as T tends to infinity. This entails that $\rho (\sqrt{T} \varDelta _{T, \varepsilon ,h}^*, {\mathcal {D}}_h)$ tends in probability to 0 as T tends to infinity. $\qquad \quad \Box$

1.2 Proof of Proposition 4.1

For all $h=1, \ldots , H$, for all $u,v \in \mathbb {R}^M$, and for all $t \in {\mathbb {Z}}$, define the complex-valued random vector function

$$\begin{aligned} \varGamma _{t, \varepsilon , h}(u,v)= & {} [e^{iu' \varepsilon _t} -\varphi (u)] \left( e^{iv' \varepsilon _t} -\varphi (v), e^{iv' \varepsilon _{t+1}} -\varphi (v), \ldots , e^{iv' \varepsilon _{t+h}} -\varphi (v) \right) '. \end{aligned}$$

It is easy to see that

$$\begin{aligned} {1 \over T} \sum _{t=1}^T \varGamma _{t, h, \varepsilon }(u,v)=(\varDelta _{T,\varepsilon , 0}^*(u,v), \varDelta _{T,\varepsilon , 1}^*(u,v), \ldots , \varDelta _{T,\varepsilon , h}^*(u,v))'. \end{aligned}$$

Then, for all $\lambda ^k=(\lambda _0^k, \ldots , \lambda _h^k)' \in \mathbb {R}^{1+h}$, $k=1, \ldots , \ell$, for some non negative integer $\ell$, denote by $\lambda =(\lambda ^{1'},\ldots , \lambda ^{\ell '})'$ and

$$\begin{aligned} \varpi _T=\left( T^{-1} \sum _{t=1}^T \varGamma _{t, \varepsilon , h}'(u_1,v_1), \ldots , T^{-1} \sum _{t=1}^T \varGamma _{t, \varepsilon , h}'(u_{\ell },v_{\ell }) \right) '. \end{aligned}$$

For all integer k, denoting by $\lambda ^{k'}= (\lambda ^k)'$, one has the linear combination

$$\begin{aligned} \lambda '\varpi _T\!=\! & {} {1 \over T} \sum _{t=1}^T \sum _{k=1}^{\ell } \lambda ^{k'} \varGamma _{t, h, \varepsilon }(u_k,v_k) \!=\! {1 \over T} \sum _{t=1}^T \sum _{k=1}^{\ell } [e^{iu_k' \varepsilon _t} -\varphi (u_k)]\\&\times \, \left\{ \lambda _0^1[ e^{iv_k' \varepsilon _t} -\varphi (v_k)]+ \lambda _1^k[ e^{iv_k' \varepsilon _{t+1}} -\varphi (v_k)]+ \cdots + \lambda _h^k[ e^{iv_k' \varepsilon _{t+h}} -\varphi (v_k)]\right\} \!. \end{aligned}$$

Since the following complex-valued random variables

$$\begin{aligned}&\sum _{k=1}^{\ell } [ e^{iu_k' \varepsilon _t} -\varphi (u_k)] \left\{ \lambda _0^1[ e^{iv_k' \varepsilon _t} -\varphi (v_k)]+ \lambda _1^k[ e^{iv_k' \varepsilon _{t+1}} -\varphi (v_k)]+ \cdots \right. \\&\quad \left. +\,\lambda _h^k[ e^{iv_k' \varepsilon _{t+h}} -\varphi (v_k)]\right\} \end{aligned}$$

are h-dependent, one has by the CLT for complex-valued h-dependent random variables that the complex random sum $\lambda ' \varpi _T$ converges in distribution to a zero-mean complex-valued Gaussian random variable with covariance and pseudo-covariance given respectively by

$$\begin{aligned} \sum _{h=1}^H\sum _{j=1}^{\ell } \sum _{k=1}^{\ell } \lambda _h^j \lambda _h^k \varTheta (u_j,u_k) \varTheta (v_j,v_k) \quad \text{ and } \quad \sum _{h=1}^H \sum _{j=1}^{\ell } \sum _{k=1}^{\ell } \lambda _h^j \lambda _h^k \varTheta (u_j,-u_k) \varTheta (v_j,-v_k). \end{aligned}$$

