Testing serial independence with functional data

Abstract

We consider tests of serial independence for a sequence of functional observations. The new methods are formulated as L2-type criteria based on empirical characteristic functions and are convenient from the computational point of view. We derive asymptotic normality of the proposed test statistics for both discretely and continuously observed functions. In a Monte Carlo study, we show that the new test is sensitive with respect to functional GARCH alternatives, investigate the choice of necessary tuning parameters, and demonstrate that critical values obtained by resampling lead to a test with good performance in any setup, whereas the asymptotic critical values may be recommended only for a sufficiently fine discretization grid. Finite-sample comparison with a distance (auto)covariance test criterion is also included, and the article concludes with application on a real data set.

A Proofs

Proof of Theorem 1

As a shorthand, we set $T_n:=\varDelta _{n,H; p}/\sqrt{H}$ for the test statistic. Also with the following simplified notation:

$$\begin{aligned} \widetilde{g}(\varvec{u},\varvec{v}; \varvec{X}_{j}, \varvec{X}_{j+h})=&g(\varvec{u}, \varvec{v}; \varvec{X}_{j},\varvec{X}_{j+h})-\mathbb {E} \left\{ g(\varvec{u},\varvec{v}; \varvec{X}_{j},\varvec{X}_{j+h})| \varvec{X}_j\right\} \\&- \mathbb {E}\left\{ g( \varvec{u},\varvec{v}; \varvec{X}_{j}, \varvec{X}_{j+h})|\varvec{X}_{j+h}\right\} +\mathbb {E} g(\varvec{u},\varvec{v};\varvec{X}_{j},\varvec{X}_{j+h}),\\ g(\varvec{u}, \varvec{v}; \varvec{X}_{j},\varvec{X}_{j+h})=&\cos (\varvec{u}^{\top } \varvec{X}_j+\varvec{v} ^{\top } \varvec{X}_{j+h}) +\sin (\varvec{u}^{\top } \varvec{X}_j+\varvec{v} ^{\top } \varvec{X}_{j+h}), \end{aligned}$$

repeated use of the simple properties $\mathbb {E} \widetilde{g}(\varvec{u}, \varvec{v}; \varvec{X}_{j_1}, \varvec{X}_{j_1+h_1})\widetilde{g}(\varvec{u}, \varvec{v};\varvec{X}_{j_2}, \varvec{X}_{j_2+h_2})=0\ \text{ if } \ j_1\ne j_2, \ \text{ or } \text{ if } \ j_1=j_2\ \text{ and } \ h_1\ne h_2$ and $\mathbb {E} \widetilde{g} ( \varvec{u},\varvec{v}; \varvec{X}_{j}, \varvec{X}_{j+h})=0$ is made. $\square$

The test statistic $T_n$ will be decomposed into several summands, some of them negligible and some others influential. Notice that under the null hypothesis $T_n$ can be rewritten as:

$$\begin{aligned}&T_n= \frac{1}{\sqrt{H}} \sum _{h=1}^H (n-h)\iint \Bigg |\frac{1}{n-h} \sum _{j=1}^{n-h}\left\{ \exp ( {\mathrm{i}}\varvec{u}^\top \varvec{X}_j) -\varphi _x(\varvec{u})\right\} \left\{ \exp ({\mathrm{i}} \varvec{v}^\top \varvec{X}_{j+h}) -\varphi _x(\varvec{v})\right\} \\&\quad -\frac{1}{(n-h)^2}\sum _{j=1}^{n-h}\left\{ \exp ({\mathrm{i}} \varvec{u}^\top \varvec{X}_j) -\varphi _x(\varvec{u})\right\} \sum _{r=1}^{n-h}\left\{ \exp ({\mathrm{i}} \varvec{v}^\top \varvec{X}_{r+h}) -\varphi _x(\varvec{v}) \right\} \Bigg |^2 w(\varvec{u}) w(\varvec{v}) \mathrm{d}\varvec{u} \mathrm{d}\varvec{v} , \end{aligned}$$

where $\varphi _x(\varvec{u})$ is CF of $\varvec{X}_j$. Since

$$\begin{aligned}&\iint \mathbb {E} \Bigg |\frac{1}{n-h} \sum _{j=1}^{n-h}\left\{ \exp ( {\mathrm{i}}\varvec{u}^\top \varvec{X}_j)-\varphi _x(\varvec{u})\right\} \sum _{s=1}^{n-h}\left\{ \exp ({\mathrm{i}} \varvec{v}^\top \varvec{X}_{s+h})-\varphi _x(\varvec{v})\right\} \Bigg |^2 w( \varvec{u}) w(\varvec{v}) \mathrm{d}\varvec{u} \mathrm{d}\varvec{v} \nonumber \\&\quad \le D\iint w( \varvec{u}) w( \varvec{v}) \mathrm{d}\varvec{u} \mathrm{d}\varvec{v}=O(1) \end{aligned}$$

