On Diagnostic Checking Autoregressive Conditional Duration Models with Wavelet-Based Spectral Density Estimators

Abstract

There has been an increasing interest recently in the analysis of financial data that arrives at irregular intervals. An important class of models is the autoregressive Conditional Duration (ACD) model introduced by Engle and Russell (Econometrica 66:1127–1162, 1998, [22]) and its various generalizations. These models have been used to describe duration clustering for financial data such as the arrival times of trades and price changes. However, relatively few evaluation procedures for the adequacy of ACD models are currently available in the literature. Given its simplicity, a commonly used diagnostic test is the Box-Pierce/Ljung-Box statistic adapted to the estimated standardized residuals of ACD models, but its asymptotic distribution is not the standard one due to parameter estimation uncertainty. In this paper we propose a test for duration clustering and a test for the adequacy of ACD models using wavelet methods. The first test exploits the one-sided nature of duration clustering. An ACD process is positively autocorrelated at all lags, resulting in a spectral mode at frequency zero. In particular, it has a spectral peak at zero when duration clustering is persistent or when duration clustering is small at each individual lag but carries over a long distributional lag. As a joint time-frequency decomposition method, wavelets can effectively capture spectral peaks and thus are expected to be powerful. Our second test checks the adequacy of an ACD model by using a wavelet-based spectral density of the estimated standardized residuals over the whole frequency. Unlike the Box-Pierce/Ljung-Box tests, the proposed diagnostic test has a convenient asymptotic “nuisance parameter-free” property—parameter estimation uncertainty has no impact on the asymptotic distribution of the test statistic. Moreover, it can check a wide range of alternatives and is powerful when the spectrum of the standardized duration residuals is nonsmooth, as can arise from neglected persistent duration clustering, seasonality, calender effects and business cycles. For each of the two new tests, we propose and justify a suitable data-driven method to choose the finest scale—the smoothing parameter in wavelet estimation. This makes the methods fully operational in practice. We present a simulation study, illustrating the merits of the wavelet-based procedures. An application with tick-by-tick trading data of Alcoa stock is presented.

Appendix

To prove Theorems 1–3, we first state some useful lemmas.

Lemma 1

Define $d_{J}(h)\equiv \sum _{j=0}^{J}\lambda (2\pi h/2^{j}),$ $h,J\in \mathbb {Z}$, where $\lambda (z)$ is as in (8). Then

• (i) $d_{J}(0)=0$ and $d_{J}(-h)=d_{J}(h)$ for all $h,J\in \mathbb {Z},J>0$;

• (ii) $|d_{J}(h)|\le C<\infty$ uniformly in $h,J\in \mathbb {Z},J>0$;

• (iii) For any given $h\in \mathbb {Z},h\ne 0$, $d_{J}(h)\rightarrow 1$ as $J\rightarrow \infty$;

• (iv) For any given $r \ge 1$, $\sum _{h=1}^{n-1}|d_{J}(h)|^{r}=O(2^{J})$ as $J,n\rightarrow \infty$.

Lemma 2

Let $V_{n}(J)$ and $V_{0}$ be defined as in Theorem 1. Suppose $J\rightarrow \infty , 2^{J}/n\rightarrow 0$. Then $2^{-J}V_{n}(J)\rightarrow V_{0}$, where $V_{0}=\int _{0}^{2\pi }|\varGamma (z)|^{2}dz$, with $\varGamma (z)=\sum _{-\infty }^{\infty }\hat{\psi }(z+2\pi m)$.

Proof

For the proofs of Lemmas 1 and 2, see [42, Proof of Lemma A.1] and [44, Proof of Lemma A.2]. $\Box$

Proof

(Theorem 1) The model under the null hypothesis is $D_t \equiv \beta _0$, $X_t = \beta _0 \varepsilon _t$, $E(\varepsilon _t)=1$, $\{ \varepsilon _t \}$ an iid process. We write $\bar{R}_X(h)= n^{-1}\sum _{t=|h|+1}^n (X_t/\bar{X}-1)(X_{t-|h|}/\bar{X}-1)$. Alternatively, $\hat{\rho }_X(h)= \bar{R}_X(h)/\bar{R}_X(0)$. Under the null hypothesis, $\bar{R}_X(h) = \bar{R}_{\varepsilon }(h)$ where $\bar{R}_{\varepsilon }(h)$ is defined similarly as $\bar{R}_X(h)$. We define $R_{\varepsilon }(h)=\text {cov}(\varepsilon _t,\varepsilon _{t-h})$. Let $u_t=\varepsilon _t-1$. We define $\tilde{R}_{\varepsilon }(h)=n^{-1}\sum _{t=|h|+1}^n u_t u_{t-h}$ and $\tilde{\rho }_{\varepsilon }(h) = \tilde{R}_{\varepsilon }(h)/R_{\varepsilon }(0)$. Define

$$ \tilde{f}_{\varepsilon }(0) \equiv (2\pi )^{-1} + \sum _{j=0}^{J}\sum _{k=1}^{2^{j}} \tilde{\alpha }_{jk}\varPsi _{jk}(0), $$

where $\tilde{\alpha }_{jk} \equiv (2\pi )^{-1/2}\sum _{h=1-n}^{n-1} \tilde{ \rho }_{\varepsilon }(h)\hat{\varPsi }_{jk}^{*}(h)$, $\hat{\varPsi } _{jk}(h)\equiv (2\pi )^{-1/2}\int _{-\pi }^{\pi } \varPsi _{jk}(\omega )e^{-ih\omega }d\omega$.

Writing $\hat{f}_X(0)-(2\pi )^{-1}=[\hat{f}_X(0)-\tilde{f}_{\varepsilon }(0)]+ [\tilde{f}_{\varepsilon }(0)-(2\pi )^{-1}],$ we shall prove Theorem 1 by showing Theorems 7–8 below.

Theorem 7

$[ V_{n}(J) ]^{-1/2}n^{1/2}[\hat{f}_X(0)-\tilde{f}_{\varepsilon }(0)]\rightarrow ^{p}0.$

Theorem 8

$[ V_{n}(J) ]^{-1/2}n^{1/2}\pi [\tilde{f}_{\varepsilon }(0)-(2\pi )^{-1}]\rightarrow ^{d} N(0,1).$

Proof

(Theorem 7) We use the following representations (see [44]):

$$\begin{aligned} \hat{f}_X(0)= & {} (2\pi )^{-1}+\pi ^{-1}\sum _{h=1}^{n-1}d_{J}(h)\hat{\rho }_X(h), \\ \tilde{f}_{\varepsilon }(0)= & {} (2\pi )^{-1}+\pi ^{-1}\sum _{h=1}^{n-1}d_{J}(h)\tilde{\rho }_{\varepsilon }(h) . \nonumber \end{aligned}$$

(20)

We obtain $\pi [\hat{f}_X(0)-\tilde{f}_{\varepsilon }(0)]=\sum _{h=1}^{n-1}d_{J}(h) [\hat{\rho }_X(h)-\tilde{\rho }_{\varepsilon }(h)]$. Because $\bar{R}_X(0)-R_{\varepsilon }(0)=O_{P}(n^{-1/2})$ given Assumption 1, it suffices to show

$$\begin{aligned}{}[ V_{n}(J) ]^{-1/2}n^{1/2}\sum _{h=1}^{n-1}d_{J}(h) [\bar{R}_X(h)-\tilde{R}_{\varepsilon }(h)]\rightarrow ^{p} 0. \end{aligned}$$

(21)

We shall show (21) for the case $J\rightarrow \infty$, where $2^{-J}V_{n}(J)\rightarrow V_{0}$ by Lemma 2. The proof for fixed J is similar, with $V_{n}(J)\rightarrow V_{0}(J)$, where $V_{0}(J)$ is defined in Lemma 2.