$\square$

1.3 Proof of Theorem 4.2

Lemma 8.1

Under ${\mathbb {H}}_0$, for any $h=1, \ldots , H$,

$$\begin{aligned} \rho \left( \sqrt{T} \varDelta _{T,\varepsilon , h}^*, \sqrt{T} \varDelta _{T,\varepsilon , h} \right) {\mathop {\longrightarrow }\limits ^{P}}0, \quad T \rightarrow \infty . \end{aligned}$$

Proof of Lemma 8.1

For all $j \ge 1$ and $\forall u,v \in \mathbb {R}^M$ such that $||(u,v)|| \le j$, careful developments yield

$$\begin{aligned}&\sqrt{T}\left[ \varDelta _{T,\varepsilon , h}^*(u,v)- \varDelta _{T,\varepsilon , h}(u,v) \right] \nonumber \\&\quad = {1 \over \sqrt{T}} \sum _{t=T-h+1}^T e^{i(u'\varepsilon _t+v'\varepsilon _{t+h})}- \varphi (u)\sqrt{T}\left[ \varphi _{T, \varepsilon }(v) -\varphi (v)\right] \nonumber \\&\qquad -\, \varphi _{T, \varepsilon }(u) \sqrt{T}\left[ \varphi (v)-\varphi _{T, \varepsilon }(v) \right] -{h \over \sqrt{T}} \varphi _{T, \varepsilon }(u)\varphi _{T, \varepsilon }(v) \nonumber \\&\qquad -\,\varphi (u){1 \over \sqrt{T}} \sum _{t=1}^h e^{iu'\varepsilon _t} + \varphi (u){1 \over \sqrt{T}} \sum _{t=T+1}^{T+h} e^{iu'\varepsilon _t} \nonumber \\&\quad = {1 \over \sqrt{T}} \sum _{t=T-h+1}^T e^{i(u'\varepsilon _t+v'\varepsilon _{t+h})}- \varphi (u){1 \over \sqrt{T}} \sum _{t=1}^h e^{iu'\varepsilon _t}+ \varphi (u){1 \over \sqrt{T}} \sum _{t=T+1}^{T+h} e^{iu'\varepsilon _t} \nonumber \\&\qquad -\, \sqrt{T}\left[ \varphi _{T, \varepsilon }(v) -\varphi (v)\right] \left[ \varphi _{T, \varepsilon }(u)-\varphi (u) \right] \nonumber \\&\qquad -\, {h \over \sqrt{T}} \varphi _{T, \varepsilon }(u)\varphi _{T, \varepsilon }(v). \end{aligned}$$

(8.1)

All the functions in the right-hand side of the above equation are continuous on compact sets on whcih they achieve their bounds. Since they all tend in probability to 0 as T goes to infinity, it follows that for all $j\ge 1$,

$$\begin{aligned} \rho _j\left( \sqrt{T}\varTheta _{T,\varepsilon , h}^*,\sqrt{T}\varDelta _{T,\varepsilon , h} \right)= & {} \sup _{||(u,v)|| \le j} |\sqrt{T}\left( \varDelta _{T,\varepsilon , h}^*(u,v)\right. \\&\left. \quad - \varDelta _{T,\varepsilon , h}(u,v) \right) | {\mathop {\longrightarrow }\limits ^{P}}0, \quad T \rightarrow \infty . \end{aligned}$$

This clearly implies that $\rho ( \sqrt{T}\varTheta _{T,\varepsilon , h}^*,\sqrt{T}\varDelta _{T,\varepsilon , h})$ tends in probability to 0 as $T \rightarrow \infty .$ $\square$

$\bullet$ For the proof of Theorem 4.2, one can write that for all $u,v \in \mathbb {R}^M$,

$$\begin{aligned} \sqrt{T} \varDelta _{T, \varepsilon ,h}(u,v)= \sqrt{T} \varDelta _{T, \varepsilon ,h}^*(u,v) + \left[ \sqrt{T} \varDelta _{T, \varepsilon ,h}(u,v)-\sqrt{T} \varDelta _{T, \varepsilon ,h}^*(u,v) \right] . \end{aligned}$$

The proof then results from an application of Lemma 8.1 and Theorem 4.1. $\square$

1.4 Proof of Corollary 4.1

Lemma 8.2

Let $\gamma$ be a complex-valued function defined on $\mathbb {R}^M$, and let $\{\zeta _t ; t=0, \pm 1, \pm 2, \ldots \}$ be any sequence of random vectors such that $\{(y_t, \zeta _t); t=0, \pm 1, \pm 2, \ldots \}$ is ergodic. Then, for all $h=0,1, \ldots , m$ and for all $u \in \mathbb {R}^M$,