(20)

for some $D>0$ then also

$$\begin{aligned} T_{n}= T_{n,1} + O\big ( H^{1/4} n^{-1/2}\big ), \end{aligned}$$

(21)

where

$$\begin{aligned} T_{n,1}=&\frac{1}{\sqrt{H}} \sum _{h=1}^H (n-h)\iint \Bigg |\frac{1}{n-h} \sum _{j=1}^{n-h}\left\{ \exp ({\mathrm{i}} \varvec{u}^\top \varvec{X}_j) -\varphi _x(\varvec{u})\right\} \left\{ \exp ({\mathrm{i}} \varvec{v}^\top \varvec{X}_{j+h}) -\varphi _x(\varvec{v})\right\} \Bigg |^2 \\&\times w(\varvec{u}) w(\varvec{v}) \mathrm{d}\varvec{u} \mathrm{d}\varvec{v}\\ =&\frac{1}{\sqrt{H}} \sum _{h=1}^H (n-h)\iint \Bigg \{\frac{1}{n-h} \sum _{j=1}^{n-h} \widetilde{g}(\varvec{u},\varvec{v}; \varvec{X}_{j}, \varvec{X}_{j+h})\Bigg \}^2 w(\varvec{u}) w(\varvec{v}) \mathrm{d}\varvec{u} \mathrm{d}\varvec{v}. \end{aligned}$$

The latter expression for $T_{n,1}$ is obtained after long but straightforward calculations using elementary properties of trigonometric functions and the assumptions on $w(\cdot )$. Further, $T_{n,1}$ can be decomposed as

$$\begin{aligned} T_{n,1}=&\frac{1}{\sqrt{H}} \sum _{h=1}^H \frac{1}{n-h}\iint \sum _{j=1}^{n-h} \widetilde{g}^2(\varvec{u},\varvec{v}; \varvec{X}_{j}, \varvec{X}_{j+h}) w(\varvec{u}) w(\varvec{v}) \mathrm{d}\varvec{u} \mathrm{d}\varvec{v}\\&+ \frac{1}{\sqrt{H}} \sum _{h=1}^H \iint \frac{1}{n-h} \sum _{j_1=1}^{n-h}\sum _{j_2=1}^{n-h} I\{|j_1-j_2|\ge H\} \widetilde{g}(\varvec{u},\varvec{v}; \varvec{X}_{j_1},\varvec{X}_{j_1+h})\\&\quad \times \widetilde{g}(\varvec{u},\varvec{v}; \varvec{X}_{j_2}, \varvec{X}_{j_2+h}) w(\varvec{u}) w(\varvec{v}) \mathrm{d}\varvec{u} \mathrm{d}\varvec{v}\\&+ \frac{1}{\sqrt{H}} \sum _{h=1}^H \frac{1}{n-h}\iint \sum _{j_1=1}^{n-h}\sum _{j_2=1}^{n-h} I\{j_1\ne j_2;|j_1-j_2|< H\} \widetilde{g}(\varvec{u},\varvec{v}; \varvec{X}_{j_1}, \varvec{X}_{j_1+h})\\&\quad \times \widetilde{g}(\varvec{u},\varvec{v};\varvec{X}_{j_2}, \varvec{X}_{j_2+h}) w(\varvec{u}) w(\varvec{v}) \mathrm{d}\varvec{u} \mathrm{d}\varvec{v} := T_{n,11} +T_{n,12}+T_{n,13}, \end{aligned}$$

where $I\{A\}$ is the indicator of the set A. We study these terms separately and show that $T_{n,11}$ and $T_{n,12}$ are influential, while $T_{n,13}$ is negligible. Their properties are formulated in the next two lemmas.

Lemma 1

Under the assumptions of Theorem 1, it holds that $\mathbb {E} T_{n,13}=0$, $\mathbb {E} T^2_{n,13}= O(H^3/n)$, and $\mathbb {E} T_{n,11}=\sqrt{H} \mathbb {E} \iint \widetilde{g}^2(\varvec{u},\varvec{v}; \varvec{X}_{1}, \varvec{X}_{2}) w(\varvec{u}) w(\varvec{v}) \mathrm{d}\varvec{u} \mathrm{d}\varvec{v} =\sqrt{H} \gamma _p +O_P(H n^{-1/2})$.