Straightforward algebra yields $\hat{R}_X(h)-\tilde{R}_{\varepsilon }(h)= (\bar{\varepsilon }^{-1}-1)\hat{A}_{1}(h)+ (\bar{\varepsilon }^{-1}-1)\hat{A}_{2}(h)+ (\bar{\varepsilon }^{-1}-1)^2\hat{A}_{3}(h)$, where

$$ \hat{A}_1(h) = n^{-1}\sum _{t=|h|+1}^n u_t \varepsilon _{t-h}, \; \hat{A}_2(h) = n^{-1}\sum _{t=|h|+1}^n u_{t-h} \varepsilon _{t}, \; \hat{A}_3(h) = n^{-1}\sum _{t=|h|+1}^n \varepsilon _t \varepsilon _{t-h}. $$

We first consider $\hat{A}_{1}(h)$. Note that $E[\hat{A}_1(h)\hat{A}_1(m)] = O(n^{-1})$, $\forall h,m$. Then expending the square, we show that $E[\sum _{h=1}^{n-1}d_J(h)\hat{A}_1(h)]^2 = O(2^J/n + 2^{2J}/n)$. This shows that $\sum _{h=1}^{n-1} d_J(h) \hat{A}_1(h) = O_P(2^J/n^{1/2})$. Proceeding similarly we show that $\sum _{h=1}^{n-1} d_J(h) \hat{A}_2(h) = O_P(2^J/n^{1/2})$. We show also easily that

$$ \sum _{h=1}^{n-1} d_J(h)\hat{A}_{3}(h) = O_P(2^J). $$

This completes the proof for Theorem 7. $\Box$

Proof

(Theorem 8) Put $\hat{W}\equiv \sum _{h=1}^{n-1}d_{J}(h)\tilde{R}_{\varepsilon }(h)/R_{\varepsilon }(0).$ Write $\hat{W}=n^{-1}\sum _{t=2}^{n}W_{t}$, where

$$ W_{t}\equiv R^{-1}_{\varepsilon }(0)u_{t}\sum _{h=1}^{t-1}d_{J}(h)u_{t-h}. $$

Observe that $\{W_{t},\mathscr {F}_{t-1}\}$ is an adapted martingale difference sequence, where $\mathscr {F}_{t}$ is the sigma field consisting of all $u_{s},s\le t.$ Thus, we obtain

$$ {\text {var}}(n^{1/2}\hat{W})= n^{-1}\sum _{t=2}^{n}E[W_{t}^{2}]= n^{-1} \sum _{t=2}^{n}\sum _{h=1}^{t-1}d_{J}^{2}(h)= \sum _{h=1}^{n}(1-h/n)d_{J}^{2}(h)=V_{n}(J). $$

By the martingale central limit theorem in [37, pp.10–11], $[ V_{n}(J) ]^{-1/2}n^{1/2}\hat{W}\rightarrow ^{d}N(0,1)$ if we can show

$$\begin{aligned}{}[V_{n}(J)]^{-1} n^{-1}\sum _{t=2}^{n}E\left\{ W_{t}^{2}\mathbf {1}[|W_{t}|>\eta n^{1/2}\{ V_{n}(J) \}^{1/2}]\right\}\rightarrow & {} \; 0 \text { for any }\eta >0, \end{aligned}$$

(22)

$$\begin{aligned}{}[V_{n}(J)]^{-1} n^{-1}\sum _{t=2}^{n}E(W_{t}^{2}|\mathscr {F}_{t-1})&\rightarrow ^{p}&\; 1. \end{aligned}$$

(23)

For space, we shall show the central limit theorem for $\hat{W}$ for large J (i.e., $J\rightarrow \infty ).$ The proof for fixed J is similar and simpler because $d_{J}(h)$ is finite and summable.

We shall verify the first condition by showing $2^{-2J}n^{-2}\sum _{t=2}^{n}E(W_{t}^{4}) \rightarrow 0$. Put $\mu _{4} \equiv E(u_{t}^{4})$. By Assumption 1, we have

$$\begin{aligned} E(W_{t}^{4})= & {} \mu _{4}R^{-4}_{\varepsilon }(0) E\left[ \sum _{h=1}^{t-1}d_{J}(h)u_{t-h}\right] ^{4}, \\= & {} \mu _{4}^{2}R^{-4}_{\varepsilon }(0)\sum _{h=1}^{t-1}d_{J}^{4}(h)+ 6\mu _{4}R^{-2}_{\varepsilon }(0)\sum _{h_1=2}^{t-1}\sum _{h_2=1}^{h_1-1}d_{J}^{2}(h_1)d_{J}^{2}(h_2) \le 3\mu _{4}^{2}R^{-4}_{\varepsilon }(0)\left[ \sum _{h=1}^{n-1}d_{J}^{2}(h)\right] ^{2}. \end{aligned}$$

It follows from Lemma 2 that $2^{-2J}n^{-2}\sum _{t=1}^{n}E(W_{t}^{4})=O(n^{-1}).$ This show (22).

Next, given Lemma 2, it suffices for expression (23) to establish the sufficient condition

$$ 2^{-2J}{\text {var}}[n^{-1}\sum _{t=2}^{n}E(W_{t}^{2}|\mathscr {F}_{t-1})]\rightarrow 0. $$

By the definition of $W_{t}$, we have

$$\begin{aligned} E(W_{t}^{2}|\mathscr {F}_{t-1})= & {} R^{-1}_{\varepsilon }(0)\left[ \sum _{h=1}^{t-1}d_{J}(h)u_{t-h}\right] ^{2}, \\= & {} E(W_{t}^{2})+R^{-1}_{\varepsilon }(0)\sum _{h=1}^{t-1}d_{J}(h)[u_{t-h}^{2}-R_{\varepsilon }(0)]\\&+ 2R^{-1}_{\varepsilon }(0)\sum _{h_1=2}^{t-1} \sum _{h_2=1}^{h_1-1}d_{J}(h_1)d_{J}(h_2)u_{t-h_1}u_{t-h_2}, \\= & {} E(W_{t}^{2})+R^{-1}_{\varepsilon }(0)A_t+2R^{-1}_{\varepsilon }(0)B_{t}. \end{aligned}$$

It follows that

$$\begin{aligned} n^{-1}\sum _{t=2}^{n}[E(W_{t}^{2}|\mathscr {F}_{t-1})-E(W_{t}^{2})]= & {} R^{-1}_{\varepsilon }(0)n^{-1}\sum _{t=2}^{n}A_{t}+ 2R^{-1}_{\varepsilon }(0)n^{-1}\sum _{t=2}^{n}B_{t} \nonumber \\= & {} R^{-1}_{\varepsilon }(0)\hat{A}+2R^{-1}_{\varepsilon }(0)\hat{B}. \end{aligned}$$

(24)

Whence, it suffices to show $2^{-2J}[{\text {var}}(\hat{A})+{\text {var}}(\hat{B})]\rightarrow 0$. First, noting that $A_{t}$ is a weighted sum of independent zero-mean variables $\{u_{t-h}^{2}-R_{\varepsilon }(0)\},$ we have $E(A_{t}^{2})=[\mu _{4}-R^2_{\varepsilon }(0)]\sum _{h=1}^{t-1}d_{J}^{4}(h).$ It follows by Minkowski’s inequality and Lemma 1(iv) that

$$\begin{aligned} E(\hat{A}^{2}) \le \left\{ n^{-1}\sum _{t=2}^{n}[E(A_{t}^{2})]^{1/2}\right\} ^{2}\le \left[ \mu _{4}-R^{2}_{\varepsilon }(0)\right] \left[ \sum _{h=1}^{n-1}d_{J}^{4}(h)\right] =O(2^{J}). \end{aligned}$$

(25)

Next, we consider ${\text {var}}(\hat{B})$. For all $t\ge s,$ we have

$$\begin{aligned} E(B_{t}B_{s})= & {} R_{\varepsilon }^{2}(0) \sum _{m_{2}=2}^{t-1}\sum _{h_{2}=1}^{m_{2}-1}\sum _{m_{1}=2}^{s-1}\sum _{h_{1}=1}^{m_{1}-1}d_{J}(m_{1})d_{2}(h_{1})d_{J}(m_{2})d_{J}(h_{2})\delta _{t-h_{1},s-h_{2}}\delta _{t-m_{1},s-m_{2}}, \\= & {} R_{\varepsilon }^{2}(0)\sum _{m=2}^{t-1}\sum _{h=1}^{l-1}d_{J}(t-s+m)d_{J}(t-s+h)d_{J}(m)d_{J}(h), \end{aligned}$$

where $\delta _{j,h}=1$ if $h=j$ and $\delta _{j,h}=0$ otherwise. It follows that

$$\begin{aligned} E(\hat{B}^{2})\le & {} 2n^{-2}\sum _{t=3}^{n}\sum _{s=2}^{t}E(B_{t}B_{s})\le 2R_{\varepsilon }^{2}(0)n^{-1} \sum _{\tau =0}^{n-1}\sum _{m=2}^{n-1}\sum _{h=1}^{m-1}|d_{J}(\tau +m)d_{J}(\tau +h)d_{J}(m)d_{J}(h)|, \nonumber \\\le & {} 2R_{\varepsilon }^{2}(0)n^{-1}\left[ \sum _{\tau =0}^{n-1}d_{J}^{2}(\tau )\right] \left[ \sum _{h=1}^{n-1}|d_{J}(h)|\right] ^{2}=O(2^{3J}/n), \end{aligned}$$

(26)

by Lemma 1(iv). Combining (24)–(26) yields $2^{-2J}[{\text {var}}(\hat{A})+{\text {var}}(\hat{B})]=O(2^{-J}+2^{J}/n)\rightarrow 0$ given $J\rightarrow \infty , 2^{J}/n\rightarrow 0.$ Thus, condition (23) holds. By [37, pp.10–11], $[ V_{n}(J) ]^{-1/2}n^{1/2}\hat{W}\rightarrow ^{d}N(0,1)$. This completes the proof of Theorem 8. $\Box$