• i. If $\zeta _t$ is independent of $\sigma (y_s, s <t)$, $E[|\gamma ( \zeta _t)|^2] < \infty$ and $E[\gamma (\zeta _t)]=0$, one has, in probability,

  $$\begin{aligned} {1 \over \sqrt{T}} \sum _{t=1}^{T-h} [u'({\widehat{\varepsilon }}_t - \varepsilon _t)] \gamma ( \zeta _t) \longrightarrow 0, \quad T \rightarrow \infty . \end{aligned}$$

• ii. If $\gamma$ is bounded, one has, in probability,

  $$\begin{aligned} {1 \over \sqrt{T}} \sum _{t=1}^{T-h} [u'({\widehat{\varepsilon }}_t - \varepsilon _t)]^2 \gamma ( \zeta _t) \longrightarrow 0, \quad T \rightarrow \infty . \end{aligned}$$

Proof of Lemma 8.2

i. For all $u \in \mathbb {R}^M$, one has from the asymptotic normality of the ${\widehat{A}}_j$’s and by ergodicity, one has in probability

$$\begin{aligned} {1 \over \sqrt{T}} \sum _{t=1}^{T-h} u'({\widehat{\varepsilon }}_t - \varepsilon _t) \gamma ( \zeta _t)\!=\! & {} u'\sum _{j=1}^p \sqrt{T}({\widehat{A}}_j - A_j) \left[ {1 \over T}\sum _{t=1}^{T-h} y_{t-j} \gamma (\zeta _t)\right] \longrightarrow 0, \quad T \rightarrow \infty . \end{aligned}$$

ii. For all $u \in \mathbb {R}^M$, one has the equality

$$\begin{aligned} {1 \over \sqrt{T}} \sum _{t=1}^{T-h} [u'({\widehat{\varepsilon }}_t - \varepsilon _t)]^2 \gamma ( \zeta _t)= & {} u\sum _{j=1}^p \sum _{k=1}^p\sqrt{T}({\widehat{A}}_j - A_j)\\&\times \, \left[ {1 \over T}\sum _{t=1}^{T-h} y_{t-j}'y_{t-k} \gamma (\zeta _t) \right] ({\widehat{A}}_k - A_k)'u'. \end{aligned}$$

By our assumptions, for all $j=1, \ldots , p$, $\sqrt{T}({\widehat{A}}_j - A_j)$ is asymptotically normal, and is tight. By ergodicity, the term in square brackets converges almost surely to a finite matrix. The result is handled since the ${\widehat{A}}_j - A_j$’s tend almost surely to zero as T tends to infinity. $\square$

Lemma 8.3

Under ${\mathbb {H}}_0$, for all $h=1, \ldots , H$, as $T \rightarrow \infty$,

• i. $\rho \left( \sqrt{T}\varphi _{T,\varepsilon }, \sqrt{T}\varphi _{T,{\widehat{\varepsilon }}} \right) {\mathop {\longrightarrow }\limits ^{P}}0$

• ii. $\rho \left( \sqrt{T} \varDelta _{T,\varepsilon , h}, \sqrt{T} \varDelta _{T,{\widehat{\varepsilon }}, h} \right) {\mathop {\longrightarrow }\limits ^{P}}0$

• iii. $\rho \left( \sqrt{T} {\widehat{\varDelta }}_{T,{\widehat{\varepsilon }}, h}, \sqrt{T} {\widehat{\varDelta }}_{T, \varepsilon , h} \right) {\mathop {\longrightarrow }\limits ^{P}}0$

• iv. $\rho \left( \sqrt{T} {\widehat{\varDelta }}_{T, \varepsilon , h}, \sqrt{T} \varDelta _{T, \varepsilon , h} \right) {\mathop {\longrightarrow }\limits ^{P}}0$

• v. $\rho \left( \sqrt{T} {\widehat{\varDelta }}_{T,{\widehat{\varepsilon }}, h}, \sqrt{T} \varDelta _{T,{\widehat{\varepsilon }}, h} \right) {\mathop {\longrightarrow }\limits ^{P}}0$.