Proof of Lemma 1

The assertions are obtained by directly calculating expectations and variances. $\square$

It remains to study $T_{n,12}$ which is the most difficult part. The aim is to prove that $T_{n,12}$ has asymptotically normal distribution with zero mean and a finite variance.

Lemma 2

Under the assumptions of Theorem 1, it holds that $\frac{ T_{n,12}}{ \sqrt{ \nu _p}} {\mathop {\rightarrow }\limits ^{{\mathcal {L}}}} {\mathcal {N}}(0,1)$ as $n\rightarrow \infty$.

Proof of Lemma 2

It suffices to investigate $\widehat{T}_{n,12}= \sum _{j_2=H}^{n-H} Q_{j_2,n}$ and

$$\begin{aligned} Q_{j_2,n}=\frac{1}{\sqrt{H}} \sum _{h=1}^H \iint \frac{1}{n-H} \sum _{j_1=1}^{j_2-H} \widetilde{g}(\varvec{u},\varvec{v}; \varvec{X}_{j_1}, \varvec{X}_{j_1+h}) \widetilde{g}(\varvec{u},\varvec{v}; \varvec{X}_{j_2}, \varvec{X}_{j_2+h}) w(\varvec{u}) w(\varvec{v}) \mathrm{d}\varvec{u} \mathrm{d}\varvec{v}. \end{aligned}$$

Since $\mathbb {E}\big ( Q_{j_2,n}|\varvec{X}_1,\ldots ,\varvec{X}_{j_2-1}\big )=0, \quad j_2=H,\dots , n-H$, $\widehat{T}_{n,12}$ is the sum of martingale differences. To show the asymptotic normality of $\widehat{T}_{n,12}$, we apply Theorem 24.3 (page 383) in Davidson (1994), which means to prove that as $n\rightarrow \infty$

$$\begin{aligned} \frac{\sum _{j_2=H}^{n-H} Q^2_{j_2,n}}{ \mathbb {E} \sum _{j_2=H}^{n-H} Q^2_{j_2,n}} {\mathop {\rightarrow }\limits ^{P}} 1 \quad {\text {and}}\quad \max _{j_2\le n-H} \frac{ Q^2_{j_2,n}}{\mathbb {E} \sum _{j_2=H}^{n-H} Q^2_{j_2,n}}{\mathop {\rightarrow }\limits ^{P}} 0. \end{aligned}$$

Towards this, we denote $\widetilde{g}_{i,j,h}=\widetilde{g}(\varvec{u}_i,\varvec{v}_i; \varvec{X}_{j},\varvec{X}_{j+h})$ and investigate

$$\begin{aligned} \sum _{j_2=2}^{n-H} Q^2_{j_2}= & {} \frac{1}{H}\frac{1}{(n-H)^2}\sum _{j_2=2}^{n-H}\sum _{j_1=1}^{j_2-H} \sum _{j_3=1}^{j_2-H} \sum _{h_1=1}^H \sum _{h_2=1}^H \iiiint \widetilde{g}_{1,j_1,h_1} \widetilde{g}_{2,j_3,h_2} \big \{\widetilde{g}_{1,j_2,h_1} \widetilde{g}_{2,j_2,h_2}\nonumber \\&\pm \mathbb {E}\left( \widetilde{g}_{1,j_2,h_1} \widetilde{g}_{2,j_2,h_2}\right) \big \} w(\varvec{u}_1) w(\varvec{v}_1) w(\varvec{u}_2) w(\varvec{v}_2) \mathrm{d}\varvec{u}_1\mathrm{d}\varvec{v}_1 \mathrm{d}\varvec{u}_2 \mathrm{d}\varvec{v}_2, \end{aligned}$$