Proof

(Theorem 2) We shall show for large J only; the proof for fixed J is similar. Here we explicitly denote $\hat{f}_X(0;J)$ as the spectral estimator (20) with the finest scale J. Recalling the definition of $\mathscr {E}(J)$, we have

$$\begin{aligned} \mathscr {E}(\hat{J})-\mathscr {E}(J)= & {} [V_{n}(\hat{J})]^{-1/2}n^{1/2}\pi \{\hat{f}_X(0;\hat{J})- (2\pi )^{-1}\}-[V_{n}(J)]^{-1/2}n^{1/2}\pi \{\hat{f}_X(0;J)-(2\pi )^{-1}\}, \\= & {} [V_{n}(\hat{J})/V_{n}(J)]^{1/2}[V_{n}(J)]^{-1/2} n^{1/2}\pi \{\hat{f}_X(0;\hat{J})-\hat{f}_X(0;J)\} +\{[V_{n}(J)/V_{n}(\hat{J})]^{-1/2}-1\}\mathscr {E}(J). \end{aligned}$$

Note that we have for any given constants $C_0 >0$ and $\varepsilon >0$,

$$\begin{aligned} P\left( |V_{n}(J)/V_{n}(\hat{J})-1|>\varepsilon \right)\le & {} P\left( |V_{n}(J)/V_{n}(\hat{J})-1|>\varepsilon ,C_0 2^{J/2}|2^{\hat{J}}/2^{J}-1|\le \varepsilon \right) \nonumber \\+ & {} P\left( C_0 2^{J/2}|2^{\hat{J}}/2^{J}-1|>\varepsilon \right) . \end{aligned}$$

(27)

We now study $|d_{\hat{J}}(h)-d_J(h)|$. We show that

$$ |d_{\hat{J}}(h)-d_J(h)| \le \sum _{j=\min (J,\hat{J})+1}^{\max (J,\hat{J})} |\lambda (2 \pi h/2^j)|. $$

Note that given $C_0 2^{J/2}|2^{\hat{J}}/2^J - 1| \le \varepsilon$, we have that

$$ |d_{\hat{J}}(h)-d_J(h)| \le \sum _{j=\log _2[2^J(1-\varepsilon /(C_0 2^{J/2}))]+1}^{\log _2[2^J(1+\varepsilon /(C_0 2^{J/2}))]} |\lambda (2 \pi h/2^j)|, $$

and the lower and upper bounds in the summation are now non stochastic. Since $|\sum _{m=-\infty }^{\infty } \psi (2\pi h/2^j + 2\pi m)| \le C$, we have that

$$ |d_{\hat{J}}(h)-d_J(h)| \le C \sum _{j=\log _2[2^J(1-\varepsilon /(C_0 2^{J/2}))]+1}^{\log _2[2^J(1+\varepsilon /(C_0 2^{J/2}))]} |\hat{\psi }(2 \pi h/2^j)|. $$

Since $\sum _{h=-\infty }^{\infty }|\hat{\psi }(2 \pi h/2^j)| \le C 2^j$, then

$$ \sum _{h=1}^{n-1} |d_{\hat{J}}(h)-d_J(h)| \le C \sum _{j=\log _2[2^J(1-\varepsilon /(C_0 2^{J/2}))]+1}^{\log _2[2^J(1+\varepsilon /(C_0 2^{J/2}))]} 2^j \le C2^{J/2}\varepsilon /C_0. $$

A similar argument allows us to show that

$$\begin{aligned} \sum _{h=1}^{n-1} |d_{\hat{J}}(h)-d_J(h)|^2 \le C2^{J/2}\varepsilon /C_0. \end{aligned}$$

(28)

We show that $V_n(\hat{J})/V_n(J) \rightarrow ^p 1$. Note that

$$ |V_n(\hat{J}) - V_n(J)| \le \sum _{h=1}^{n-1} |d_{\hat{J}}(h)-d_J(h)|^2 + 2\left( \sum _{h=1}^{n-1} d_J^2(h) \right) ^{1/2} \left( \sum _{h=1}^{n-1} \left[ d_{\hat{J}}(h)-d_J(h) \right] ^2 \right) ^{1/2}. $$

Since $\sum _{h=1}^{n-1} d_J^2(h) = O(2^J)$, by (27) and (28), we have the announced result that $V_n(\hat{J})/V_n(J) \rightarrow ^p 1$.

Because $\mathscr {E}(J)=O_{P}(1)$ by Theorem 1 and since $V_{n}(\hat{J})/V_{n}(J)\rightarrow ^{p} 1$, we have $\mathscr {E}(\hat{J})-\mathscr {E}(J)\rightarrow ^{p} 0$ provided $[V_{n}(J)]^{-1/2} n^{1/2}\pi \{\hat{f}_{\hat{J}}(0)-\hat{f}_{J}(0)\} \rightarrow ^{p}0,$ which we shall show below. The asymptotic normality of $\mathscr {E}(\hat{J})$ follows from a standard application of Slutsky’s theorem and Theorem 1.

Because $V_{n}(J)=O(2^{J}),$ it suffices to show $\hat{f}_{X}(0;\hat{J})-\hat{f}_{X}(0;J)=o_{P}(2^{J/2}/n^{1/2}).$ Write

$$ \pi \{\hat{f}_{X}(0;\hat{J})-\hat{f}_{X}(0;J)\}=\hat{R}_{X}^{-1}(0)\sum _{h=1}^{n-1}[ d_{\hat{J}}(h)-d_{J}(h) ] R_{X}(h). $$

Given $|d_{\hat{J}}(h)-d_{J}(h)|\le \sum _{j=\min (\hat{J},J)}^{\max (\hat{J},J)}|\lambda (2\pi h/2^{j})|$, we have, under the null hypothesis, the following inequality

$$ E\sum _{h=1}^{n-1}\left| d_{\hat{J}}(h)-d_{J}(h)\right| \left| R_{X}(h)\right| \le \sup _{h}E(R_{X}^{2}(h))^{1/2}\sum _{h=1}^{n-1}\left( E|d_{\hat{J}}(h)-d_{J}(h)\right| ^{2})^{1/2}= o(2^{J/2}/n^{1/2}). \nonumber $$

We obtain $\hat{f}_{X}(0;\hat{J})-\hat{f}_{X}(0;J)=o_{P}(2^{J/2}/n^{1/2}).$ This completes the proof of Theorem 2. $\Box$

Proof

(Theorem 3) Recall $\hat{R}_X(h)= n^{-1}\sum _{t=|h|+1}^n (X_t-\bar{X})(X_{t-|h|}-\bar{X})$ and $\tilde{R}_X(h)= n^{-1}\sum _{t=|h|+1}^n (X_t-\mu )(X_{t-|h|}-\mu )$. We study $\hat{f}_X(0;J) - f_X(0)$. Write

$$\begin{aligned} \hat{f}_X(0;J) - f_X(0) = \{\hat{f}_X(0;J) - \tilde{f}_X(0;J)\} + \{ \tilde{f }_X(0;J) - E[\tilde{f}_X(0;J)] \} + \{ E[\tilde{f}_X(0;J)] - f_X(0) \}, \end{aligned}$$

(29)

where

$$\begin{aligned} \tilde{f}_X(0;J) = \frac{\tilde{R}_X(0)}{2\pi }+ \frac{1}{\pi } \sum _{h=1}^{n-1} d_J(h) \tilde{R}_X(h). \end{aligned}$$

(30)

We can show that $E[ \hat{f}_X(0;J) - \tilde{f}_X(0;J) ]^2 = O(n^{-2} + 2^{2J}/n^2)$, meaning that replacing $\bar{X}$ by $\mu$ has to impact asymptotically. For the second term in (29), we show that

$$\begin{aligned} E[\tilde{f}_X(0;J) - E\tilde{f}_X(0;J)]^2= & {} \frac{{\text {var}}[ \tilde{R}_X(0)]}{4 \pi ^2} + \pi ^{-2} \sum _{h=1}^{n-1}\sum _{m=1}^{n-1} d_J(h)d_J(m) {\text {cov}}(\tilde{R}_X(h),\tilde{R}_X(m)) \nonumber \\&+ \pi ^{-2} \sum _{h=1}^{n-1} d_J(h) {\text {cov}}(\tilde{R}_X(0),\tilde{R}_X(h)). \end{aligned}$$

(31)