Proof of Lemma 8.3

$\bullet$ Part i—For all $j \ge 1$ and $\forall u, {\tilde{u}} \in \mathbb {R}^M$ such that $||(u,{\tilde{u}})|| \le j$, considering $\varphi$ as a function of $(u, {\widetilde{u}})$, by a Taylor expansion, one has, for some ${\widetilde{\varepsilon }}_t$ lying between $\varepsilon _t$ and ${\widehat{\varepsilon }}_t$,

$$\begin{aligned} \sqrt{T}\left[ \varphi _{T,\varepsilon }(u)- \varphi _{T,{\widehat{\varepsilon }}}(u) \right]= & {} {1 \over \sqrt{T}} \sum _{t=1}^Tiu'(\varepsilon _t - {\widehat{\varepsilon }}_t)-{1 \over 2\sqrt{T}} \sum _{t=1}^T[u'(\varepsilon _t - {\widehat{\varepsilon }}_t)]^2e^{iu'{\widetilde{\varepsilon }}_t}. \end{aligned}$$

As ${\widetilde{\varepsilon }}_t$ is a function of the $y_t$’s, the sequence $\{(y_t, {\widetilde{\varepsilon }}_t); t=\pm 1, \pm 2, \ldots ,\}$ is ergodic. Then Lemma 8.2 entails that the right-hand side of the last equality converges in probability to 0, as T tends to infinity. Whence, since by continuity $\sqrt{T}\left[ \varphi _{T,\varepsilon }(u)- \varphi _{T,{\widehat{\varepsilon }}}(u) \right]$ achieves its bound on any compact set, it results that

$$\begin{aligned} \sup _{||(u,{\tilde{u}})|| \le j} \sqrt{T}| \varphi _{T,\varepsilon }(u)- \varphi _{T,{\widehat{\varepsilon }}}(u) | {\mathop {\longrightarrow }\limits ^{P}}0, \quad T \rightarrow \infty . \end{aligned}$$

$\bullet$ Part ii—Let $x,y,x_0$ and $y_0$ be complex numbers, then using a Taylor expansion of order one of the complex-valued function $(u,v) \mapsto uv$, one easily has the identity

$$\begin{aligned} xy-x_0y_0=(x-x_0)y_0+(y-y_0)x_0+(x-x_0)(y-y_0). \end{aligned}$$

(8.2)

Now, for all $u,v \in \mathbb {R}^M$, using (8.2), one can write

$$\begin{aligned}&\sqrt{T}\left[ \varDelta _{T, \varepsilon , h}(u,v)- \varDelta _{T,{\widehat{\varepsilon }}, h}(u,v) \right] \nonumber \\&\quad =\, {1 \over \sqrt{T}} \sum _{t=1}^{T-h} \left[ e^{i(u'\varepsilon _t+v'\varepsilon _{t+h})}-e^{i(u' {\widehat{\varepsilon }}_t+v' {\widehat{\varepsilon }}_{t+h})} \right] \nonumber \\&\qquad +\, \sqrt{T}\left[ \varphi _{T, {\widehat{\varepsilon }}}(u) -\varphi _{T, \varepsilon }(u) \right] \varphi _{T, \varepsilon }(v) + \sqrt{T}\left[ \varphi _{T, {\widehat{\varepsilon }}}(v) -\varphi _{T, \varepsilon }(v) \right] \varphi _{T, \varepsilon }(u)\nonumber \\&\qquad +\, \sqrt{T}\left[ \varphi _{T, {\widehat{\varepsilon }}}(u) -\varphi _{T, \varepsilon }(u) \right] \left[ \varphi _{T, {\widehat{\varepsilon }}}(v) -\varphi _{T, \varepsilon }(v) \right] . \end{aligned}$$

(8.3)

Now, all the functions in the right-hand side of the above equality are continuous an achieve their bounds on any ball with radius j. By a first-order Taylor expansion, the first can be treated as in the proof of Part i, and its sup which is reached over any ball with radius j tends to 0 as T tends to infinity.