(22)

which will be again decomposed into several summands some of them negligible, while others are influential. Particularly, we investigate separately the terms $L_{1,n},L_{2,n},L_{3,n},L_{4n}$ defined below. Defining

$$\begin{aligned} L_{1,n}= & {} \frac{1}{H}\frac{1}{(n-H)^2}\sum _{j_2=2}^{n-H} \sum _{j_1=j_3=1}^{j_2-H} \sum _{h_1=1}^H \sum _{h_2=1}^H \iiiint \widetilde{g}_{1,j_1,h_1} \widetilde{g}_{2,j_1,h_2} \big \{\widetilde{g}_{1,j_2,h_1} \widetilde{g}_{2,j_2,h_2}\\&-\mathbb {E}(\widetilde{g}_{1,j_2,h_1} \widetilde{g}_{2,j_2,h_2})\big \} w(\varvec{u}_1) w(\varvec{v}_1) w(\varvec{u}_2) w(\varvec{v}_2) \mathrm{d}\varvec{u}_1\mathrm{d}\varvec{v}_1 \mathrm{d}\varvec{u}_2 \mathrm{d}\varvec{v}_2 \end{aligned}$$

it can be shown by straightforward but long calculations that $\mathbb {E}L_{1,n}= 0$, $\mathbb {E}L_{1,n}^2=O\Big ( \frac{H n^2}{ H^2 n^4}\Big )$. Next, for

$$\begin{aligned} L_{2,n}= & {} \frac{1}{H}\frac{1}{(n-H)^2} \sum _{j_2=2}^{n-H}\sum _{j_1=j_3=1}^{j_2-H} \sum _{h=1}^H \iiiint \widetilde{g}_{1,j_1,h} \widetilde{g}_{2,j_1,h} \mathbb {E}(\widetilde{g}_{1,j_2,h} \widetilde{g}_{2,j_2,h})\\&\times w(\varvec{u}_1) w(\varvec{v}_1) w(\varvec{u}_2) w(\varvec{v}_2) \mathrm{d}\varvec{u}_1\mathrm{d}\varvec{v}_1 \mathrm{d}\varvec{u}_2 \mathrm{d}\varvec{v}_2 \end{aligned}$$

we have that

$$\begin{aligned} \mathbb {E} L_{2,n}= & {} \frac{1}{H}\frac{1}{(n-H)^2}\sum _{j_2=2}^{n-H} \sum _{j_1=j_3=1}^{j_2-H} \sum _{h=1}^H \iiiint \mathbb {E} ( \widetilde{g}_{1,j_1,h} \widetilde{g}_{2,j_1,h}) \mathbb {E}(\widetilde{g}_{1,j_2,h} \widetilde{g}_{2,j_2,h})\\&\times w(\varvec{u}_1) w(\varvec{v}_1) w(\varvec{u}_2) w(\varvec{v}_2) \mathrm{d}\varvec{u}_1\mathrm{d}\varvec{v}_1 d \varvec{u}_2 \mathrm{d}\varvec{v}_2,\\ \mathbb {E}( L_{2,n}-\mathbb {E}L_{2,n})^2= & {} O\Big (\frac{1}{H^2n^4}n^3 H\Big )=O\Big (\frac{1}{nH}\Big ). \end{aligned}$$

Likewise for

$$\begin{aligned} L_{3,n}= & {} \frac{1}{H}\frac{1}{(n-H)^2}\sum _{j_2=3}^{n-H}\sum _{j_1=1}^{j_3-1} \sum _{j_3=1}^{j_2-H} \sum _{h_1=1}^H \iiiint \widetilde{g}_{1,j_1,h_1} \widetilde{g}_{2,j_3,h_1} \{\widetilde{g}_{1,j_2,h_1} \widetilde{g}_{2,j_2,h_1}\\&- \mathbb {E} (\widetilde{g}_{1,j_2,h_1} \widetilde{g}_{2,j_2,h_1})\} w(\varvec{u}_1) w(\varvec{v}_1) w(\varvec{u}_2) w(\varvec{v}_2) \mathrm{d}\varvec{u}_1\mathrm{d}\varvec{v}_1 \mathrm{d}\varvec{u}_2 \mathrm{d}\varvec{v}_2, \end{aligned}$$

it may be shown that $\mathbb {E} L_{3,n}=0$ and $\mathbb {E} L^2_{3,n} =O\Big ( \frac{1}{H^2 n^4} n^3 H^2\Big )$. Finally for

$$\begin{aligned} L_{4,n}= & {} \frac{1}{H}\frac{1}{(n-H)^2}\sum _{j_2=3}^{n-H} \sum _{j_1=1}^{j_3-1} \sum _{j_3=1}^{j_2-H} \sum _{h=1}^H \iiiint \widetilde{g}_{1,j_1,h} \widetilde{g}_{2,j_3,h} \mathbb {E} (\widetilde{g}_{1,j_2,h} \widetilde{g}_{2,j_2,h})\\&\times w(\varvec{u}_1) w(\varvec{v}_1) w(\varvec{u}_2) w(\varvec{v}_2) \mathrm{d}\varvec{u}_1\mathrm{d}\varvec{v}_1 \mathrm{d}\varvec{u}_2 \mathrm{d}\varvec{v}_2 \end{aligned}$$

we have $\mathbb {E} L_{4,n}=0$, $\mathbb {E} L_{4,n}^2 =O\Big (\frac{n^2}{H^2n^4} H n^2\Big )=O(H^{-1})$. More detailed (long but straightforward) calculations give that $\mathbb {E} L_{2,n}=\frac{1}{2} \nu (1+o_P(1))$.