From [38, p. 313], we have

$$\begin{aligned} \frac{(n-l)(n-m)}{n^2} {\text {cov}}\left[ \tilde{R}_X(h),\tilde{R} _X(m)\right]= & {} n^{-1}\sum _{u=-\infty }^{\infty }w_{n}(u,h,m) [R_X(u) R_X(u+m-h) \nonumber \\&+ R_X(u+m) R_X(u-h)+\kappa (0,h,u,u+m)], \nonumber \end{aligned}$$

where the function $w_{n}(u,h,m)$ is defined in [38]. We write

$$\begin{aligned} E\left\{ \tilde{f}_X(0;J) - E[\tilde{f}_X(0;J)] \right\} ^2 = \frac{{\text {var}}( \tilde{R}_X(0))}{4\pi ^2} + \frac{A_n}{\pi ^2} + \frac{B_n}{\pi ^2}, \end{aligned}$$

(32)

where

$$\begin{aligned} A_n= & {} E\left\{ [\tilde{R}_X(0)-E(\tilde{R}_X(0))] \sum _{h=1}^{n-1} d_J(h)[\tilde{R}_X(h)-E(\tilde{R}_X(h))] \right\} , \\ B_n= & {} \sum _{h=1}^{n-1}\sum _{m=1}^{n-1} d_J(h) d_J(m) {\text {cov}}[\tilde{R}_X(h),\tilde{R}_X(m)]. \end{aligned}$$

It follows that

$$\begin{aligned} (n/2^{J+1})B_n= & {} 2^{-(J+1)}\sum _{h=1}^{n-1}\sum _{m=1}^{n-1} d_{J}(h)d_{J}(m)\sum _{u=-\infty }^{\infty }w_{n}(u,h,m)R_X(u)R_X(u+m-h) \nonumber \\&+2^{-(J+1)}\sum _{h=1}^{n-1}\sum _{m=1}^{n-1}d_J(h)d_J(m) \sum _{u=-\infty }^{\infty }w_{n}(u,h,m)R_X(u+m)R_X(u-h) \nonumber \\&+2^{-(J+1)}\sum _{h=1}^{n-1}\sum _{m=1}^{n-1}d_J(h)d_J(m) \sum _{u=-\infty }^{\infty }w_{n}(u,h,m)\kappa (0,h,u,u+m), \nonumber \\= & {} B_{1n}+B_{2n}+B_{3n}. \end{aligned}$$

(33)

Following a reasoning similar to [42, proof of Theorem 4.2], we can show that

$$ B_{1n} = 2^{-1}D_{\psi }(2\pi )^{2}f_X^2(0)[1+o(1)], \; B_{2n}\rightarrow 0, \; B_{3n} \rightarrow 0, $$

and $\left| A_n \right| =O(2^{J/2}/n)=o(2^{J}/n)$. It follows that

$$\begin{aligned} n/2^{J+1}B_n \rightarrow 2\pi ^{2}D_{\psi }f_X^2(0), \end{aligned}$$

(34)

and

$$\begin{aligned} (n/2^{J+1}) E[\tilde{f}_X(0;J) - E\tilde{f}_X(0;J)]^2 \rightarrow 2 D_{\psi }f_X^2(0). \end{aligned}$$

(35)

We consider the bias term $E[\tilde{f}_{X}(J)]-f_{X}(0)$ in (29). Using the definition of $\tilde{f}_{X}(0;J)$ in (30), we can decompose

$$\begin{aligned} E[\tilde{f}_{X}(J)]-f_{X}(0)= & {} \pi ^{-1}\sum _{h=1}^{n-1}d_{J}(h)(1-h/n)R_{X}(h)-\pi ^{-1}\sum _{h=1}^{\infty }R_{X}(h), \nonumber \\= & {} \pi ^{-1}\sum _{h=1}^{n-1}(1-h/n)\left[ d_{J}(h)-1\right] R_{X}(h)-\pi ^{-1}\sum _{h=1}^{n-1}(h/n)R_{X}(h)-\pi ^{-1}\sum _{h=n}^{\infty }R_{X}(h), \nonumber \\= & {} B_{4n}-B_{5n}-B_{6n},\text { say.} \end{aligned}$$

(36)

Following a reasoning similar to [42, proof of Theorem 4.2], we can show that

$$ B_{4n}=-2^{-q(J+1)}\lambda _{q}f_{X}^{(q)}(0)[1+o(1)], $$

$\left| B_{5n}\right| \le n^{-\min (1,q)}\sum _{h=1}^{n-1}l^{q}|R_{X}(h)|=O(n^{-\min (1,q)})$ and also that $\left| B_{6n}\right| \le 2n^{-q}\sum _{h=n}^{\infty }h^{q}|R_{X}(h)|=o(n^{-q})$. The bias term is then

$$\begin{aligned} E\tilde{f}_{X}(0;J)-f_{X}(0)=-2^{-q(J+1)}\lambda _{q}f_{X}^{(q)}(0)+o(2^{-qJ})+O(n^{-\min (1,q)}). \end{aligned}$$

(37)

Now, combining (29), (35), (37) we obtain

$$ E\{[\hat{f}_{X}(0;J)-f_{X}(0)]^{2}\}=\frac{2^{J+1}}{n}2D_{\psi }f_{X}^{2}(0)+2^{-2q(J+1)}\lambda _{q}^{2}[f_{X}^{(q)}(0)]^{2}+o(2^{J}/n+2^{-2qJ}). $$

The desired result follows by using $2^{J+1}/n^{\frac{1}{2q+1}}\rightarrow c$. This completes the proof of Theorem 3. $\Box$

Proof

(Corollary 1) The result follows immediately from Theorem 2 because the conditions of Corollary 1 imply $2^{\hat{J}}/2^{J}-1=o_{P}(n^{-1/2(2q+1)})=o_{P}(2^{-J/2}),$ where the nonstochastic finest scale J is given by $2^{J+1}\equiv \max \{[q\lambda _{q}^{2}\alpha (q)n/D_{\psi }]^{1/(2q+1)},0\}.$ The latter satisfies the conditions of Theorem 2. $\Box$

To prove Theorems 4–6, we first state a useful lemma.

Lemma 3

Suppose Assumptions 1 and 2 hold, $J\rightarrow \infty$, and $2^{J}/n\rightarrow 0$. Define

$$ b_{J}(h,m)=a_{J}(h,m)+a_{J}(-h,-m)+a_{J}(h,-m)+a_{J}(-h,m), $$

where $a_{J}(h,m)=2\pi \sum _{j=0}^{J}\sum _{k=1}^{2^{j}}\hat{\psi }_{jk}(2\pi h)\hat{\psi }_{jk}^{*}(2\pi m).$ Then

• (i) $b_{J}(h,m)$ is real-valued, $b_{J}(0,m)=b_{J}(h,0)=0$ and $b_{J}(h,m)=b_{J}(m,h)$;

• (ii) $\sum _{h=1}^{n-1}\sum _{m=1}^{n-1}h^{\nu }|b_{J}(h,m)|=O(2^{(1+\nu )J})$ for $0\le \nu \le \frac{1}{2}$;

• (iii) $\sum _{h=1}^{n-1}\{\sum _{m=1}^{n-1}|b_{J}(h,m)|\}^{2}=O(2^{J})$;

• (iv) $\sum _{h_{1}=1}^{n-1}\sum _{h_{2}=1}^{n-1}\{\sum _{m=1}^{n-1}|b_{J}(h_{1},m)b_{J}(h_{2},m)|\}^{2}=O\{(J+1)2^{J})$;

• (v) $\sum _{h=1}^{n-1}b_{J}(h,h)=(2^{J+1}-1) \{1$+$O((J+1)/2^{J}$+$2^{J(2\tau -1)}/n^{2\tau -1})\},$ where $\tau$ is in Assumption 3;

• (vi) $\sum _{h=1}^{n-1}\sum _{m=1}^{n-1}b_{J}^{2}(h,m)=2(2^{J+1}-1)\{1+o(1)\}$;

• (vii) $\sup _{1\le h,m\le n-1}|b_{J}(h,m)|\le C(J+1)$;

• (viii) $\sup _{1\le h\le n-1}\sum _{m=1}^{n-1}|b_{J}(h,m)|\le C(J+1)$.