Next, using Part i, since over any ball with radius j, $\sqrt{T}\left( \varphi _{T, {\widehat{\varepsilon }}} - \varphi _{T, \varepsilon }\right)$ tends in probability to 0 as T tends to infinity, it is easy to see that over any ball with radius j the remaining terms converge in probability to 0 as T tends to infinity. Whence, for all $j \ge 1$,

$$\begin{aligned} \rho _j(\sqrt{T}\left( \varDelta _{T, {\widehat{\varepsilon }}, h}, \varDelta _{T,\varepsilon , h}\right) = \sup _{||(u,v)|| \le j}|\sqrt{T}\left( \varDelta _{T, {\widehat{\varepsilon }}, h}(u,v)- \varDelta _{T,\varepsilon , h}(u,v) \right) | \longrightarrow 0, \ T \rightarrow \infty . \end{aligned}$$

$\bullet$ Part iii—For all $u,v \in \mathbb {R}^M$, one has :

$$\begin{aligned} \sqrt{T}\left[ {\widehat{\varDelta }}_{T, {\widehat{\varepsilon }}, h}(u,v)- {\widehat{\varDelta }}_{T, \varepsilon , h}(u,v) \right]= & {} {T \over T -h} \sqrt{T}\left[ \varDelta _{T, {\widehat{\varepsilon }}, h}(u,v)- \varDelta _{T, \varepsilon , h}(u,v) \right] . \end{aligned}$$

The result follows immediately from Part ii.

$\bullet$ Part iv—For all $u,v \in \mathbb {R}^M$ one can easily check that

$$\begin{aligned} \sqrt{T}\left[ {\widehat{\varDelta }}_{T, \varepsilon , h}(u,v)- \varDelta _{T, \varepsilon , h}(u,v) \right]= & {} -{h\sqrt{T} \over T -h} \varDelta _{T, \varepsilon , h}(u,v). \end{aligned}$$

The result is handled from the fact that $\sup _{||(u,v)|| \le j}|\varDelta _{T, \varepsilon , h}(u,v)|$ converges almost surely to $\sup _{||(u,v)|| \le j}|\varDelta _{\varepsilon , h}(u,v)|$ and the fact that $h\sqrt{T} /(T -h)$ tends to 0.

$\bullet$ Part v—For all $u,v \in \mathbb {R}^M$ one can write the decomposition

$$\begin{aligned} \sqrt{T} {\widehat{\varDelta }}_{T,{\widehat{\varepsilon }}, h}(u,v)- \sqrt{T} \varDelta _{T, {\widehat{\varepsilon }}, h} (u,v)= & {} \sqrt{T}\left[ {\widehat{\varDelta }}_{T, {\widehat{\varepsilon }}, h}(u,v)-{\widehat{\varDelta }}_{T, \varepsilon , h}(u,v)\right] \\&+\, \sqrt{T} \left[ {\widehat{\varDelta }}_{T,\varepsilon , h}(u,v)-\varDelta _{T, \varepsilon , h} (u,v)\right] \\&+\,\sqrt{T} \left[ \varDelta _{T, \varepsilon , h}(u,v)-\varDelta _{T, {\widehat{\varepsilon }}, h}(u,v) \right] . \end{aligned}$$

The result follows by the triangle inequality and the application of Parts ii–iv. $\square$

For the proof of Part i of Corollary 4.1, one has from the second point of Lemma 8.3 that $\sqrt{T} \varDelta _{T, {\widehat{\varepsilon }}, h}$ has the same asymptotic distribution as $\sqrt{T} \varDelta _{T, \varepsilon , h}^*$, which, by Theorem 4.2 is that of any zero-mean complex-valued Gaussian random variable with covariance kernel $\varLambda _h$ and pseudo-covariance kernel ${\widetilde{\varLambda }}_h$.

• The second part of Corollary 4.1 is a direct consequence of its Part i and Parts iii–v of Lemma 8.3.

• The last point of Corollary 4.1 results from the fact that, from Lemma 8.1 and Theorem 4.1, one has, for all $h=0,1, \ldots ,H$, and $u_h, v_h \in \mathbb {R}^M$,

  $$\begin{aligned} \sqrt{T} {\widehat{\varDelta }}_{T,{\widehat{\varepsilon }}, h}(u_h,v_h) =\sqrt{T} \varDelta _{T, \varepsilon , h}^*(u_h,v_h)+o_p(1). \end{aligned}$$