Combining the above properties of $L_{j,n}, \, j=1,\ldots ,4$, we obtain $\sum _{j_2=H}^{n-H} Q_{j_2}^2= \nu (1+o_P(1))$ and $\sum _{j_2=H}^{n-H}\mathbb {E} Q_{j_2}^2= \nu (1+o(1))$ and that for any $c>0$,

$$\begin{aligned}&\max _{j_2\le n-H}\frac{\mathbb {E} Q^2_{j_2,n}}{ \mathbb {E} \sum _{j_2=H}^{n-H} Q^2_{j_2,n}}=o_P(1),\\&\quad P\Big (\max _{j_2\le n-H}\frac{| Q^2_{j_2,n}- \mathbb {E} Q^2_{j_2,n}| }{ \mathbb {E} \sum _{j_2=H}^{n-H}\mathbb {E} Q^2_{j_2,n}}\ge c\Big )\\&\quad \le \sum _{j_2=H}^{n-H}\frac{1}{c^2 \big (\mathbb {E} \sum _{j_2=H}^{n-H} Q^2_{j_2,n}\big )^2} \mathbb {E} \big (Q^2_{j_2,n}- Q^2_{j_2,n}\big )^2=o_P(1). \end{aligned}$$

So the requirements (22) are fulfilled and going through the whole proof we can conclude that the assertion of Lemma 2 holds true. $\square$

Proof of Theorem 1, continuation

Combining (21), Lemma 1 and Lemma 2 imply the assertion of Theorem 1. $\square$

Proof of Theorem 2

Since under the assumptions of stationarity and ergodicity (20), the next properties hold true

$$\begin{aligned}&\frac{1}{\sqrt{H}}\sum _{h=1}^H (n-h)\iint \mathbb {E}\Big |\frac{1}{n-h} \sum _{j=1}^{n-h}\big \{\exp ({\mathrm{i}} (\varvec{u}^\top \varvec{X}_j +\varvec{v}^\top \varvec{X}_{j+h}))\\&\quad - \mathbb {E}\exp ({\mathrm{i}} (\varvec{u}^\top \varvec{X}_j +\varvec{v}^\top \varvec{X}_{j+h}) )\big \} \Big |^2 w( \varvec{u}) w(\varvec{v}) \mathrm{d}\varvec{u} \mathrm{d}\varvec{v} =O(1), \\&\quad \frac{1}{\sqrt{H}}\sum _{h=1}^H (n-h)\iint \mathbb {E} \Big |\frac{1}{(n-h)^2}\sum _{j=1}^{n-h}\big \{\exp ({\mathrm{i}} \varvec{u}^\top \varvec{X}_j) -\varphi _x(\varvec{u})\big \} \\&\quad \times \sum _{r=1}^{n-h}\big \{\exp ({\mathrm{i}} \varvec{v}^\top \varvec{X}_{r+h}) -\varphi _x(\varvec{v})\big \} \Big |^2 w( \varvec{u}) w( \varvec{v}) \mathrm{d}\varvec{u} \mathrm{d}\varvec{v}=O(1) \end{aligned}$$

and since by assumption (16) we also have

$$\begin{aligned}&\frac{1}{\sqrt{H}} \sum _{h=1}^H (n-h)\iint \Big |\mathbb {E}\exp \{{\mathrm{i}} (\varvec{u}^\top \varvec{X}_j +\varvec{v}^\top \varvec{X}_{j+h}) \}\\&- \varphi _{x_1}(\varvec{u})\varphi _{x_{1+h}} (\varvec{v})\Big |^2 w(\varvec{u}) w( \varvec{v}) \mathrm{d}\varvec{u} \mathrm{d}\varvec{v}=O(n), \end{aligned}$$

the assertion of Theorem 2 directly follows. $\square$

Proof of Theorem 3

Going through the proof of Theorem 1 taking into account the considered setup, we can directly conclude Theorem 3. $\square$

Proof of Theorem 4

It is omitted since it follows the same line as that of Theorem 2. $\square$