Proof

(Lemma 3) See [50, Appendix B] for (i)–(vi) and [43, Lemma A.1] for (vii) and (viii). $\Box$

Proof

(Theorem 4) Let $Z_{t}=\varepsilon _{t}-1$ be such that $E(Z_{t})=0$. Under $\mathbb {H}_{0}^{\mathscr {A}}$, $\varepsilon _{t}=e_{t}$. We consider $\bar{R}_{Z}(h)=n^{-1}\sum _{t=|h|+1}^{n}Z_{t}Z_{t-|h|}$ and $\bar{\alpha }_{ejk}=\sum _{h=1-n}^{n-1}\bar{R}_{Z}(h)\hat{\psi }_{jk}^{*}(2\pi h)$. Let $\bar{\mathscr {A}}(J)$ defined as $\mathscr {A}(J)$ but using $\bar{\alpha }_{ejk}$. Let $\bar{\rho }_{Z}(h)=\bar{R}_{Z}(h)/\bar{R}_{Z}(0)$. Because $\bar{\rho }_{Z}(-h)=\bar{\rho }_{Z}(h)$ and $\bar{\alpha }_{ejk}$ is real-valued, we have

$$ 2\pi n\sum _{j=0}^{J}\sum _{k=1}^{2^{j}}\bar{\alpha }_{ejk}^{2}=n \sum _{h=1}^{n-1}\sum _{m=1}^{n-1}b_{J}(h,m)\bar{\rho }_{Z}(h)\bar{\rho }_{Z}(m), $$

where the equality follows from re-indexing and the definition of $b_{J}(h,m)$. We have $\bar{R}_Z(0)-\sigma _Z^{2}=O_{P}(n^{-1/2})$, since under $\mathbb {H}_{0}^{\mathscr {A}}$ we have that $\{ e_t \}$ is iid. Also, we have

$$\begin{aligned} n\sum _{h=1}^{n-1}\sum _{m=1}^{n-1}b_{J}(h,m)\bar{\rho }_Z(h)\bar{\rho }_Z(m)= & {} \sigma _Z^{-4}n\sum _{h=1}^{n-1}\sum _{m=1}^{n-1}b_{J}(h,m)\bar{R}_Z(h)\bar{R}_Z(m) \nonumber \\&+[ \bar{R}^{-2}_Z(0)-\sigma _Z^{-4} ] n\sum _{h=1}^{n-1}\sum _{m=1}^{n-1}b_{J}(h,m)\bar{R}_Z(h)\bar{R}_Z(m), \\= & {} \sigma _Z^{-4}n\sum _{h=1}^{n-1}\sum _{m=1}^{n-1}b_{J}(h,m)\bar{R}_Z(h)\bar{R}_Z(m)+ O_{P}(2^{J}/n^{1/2}), \nonumber \end{aligned}$$

(38)

where the second term is of the indicated order of magnitude because

$$ E\left[ \sum _{h=1}^{n-1}\sum _{m=1}^{n-1}|b_J(h,m)\bar{R}_Z(h)\bar{R}_Z(m)| \right] \le Cn^{-1}\sum _{h=1}^{n-1}\sum _{m=1}^{n-1}|b_J(h,m)|=O(2^J/n). $$

given $E[\bar{R}_Z^{2}(h)] \le Cn^{-1}$ and Lemma 3(ii). We now focus on the first term in (38). We have,

$$\begin{aligned} n\sum _{h=1}^{n-1}\sum _{m=1}^{n-1}b_{J}(h,m)\bar{R}_Z(h)\bar{R}_Z(m)= & {} n^{-1}\sum _{h=1}^{n-1}\sum _{m=1}^{n-1}b_{J}(h,m)\sum _{t=h+1}^{n}\sum _{s=m+1}^{n}Z_{t}Z_{t-h}Z_{s}Z_{s-m}, \nonumber \\= & {} n^{-1}\sum _{h=1}^{n-1}\sum _{m=1}^{n-1}b_{J}(h,m)\left( \sum _{t=1}^{n}\sum _{s=1}^{n}-\sum _{t=1}^{h}\sum _{s=m+1}^{n} -\sum _{t=1}^{n}\sum _{s=1}^{m}\right) Z_{t}Z_{t-h}Z_{s}Z_{s-m}, \nonumber \\= & {} \hat{C}_{n}+\hat{D}_{1n}-\hat{D}_{2n}-\hat{D}_{3n}, \end{aligned}$$

(39)

where

$$\begin{aligned} \hat{C}_{n}= & {} n^{-1}\sum _{h=1}^{n-1}\sum _{m=1}^{n-1}b_{J}(h,m)\left( \sum _{t=2}^{n}\sum _{s=1}^{t-1}+\sum _{s=2}^{n}\sum _{t=1}^{s-1}\right) Z_{t}Z_{t-h}Z_{s}Z_{s-m}, \\= & {} 2n^{-1}\sum _{h=1}^{n-1}\sum _{m=1}^{n-1}b_{J}(h,m)\sum _{t=2}^{n}\sum _{s=1}^{t-1}Z_{t}Z_{t-h}Z_{s}Z_{s-m}, \text { given }b_{J}(h,m)=b_{J}(m,h), \\ \hat{D}_{1n}= & {} n^{-1}\sum _{h=1}^{n-1}\sum _{m=1}^{n-1}b_{J}(h,m)\sum _{t=1}^{n}Z_{t}^{2}Z_{t-h}Z_{t-m}, \\ \hat{D}_{2n}= & {} n^{-1}\sum _{h=1}^{n-1}\sum _{m=1}^{n-1}b_{J}(h,m)\sum _{t=1}^{h}\sum _{s=m+1}^{n}Z_{t}Z_{t-h}Z_{s}Z_{s-m}, \\ \hat{D}_{3n}= & {} n^{-1}\sum _{h=1}^{n-1}\sum _{m=1}^{n-1}b_{J}(h,m)\sum _{t=1}^{n}\sum _{s=1}^{m}Z_{t}Z_{t-h}Z_{s}Z_{s-m}. \end{aligned}$$

In order to prove Theorem 4, we first state Proposition 1 that shows that $\hat{C}_{n}$ represents the dominant term.

Proposition 1

Suppose Assumptions 1–3 hold, $J\rightarrow \infty$, and $2^{2J}/n\rightarrow 0$. Then

$$ 2^{-J/2}\left\{ 2\pi n\sum _{j=0}^{J}\sum _{k=1}^{2^{j}}\bar{\alpha }_{jk}^{2}-(2^{J+1}-1)\right\} =2^{-J/2}\sigma ^{-4}\hat{C}_{n}+o_{P}(1). $$

We now decompose $\hat{C}_{n}$ into the terms with $t-s>q$ and $t-s\le q$, for some $q\in \mathbb {Z}^{+}$:

$$\begin{aligned} \hat{C}_{n}= & {} 2n^{-1}\sum _{h=1}^{n-1}\sum _{m=1}^{n-1}b_{J}(h,m)\left( \sum _{t=q+2}^{n}\sum _{s=1}^{t-q-1}+\sum _{t=2}^{n}\sum _{s=\max (t-q,1)}^{t-1}\right) Z_{t}Z_{t-h}Z_{s}Z_{s-m}, \nonumber \\= & {} \hat{D}_{n}+\hat{D}_{4n}. \end{aligned}$$

(40)

Furthermore, we decompose

$$\begin{aligned} \hat{D}_{n}= & {} 2n^{-1}\left( \sum _{h=1}^{q}\sum _{m=1}^{q}+\sum _{h=1}^{q} \sum _{m=q+1}^{n-1}+\sum _{h=q+1}^{n-1}\sum _{m=1}^{n-1}\right) b_{J}(h,m)\sum _{t=q+2}^{n}\sum _{s=1}^{t-q-1}Z_{t}Z_{t-h}Z_{s}Z_{s-m}, \nonumber \\= & {} \hat{U}_{n}+\hat{D}_{5n}+\hat{D}_{6n},\text { say,} \end{aligned}$$

(41)

where $\hat{D}_{5n}$ and $\hat{D}_{6n}$ are the contributions from $m>q$ and $h>q$, respectively.

Proposition 2 shows that $\hat{C}_{n}$ can be approximated arbitrarily well by $\hat{U}_{n}$ under a proper condition on q.

Proposition 2

Suppose Assumptions 1–3 hold, $J\rightarrow \infty ,2^{2J}/n\rightarrow 0$, and $q\equiv q_{n}\rightarrow \infty ,q/2^{J}\rightarrow \infty ,q^{2}/n\rightarrow 0$. Then $2^{-J/2}\hat{C}_{n}=2^{-J/2}\hat{U}_{n}+o_{P}(1).$

It is much easier to show the asymptotic normality of $\hat{U}_{n}$ than of $\hat{C}_{n},$ because for $\hat{U}_{n},$ $\{Z_{t}Z_{t-h}\}$ and $\{Z_{s}Z_{s-m}\}$ are independent given $t-s>q$ and $0<h,m\le q$.

Proposition 3

Suppose Assumptions 1–3 hold, and $J\rightarrow \infty ,2^{2J}/n\rightarrow 0,q/2^{J}\rightarrow \infty , q^{2}/n\rightarrow 0$. Let $\lambda _{n}^{2}=E(\hat{U}_{n}^{2})$. Then $4(2^{J+1}-1)\sigma ^{8}/\lambda _{n}^{2}\rightarrow 1$, and $\lambda _{n}^{-1}\hat{U}_{n}\rightarrow ^{d}N(0,1)$.

Propositions 1–3 and Slutsky’s Theorem imply $\bar{\mathscr {A}}(J) \rightarrow ^{d} N(0,1)$. Propositions 4 and 5 show that parameter estimation does not have impact on the asymptotic distribution of the test statistic.