$\square$

1.5 Proof of Corollary 4.2

We first show that under ${\mathbb {H}}_0$,

$$\begin{aligned}&\int _{\mathbb {R}^M} \int _{\mathbb {R}^M} T \left| {\widehat{\varDelta }}_{T, {\widehat{\varepsilon }}, h}(u,v) \right| ^2 dW(u,v) \nonumber \\&\quad =\int _{\mathbb {R}^M} \int _{\mathbb {R}^M}T \left| \varDelta _{T, \varepsilon , h}^*(u,v) \right| ^2 dW(u,v)+o_P(1). \end{aligned}$$

(8.4)

In this purpose, we show the following:

Lemma 8.4

Under ${\mathbb {H}}_0$,

• i. $\displaystyle \int _{\mathbb {R}^M} \int _{\mathbb {R}^M} T \left| \varDelta _{T, \varepsilon , h}^*(u,v) \right| ^2 dW(u,v)$ is tight.

• ii. $\displaystyle \int _{\mathbb {R}^M} \int _{\mathbb {R}^M} T \left| {\widehat{\varDelta }}_{T, {\widehat{\varepsilon }}, h}(u,v) - \varDelta _{T, \varepsilon , h}^*(u,v) \right| ^2 dW(u,v)$ tends in probability to 0.

Proof

For the point i, it is easy to see that for all $u,v \in \mathbb {R}^M$, under ${\mathbb {H}}_0$,

$$\begin{aligned} E(| \sqrt{T}\varDelta _{T, \varepsilon , h}^*(u,v)|^2)= \varTheta (u,v). \end{aligned}$$

This entails by Fubini–Tonelli theorem that

$$\begin{aligned} E \left[ \int _{\mathbb {R}^M} \int _{\mathbb {R}^M} T \left| \varDelta _{T, \varepsilon , h}^*(u,v) \right| ^2 dW(u,v) \right] =\int _{\mathbb {R}^M} \int _{\mathbb {R}^M} \varTheta (u,v)dW(u,v) < \infty . \end{aligned}$$

For the point ii, adding and subtracting appropriate terms, one can write that for all $u,v \in \mathbb {R}^M$,

$$\begin{aligned}&\sqrt{T} \left[ {\widehat{\varDelta }}_{T, {\widehat{\varepsilon }}, h}(u,v) -\varDelta _{T, \varepsilon , h}^*(u,v) \right] \\&\quad =\, {T \over T-h}\left\{ \sqrt{T} \left[ \varDelta _{T, {\widehat{\varepsilon }}, h}(u,v) - \varDelta _{T, \varepsilon , h}(u,v) \right] \right\} \\&\qquad +\, {h \sqrt{T} \over T-h}\varDelta _{T, \varepsilon , h}(u,v) + \sqrt{T} \left[ \varDelta _{T, \varepsilon , h}(u,v) - \varDelta _{T, \varepsilon , h}^*(u,v) \right] . \end{aligned}$$

It is now easy to see that

$$\begin{aligned}&\int _{\mathbb {R}^M} \int _{\mathbb {R}^M} T \left| {\widehat{\varDelta }}_{T, {\widehat{\varepsilon }}, h}(u,v) - \varDelta _{T, \varepsilon , h}^*(u,v) \right| ^2 dW(u,v) \\&\quad \le \left( {T \over T-h}\right) ^2 \int _{\mathbb {R}^M} \int _{\mathbb {R}^M} T \left| \varDelta _{T, {\widehat{\varepsilon }}, h}(u,v) - \varDelta _{T, \varepsilon , h}(u,v) \right| ^2 dW(u,v) \nonumber \\&\qquad + \left( {h\sqrt{T} \over T-h}\right) ^2 \int _{\mathbb {R}^M} \int _{\mathbb {R}^M} \left| \varDelta _{T, \varepsilon , h}(u,v) \right| ^2 dW(u,v) \nonumber \\&\qquad + \int _{\mathbb {R}^M} \int _{\mathbb {R}^M} T \left| \varDelta _{T, \varepsilon , h}(u,v) - \varDelta _{T, \varepsilon , h}^*(u,v) \right| ^2 dW(u,v) \nonumber . \end{aligned}$$

(8.5)

Looking close to the proofs of Parts i–ii of Lemma 8.3, one can see that the integral in the first term in the right-hand side of (8.5) can be bounded by

$$\begin{aligned} Z_T \int _{\mathbb {R}^M} \int _{\mathbb {R}^M} || (u,v)||^2 dW(u,v), \end{aligned}$$

where $\{Z_T; T \ge 1 \}$ is a sequence of random variables which tends in probability to 0 as T tends to infinity. Thus, the first term in the right-hand side of (8.5) tends in probability to 0, as T tends to infinity.