Proposition 4

$n \sum _{j=0}^{J}\sum _{k=1}^{2^{j}} (\hat{\alpha }_{jk}-\bar{\alpha }_{jk})^{2}= O_P(2^J/n) + O_P(1).$

Proposition 5

$n \sum _{j=0}^{J}\sum _{k=1}^{2^{j}} (\hat{\alpha }_{jk}-\bar{\alpha }_{jk}) \bar{\alpha }_{jk} = o_P(2^{J/2}).$

The proof of Theorem 4 will be completed provided Propositions 1–5 are shown. The proofs of Propositions 1–3 are very similar to the proofs of Propositions 1–3 in [50], for proving the asymptotic normality of a wavelet-based test statistic for serial correlation. These proofs are then omitted (but for the interested reader all the detailed proofs are available from the authors).

Proof

(Proposition 4) A standard Taylor’s expansion gives

$$ D_t^{-1}(\hat{{\varvec{\theta }}}) = D_t^{-1}({\varvec{\theta }}_0) + \left\{ \frac{\partial }{\partial {\varvec{\theta }}} D_t^{-1}({\varvec{\theta }}_0) \right\} ^{\top } (\hat{{\varvec{\theta }}}- {\varvec{\theta }}_0) + \frac{1}{2} (\hat{{\varvec{\theta }}}- {\varvec{\theta }}_0)^{\top }\frac{\partial ^2}{\partial {\varvec{\theta }} \partial {\varvec{\theta }}^{\top }} D_t^{-1}(\bar{{\varvec{\theta }}})(\hat{{\varvec{\theta }}}-{\varvec{\theta }}_0), $$

where $\bar{{\varvec{\theta }}}$ lies between $\hat{ {\varvec{\theta }}}$ and ${\varvec{\theta }}_0$. We have

$$\begin{aligned} \hat{R}_Z(h)-\tilde{R}_Z(h)= & {} n^{-1} \sum _{t=|h|+1}^n (\hat{Z}_t - Z_t)( \hat{Z}_{t-|h|} - Z_{t-|h|}) + n^{-1} \sum _{t=|h|+1}^n Z_t (\hat{Z}_{t-|h|} - Z_{t-|h|}) \\&+ n^{-1} \sum _{t=|h|+1}^n (\hat{Z}_t - Z_t) Z_{t-|h|}, \\= & {} \hat{A}_1(h) + \hat{A}_2(h)+\hat{A}_3(h). \end{aligned}$$

We write $\hat{\alpha }_{jk} - \tilde{\alpha }_{jk} = \hat{B}_{1jk} + \hat{B} _{2jk}+\hat{B}_{3jk}$ where

$$ \hat{B}_{1jk} = \sum _{h=1-n}^{n-1} \hat{A}_1(h) \hat{\psi }_{jk}(2\pi h), \; \hat{B}_{2jk} = \sum _{h=1-n}^{n-1} \hat{A}_2(h) \hat{\psi }_{jk}(2\pi h), \; \hat{B}_{3jk} = \sum _{h=1-n}^{n-1} \hat{A}_3(h) \hat{\psi }_{jk}(2\pi h). $$

Then $(\hat{\alpha }_{jk} - \tilde{\alpha }_{jk})^2 \le 4(\hat{B}_{1jk}^2 + \hat{B}_{2jk}^2+ \hat{B}_{3jk}^3)$ We first study the term involving $\hat{B}_{1jk}$. We write

$$ 2\pi \sum _{j=0}^J \sum _{k=1}^{2^j} \hat{B}_{1jk}^2 = \sum _{h=1-n}^{n-1}\sum _{m=1-n}^{n-1} a_J(h,m) \hat{A}_1(h)\hat{A}_1(m) = \sum _{h=1}^{n-1}\sum _{m=1}^{n-1} b_J(h,m) \hat{A}_1(h)\hat{A}_1(m). $$

We have

$$ 2\pi \sum _{j=0}^J \sum _{k=1}^{2^j} \hat{B}_{1jk}^2 \le \left( \sup \hat{A} _1(h) \right) ^2 | \sum _{h=1}^{n-1}\sum _{m=1}^{n-1} b_J(h,m) | = O_P(2^J/n). $$

We now study the term involving $\hat{B}_{2jk}$. Let

$$\begin{aligned} \hat{a}_{21}(h)= & {} n^{-1} \sum _{t=|h|+1}^n Z_t X_{t-|h|} \left\{ \frac{\partial }{\partial {\varvec{\theta }}} D_{t-|h|}^{-1}({\varvec{\theta }}_0) \right\} ^{\top }, \\ \hat{a}_{22}(h)= & {} n^{-1} \sum _{t=|h|+1}^n Z_t X_{t-|h|} \frac{\partial ^2}{\partial {\varvec{\theta }} \partial {\varvec{\theta }}^{\top }}D_{t-|h|}^{-1}(\bar{{\varvec{\theta }}}_0). \end{aligned}$$

We write $\hat{A}_2(h) = \hat{A}_{21}(h) + \hat{A}_{22}(h)$, where

$$\begin{aligned} \hat{A}_{21}(h)= & {} \hat{a}_{21}(h) (\hat{{\varvec{\theta }}} - {\varvec{\theta }}_0), \\ \hat{A}_{22}(h)= & {} \frac{1}{2} (\hat{{\varvec{\theta }}} - {\varvec{\theta }}_0)^{\top }\hat{a}_{22}(h) (\hat{{\varvec{\theta }}} - {\varvec{\theta }}_0). \end{aligned}$$

Then

$$\begin{aligned} \hat{B}_{2jk}= & {} \sum _{h=1-n}^{n-1} \hat{A}_{2}(h)\hat{\psi }_{jk}(2\pi h), \\= & {} \left[ \sum _{h=1-n}^{n-1} \hat{a}_{21}(h)\hat{\psi }_{jk}(2\pi h) \right] (\hat{{\varvec{\theta }}} - {\varvec{\theta }}_0) + \frac{1}{2}(\hat{{\varvec{\theta }}} - {\varvec{\theta }}_0)^{\top } \left[ \sum _{h=1-n}^{n-1} \hat{a}_{22}(h) \hat{\psi }_{jk}(2\pi h) \right] (\hat{{\varvec{\theta }}} - {\varvec{\theta }}_0). \end{aligned}$$

We obtain

$$\begin{aligned} 2\pi n \sum _{j=0}^J \sum _{k=1}^{2^j} B_{2jk}^2\le & {} 4 n ||\hat{ {\varvec{\theta }}} - {\varvec{\theta }}_0||^2 \sum _{j=0}^J \sum _{k=1}^{2^j} \left\| \sum _{h=1-n}^{n-1} \hat{a}_{21}(h)\hat{\psi }_{jk}(2\pi h) \right\| ^2 \\&+ 2 n||\hat{{\varvec{\theta }}} - {\varvec{\theta }}_0||^4 \sum _{j=0}^J \sum _{k=1}^{2^j} \left\| \sum _{h=1-n}^{n-1} \hat{a}_{22}(h) \hat{\psi }_{jk}(2\pi h) \right\| ^2 = O_P(2^J/n), \end{aligned}$$

since

$$\begin{aligned} 2\pi \sum _{j=0}^J \sum _{k=1}^{2^j} \left\| \sum _{h=1-n}^{n-1} \hat{a}_{21}(h) \hat{\psi }_{jk}(2\pi h) \right\| ^2= & {} \left| \sum _{h=1}^{n-1}\sum _{m=1}^{n-1} b_J(h,m) \hat{a}_{21}(h)\hat{a}_{21}^{\top }(m) \right| , \\\le & {} \left( \sup || \hat{a}_{21}(h)|| \right) ^2 \left( \sum _{h=1}^{n-1}\sum _{m=1}^{n-1} |b_J(h,m)| \right) = O_P(2^J/n). \end{aligned}$$

We now study the term involving $\hat{B}_{3jk}$.

$$\begin{aligned} \hat{a}_{31}(h)= & {} n^{-1} \sum _{t=|h|+1}^n Z_{t-|h|} X_t \left( \frac{\partial }{\partial {\varvec{\theta }}} D_{t}^{-1}({\varvec{\theta }}_0) \right) ^{\top }, \\ \hat{a}_{32}(h)= & {} n^{-1} \sum _{t=|h|+1}^n Z_{t-|h|} X_t \frac{\partial ^2}{\partial {\varvec{\theta }} \partial {\varvec{\theta }}^{\top }} D_{t}^{-1}(\bar{{\varvec{\theta }}}_0). \end{aligned}$$

We write $\hat{A}_3(h) = \hat{A}_{31}(h) + \hat{A}_{32}(h)$, where

$$\begin{aligned} \hat{A}_{31}(h)= & {} \hat{a}_{31}(h) (\hat{{\varvec{\theta }}} - {\varvec{\theta }}_0), \\ \hat{A}_{32}(h)= & {} \frac{1}{2} (\hat{{\varvec{\theta }}} - {\varvec{\theta }}_0)^{\top }\hat{a}_{32}(h) (\hat{{\varvec{\theta }}} - {\varvec{\theta }}_0). \end{aligned}$$

We obtain

$$ 2\pi n \sum _{j=0}^J \sum _{k=1}^{2^j} B_{3jk}^2 \le 4 n \sum _{j=0}^J \sum _{k=1}^{2^j} \left[ \sum _{h=1-n}^{n-1} \hat{A}_{31}(h)\hat{\psi }_{jk}(2\pi h) \right] ^2 + 4 n \sum _{j=0}^J \sum _{k=1}^{2^j} \left[ \sum _{h=1-n}^{n-1} \hat{A}_{32}(h)\hat{\psi }_{jk}(2\pi h) \right] ^2. $$

We write $\hat{a}_{31}(h) = E[\hat{a}_{31}(h)] + \left\{ \hat{a}_{31}(h)-E[\hat{a}_{31}(h)]\right\}$. We have

$$\begin{aligned} n \sum _{j=0}^J \sum _{k=1}^{2^j} \left[ \sum _{h=1-n}^{n-1} \hat{A}_{31}(h)\hat{\psi }_{jk}(2\pi h) \right] ^2\le & {} 2 n ||\hat{{\varvec{\theta }}} - {\varvec{\theta }}_0||^2 \left\{ \sum _{j=0}^J \sum _{k=1}^{2^j} \left\| \sum _h E[\hat{a}_{31}(h)] \hat{\psi }_{jk}(2\pi h) \right\| ^2 \right. \\&\left. + \sum _{j=0}^J \sum _{k=1}^{2^j} \left\| \sum _h [\hat{a}_{31}(h)-E\hat{a}_{31}(h)] \hat{\psi }_{jk}(2\pi h) \right\| ^2 \right\} . \\ \end{aligned}$$

Since we can interpret $E\hat{a}_{31}(h) = {\text {cov}}(Z_{t-|h|}, X_t \frac{\partial }{\partial {\varvec{\theta }}} D_t^{-1}({\varvec{\theta }}_0))$ as a cross-correlation function, we have that

$$ \sum _{j=0}^J \sum _{k=1}^{2^j} \left\| \sum _h E[\hat{a}_{31}(h)] \hat{\psi }_{jk}(2\pi h) \right\| ^2 = O(1). $$

Also,

$$\begin{aligned}&\sum _{j=0}^J \sum _{k=1}^{2^j} E \left\| \sum _h [\hat{a}_{31}(h)-E\hat{a}_{31}(h)] \hat{\psi }_{jk}(2\pi h) \right\| ^2 \\= & {} | \sum _{h=1}^{n-1}\sum _{m=1}^{n-1} b_J(h,m) E [ (\hat{a}_{31}(h)-E\hat{a}_{31}(h))(\hat{a}_{31}(m)-E\hat{a}_{31}(m))^{\top }] |, \\\le & {} \sum _{h=1}^{n-1}\sum _{m=1}^{n-1} |b_J(h,m)| \left\{ E\Vert \hat{a}_{31}(h)-E\hat{a}_{31}(h) \Vert ^2 \right\} ^{1/2} \left\{ E\Vert \hat{a}_{31}(m)-E\hat{a}_{31}(m)\Vert ^2 \right\} ^{1/2}, \\\le & {} \sup _h E\Vert \hat{a}_{31}(h)-E\hat{a}_{31}(h) \Vert ^2 \sum _h \sum _m |b_J(h,m)| = O(2^J/n). \end{aligned}$$

This shows Proposition 4. $\Box$

Remark 1

Proposition 4 is established under a general stationary process for $\{ Z_t \}$, that is, the result is established without assuming the null hypothesis.

Proof

(Proposition 5) We write $(\hat{\alpha }_{jk}-\tilde{\alpha }_{jk})\tilde{\alpha }_{jk}=\hat{C}_{1jk}+\hat{C}_{2jk}+\hat{C}_{3jk}$, where

$$\begin{aligned} \hat{C}_{1jk}= & {} \sum _{h=1-n}^{n-1}\hat{A}_{1}(h)\hat{\psi }_{jk}(2\pi h) \tilde{\alpha }_{jk}, \end{aligned}$$

(42)

$$\begin{aligned} \hat{C}_{2jk}= & {} \sum _{h=1-n}^{n-1}\hat{A}_{2}(h)\hat{\psi }_{jk}(2\pi h) \tilde{\alpha }_{jk}, \end{aligned}$$

(43)

$$\begin{aligned} \hat{C}_{3jk}= & {} \sum _{h=1-n}^{n-1}\hat{A}_{3}(h)\hat{\psi }_{jk}(2\pi h) \tilde{\alpha }_{jk}. \end{aligned}$$

(44)

By the Cauchy–Schwarz inequality and the fact that $n\sum _{j=0}^{J}\sum _{k=1}^{2^{j}}\tilde{\alpha }_{jk}^{2}=O_{P}(2^{J})$ under the null hypothesis, we have that

$$ n\sum _{j=0}^{J}\sum _{k=1}^{2^{j}}\sum _{h=1-n}^{n-1}\hat{A}_{1}(h)\hat{\psi }_{jk}(2\pi h)\tilde{\alpha }_{jk}=O_{P}(2^{J}/n^{1/2}). $$

Since $n\sum _{j=0}^{J}\sum _{k=1}^{2^{j}}B_{2jk}^{2}=O_{P}(2^{J}/n)$, we have that

$$ n\sum _{j=0}^{J}\sum _{k=1}^{2^{j}}\hat{C}_{2jk}=O_{P}(2^{J}/n^{1/2}). $$

We write

$$\begin{aligned} \hat{C}_{3jk}= & {} \left[ \sum _{h=1-n}^{n-1}\hat{a}_{31}(h)\hat{\psi }_{jk}(2\pi h)\tilde{\alpha }_{jk} \right] (\hat{{\varvec{\theta }}}-{\varvec{\theta }}_{0}) \\+ & {} \frac{1}{2}(\hat{{\varvec{\theta }}}-{\varvec{\theta }}_{0})^{\top } \left\{ \sum _{h=1-n}^{n-1}\hat{a}_{32}(h)\hat{\psi }_{jk}(2\pi h)\tilde{\alpha }_{jk}\right\} (\hat{{\varvec{\theta }}}-{\varvec{\theta }}_{0}). \end{aligned}$$

We have

$$ 2\pi n\sum _{j=0}^{J}\sum _{k=1}^{2^{j}}\sum _{h=1-n}^{n-1}E[ \hat{a}_{31}(h)] \hat{\psi }_{jk}(2\pi h)\tilde{\alpha }_{jk}= n\sum _{h=1}^{n-1}\sum _{m=1}^{n-1}E[\hat{a}_{31}(h)]\tilde{R}(m)b_{J}(h,m). $$

Since

$$\begin{aligned}&n^{2}E\left\| \sum _{h=1}^{n-1}\sum _{m=1}^{n-1}E[\hat{a}_{31}(h)]\tilde{R}(m)b_{J}(h,m)\right\| ^{2} \\\le & {} Cn\sum _{h_{1}=1}^{n-1}\sum _{h_{2}=1}^{n-1}\sum _{m=1}^{n-1}E[\hat{a}_{31}(h_{1})]E[\hat{a}_{31}(h_{2})]^{\top }|b_{J}(h_{1},m)b_{J}(h_{2},m)|, \\\le & {} Cn\left( \sum _{h_{1}=1}^{n-1}\sum _{h_{2}=1}^{n-1}[\sum _{m=1}^{n-1}|b_{J}(h_{1},m)b_{J}(h_{2},m)|]^{2}\right) ^{1/2}=O(n(J+1)^{1/2}2^{(J+1)/2}). \end{aligned}$$

Then

$$ n\sum _{h=1}^{n-1}\sum _{m=1}^{n-1}E[\hat{a}_{31}(h)]\tilde{R}(m)b_{J}(h,m)=O_{P}(n^{1/2}J^{1/4}2^{J/4}). $$