Since $| \varDelta _{T, \varepsilon , h}(u,v) |$ is bounded by 4 over $\mathbb {R}^M \times \mathbb {R}^M$, the second term is bounded by

$$\begin{aligned} \left( {h\sqrt{T} \over T-h}\right) ^2 \int _{\mathbb {R}^M} \int _{\mathbb {R}^M} dW(u,v), \end{aligned}$$

which tends in probability to 0, as T tends to infinity.

For the last integral in the right-hand side of (8.5), as for the first term, one can deduce from (8.1) that, since $\sqrt{T} (\varphi _{T, \varepsilon , h}-\varphi )$ is tight, using Cauchy–Schwarz inequality, the convergence in probability to 0 of this term is easily obtained by the dominated convergence theorem.

Now, for all $u,v \in \mathbb {R}^M$, developing the right-hand side of

$$\begin{aligned} | {\widehat{\varDelta }}_{T, {\widehat{\varepsilon }}, h}(u,v)|^2 = \left| \varDelta _{T, \varepsilon , h}^*(u,v)+ \left[ {\widehat{\varDelta }}_{T, {\widehat{\varepsilon }}, h}(u,v) - \varDelta _{T, \varepsilon , h}^*(u,v) \right] \right| ^2 \end{aligned}$$

and integrating both sides with respect to the weight function W, one has

$$\begin{aligned}&\int _{\mathbb {R}^M} \int _{\mathbb {R}^M} T \left| {\widehat{\varDelta }}_{T, {\widehat{\varepsilon }}, h}(u,v) \right| ^2 dW(u,v) \\&\quad =\,\int _{\mathbb {R}^M} \int _{\mathbb {R}^M}T \left| \varDelta _{T, \varepsilon , h}^*(u,v) \right| ^2 dW(u,v)\\&\qquad +\, \int _{\mathbb {R}^M} \int _{\mathbb {R}^M}T \overline{\varDelta _{T, \varepsilon , h}^*(u,v)} \left[ {\widehat{\varDelta }}_{T, {\widehat{\varepsilon }}, h}(u,v) - \varDelta _{T, \varepsilon , h}^*(u,v) \right] dW(u,v) \\&\qquad +\, \int _{\mathbb {R}^M} \int _{\mathbb {R}^M}T \varDelta _{T, \varepsilon , h}^*(u,v)\overline{ \left[ {\widehat{\varDelta }}_{T, {\widehat{\varepsilon }}, h}(u,v) - \varDelta _{T, \varepsilon , h}^*(u,v) \right] } dW(u,v)\\&\qquad +\int _{\mathbb {R}^M} \int _{\mathbb {R}^M}T \left| {\widehat{\varDelta }}_{T, {\widehat{\varepsilon }}, h}(u,v) - \varDelta _{T, \varepsilon , h}^*(u,v) \right| ^2dW(u,v). \end{aligned}$$

Then applying Cauchy-Schwarz inequality joint with Lemma 8.4 to the second and third terms in the right-hand side of the above equality, one has (8.4).

For the complete proof of Corollary 4.2, it remains to show that, as T tends to infinity,

$$\begin{aligned} \int _{\mathbb {R}^M} \int _{\mathbb {R}^M}T \left| \varDelta _{T, \varepsilon , h}^*(u,v) \right| ^2 dW(u,v) \longrightarrow \int _{\mathbb {R}^M} \int _{\mathbb {R}^M}T \left| {\mathcal {D}}_h(u,v) \right| ^2 dW(u,v). \end{aligned}$$

It suffices to show that the condition in Theorem 2.3 of Bilodeau and Lafaye de Micheaux (2005) holds. This is immediate since, as already proved above

$$\begin{aligned}&\lim _{j \rightarrow \infty } \int _{||(u,v)|| \ge j} E \left[ T \left| \varDelta _{T, \varepsilon , h}^*(u,v) \right| ^2 \right] dW(u,v) \\&\quad =\lim _{j \rightarrow \infty } \int _{||(u,v)|| \ge j} \varTheta (u,v)dW(u,v)= 0. \end{aligned}$$

$\square$