We have

$$\begin{aligned}&E\Vert n\sum _{j=0}^{J}\sum _{k=1}^{2^{j}}\sum _{h}[\hat{a}_{31}(h)-E\hat{a}_{31}(h)]\hat{\psi }_{jk}(h)\tilde{\alpha }_{jk}\Vert \\\le & {} \frac{n}{2\pi }\sum _{h=1}^{n-1}\sum _{m=1}^{n-1} E \left[ \Vert \hat{a}_{31}(h)-E\hat{a}_{31}(h) \Vert |\tilde{R}(m)| \right] |b_{J}(h,m)|, \\\le & {} \frac{n}{2\pi }\sum _{h=1}^{n-1}\sum _{m=1}^{n-1}\left[ E\Vert \hat{a}_{31}(h)-E\hat{a}_{31}(h)\Vert ^{2}\right] ^{1/2}[E\tilde{R}^{2}(m)]^{1/2}|b_{J}(h,m)|, \\= & {} O(n\;n^{-1/2}\;n^{-1/2}\;2^{J+1})=O(2^{J+1}). \end{aligned}$$

This completes the proof of Proposition 5 and so Theorem 4. $\Box$

Proof

(Theorem 5) We write

$$ \mathscr {A}(\hat{J})-\mathscr {A}(J)= [D_n(\hat{J})]^{-1/2}\left\{ 2\pi n\sum _{j=J}^{\hat{J}}\sum _{k=1}^{2^{j}}\hat{\alpha }_{jk}^{2}- [C_{n}(\hat{J})-C_{n}(J)]\right\} -\left\{ 1-[D_{n}(J)]^{1/2}/[D_{n}(\hat{J})]^{1/2} \right\} \mathscr {A}(J), $$

where $C_{n}(J)=2^{J+1}-1$, $D_{n}(J)=4(2^{J+1}-1)$. Given $\mathscr {A}(J)=O_{P}(1)$ by Theorem 4, it suffices for $\mathscr {A}(\hat{J})-\mathscr {A}(J)\rightarrow ^p 0$ and $\mathscr {A}(\hat{J}) \rightarrow ^d N(0,1)$ to establish

$$\begin{aligned} (i)&[D_{n}(J)]^{-1/2}\left\{ 2\pi n\sum _{j=J}^{\hat{J}}\sum _{k=1}^{2^{j}} \hat{\alpha }_{jk}^{2}-[C_{n}(\hat{J})-C_{n}(J)]\right\} \rightarrow ^{p} 0, \\ (ii)&D_{n}(\hat{J})/D_{n}(J)\rightarrow ^{p} 1. \end{aligned}$$

We first show (i). Decompose

$$\begin{aligned} 2\pi n\sum _{j=J}^{\hat{J}}\sum _{k=1}^{2^{j}}\hat{\alpha }_{jk}^{2}= & {} 2 \pi n\sum _{j=J}^{\hat{J}}\sum _{k=1}^{2^{j}}(\hat{\alpha }_{jk}-\tilde{\alpha }_{jk})^{2}+2\pi n\sum _{j=J}^{\hat{J}}\sum _{k=1}^{2^{j}}\tilde{\alpha }_{jk}^{2}+ 4\pi n\sum _{j=J}^{\hat{J}}\sum _{k=1}^{2^{j}}(\hat{\alpha }_{jk}-\tilde{\alpha }_{jk})\tilde{\alpha }_{jk}, \nonumber \\= & {} \hat{G}_{1}+\hat{G}_{2}+2\hat{G}_{3}. \end{aligned}$$

(45)

For the first term in (45), we write

$$\begin{aligned} \hat{G}_{1}=2\pi n\sum _{j=0}^{\hat{J}}\sum _{k=1}^{2^{j}}(\hat{\alpha }_{jk}-\tilde{\alpha }_{jk})^{2}- 2\pi n\sum _{j=0}^{J}\sum _{k=1}^{2^{j}}(\hat{\alpha }_{ijk}-\tilde{\alpha }_{jk})^{2}=\hat{G}_{11}-\hat{G}_{12}. \end{aligned}$$

(46)

By Proposition 4, we have $[D_{n}(J)]^{-1/2}\hat{G}_{12}\rightarrow ^p 0$. For the first term in (46), we have for any given constants $M>0$ and $\varepsilon >0$,

$$\begin{aligned} P\left( \hat{G}_{11}>\varepsilon \right) \le P\left( \hat{G}_{11}>\varepsilon , C_{0}2^{J/2}|2^{\hat{J}}/2^{J}-1|\le \varepsilon \right) +P\left( C_{0}2^{J/2}|2^{\hat{J}}/2^{J}-1|>\varepsilon \right) . \end{aligned}$$

(47)

For any given constants $C_{0},\varepsilon >0,$ the second term in (47) vanishes to 0 as $n \rightarrow \infty$ given $2^{J/2}|2^{\hat{J}}/2^{J}-1|\rightarrow ^{p} 0.$ For the first term, given $C_{0}2^{J/2}|2^{\hat{J}}/2^{J}-1|\le \varepsilon$, we have for n sufficiently large,

$$\begin{aligned}{}[D_{n}(J)]^{-1/2}\hat{G}_{11}\le & {} [D_{n}(J)]^{-1/2}2\pi n\sum _{j=0}^{[\log _{2}2^{J}(1+\varepsilon /(C_{0}2^{J/2}))]}\sum _{k=1}^{2^{j}}(\hat{\alpha }_{jk}-\tilde{\alpha }_{jk})^{2}, \\\le & {} [D_{n}(J)]^{-1/2}2\pi n\sum _{j=0}^{J+1}\sum _{k=1}^{2^{j}}(\hat{\alpha }_{jk}-\tilde{\alpha }_{jk})^{2}=o_{P}(1). \end{aligned}$$

by Proposition 4. Therefore, we have

$$\begin{aligned}{}[D_{n}(J)]^{-1/2}\hat{G}_{1}=o_{P}(1). \end{aligned}$$

(48)

Next, we consider $\hat{G}_{2}$ in (45). We write

$$ \hat{G}_{2}=n\sum _{h=1}^{n-1}\sum _{m=1}^{n-1}\tilde{R}_{e}(h)\tilde{R}_{e}(m)\left[ b_{\hat{J}}(h,m)-b_{J}(h,m)\right] . \nonumber $$

Since for n sufficiently large,

$$\begin{aligned}&\sum _{h=1-n}^{n-1}\sum _{m=1-n}^{n-1}\left| a_{J}(h,m)-a_{\hat{J}}(h,m)\right| \\\le & {} C\sum _{h=1-n}^{n-1}\sum _{m=1-n}^{n-1}\sum _{j=[\log _{2}2^{J}(1-\varepsilon /(C_{0}2^{J/2}))]}^{[\log _{2}2^{J}(1+\varepsilon /(C_{0}2^{J/2}))]}\left| c_{j}(h,m)\hat{\psi }(2\pi h/2^{j})\hat{\psi }^{*}(2\pi m/2^{j})\right| , \\\le & {} C\sum _{j=[\log _{2}2^{J}(1-\varepsilon /(C_{0}2^{J/2}))]}^{[\log _{2}2^{J}(1+\varepsilon /(C_{0}2^{J/2}))]}2^{j}\left[ 2^{-j}\sum _{h=1-\bar{T}}^{\bar{T}-1}|\hat{\psi }(2\pi h/2^{j})|\right] , \\&\times \left[ \sum _{r=-\infty }^{\infty }|\hat{\psi }(2\pi h/2^{j}+2\pi r)|\right] , \\\le & {} C2^{J}\varepsilon /(C_{0}2^{J/2}), \end{aligned}$$

given Assumption 3 and

$$\begin{aligned} c_{j}(h,m)=\left\{ \begin{array}{ll} 1 &{} \text {if }m-h=2^{j}r\text { for some }r\in \mathbb {Z}, \\ 0 &{} \text {otherwise.} \end{array} \right. \end{aligned}$$

(49)

Cf. [61, (6.19), p.392]. Therefore

$$\begin{aligned} E|\hat{G}_{2}|\le & {} n\sum _{h=1-n}^{n-1}\sum _{m=1-n}^{n-1}E|\tilde{R}_{e}(h)\tilde{R}_{e}(m)||a_{\hat{J}}(h,m)-a_{J}(h,m)|, \\\le & {} n\sup _{h}{\text {var}}{\tilde{R}_{e}(h)}\sum _{h=1-n}^{n-1}\sum _{m=1-n}^{n-1}|a_{\hat{J}}(h,m)-a_{J}(h,m)|, \\\le & {} C2^{J/2}\varepsilon /C_{0}. \end{aligned}$$

Then $[D_{n}(J)]^{-1/2}\left\{ \hat{G}_{2}-[C(\hat{J})-C(J)]\right\} \rightarrow ^p 0$. Note that $[D_{n}(J)]^{-1/2}[C_{n}(\hat{J})-C_{n}(J)]=o_{P}(1)$. Next, by the Cauchy-Schwarz inequality and (48), we have

$$ [D_{n}(J)]^{-1/2}|\hat{G}_{3}|\le \left( [D_{n}(J)]^{-1/2}\hat{G}_{1}\right) ^{1/2}\left( [D_{n}(J)]^{-1/2}|\hat{G}_{2}|\right) ^{1/2}=o_{P}(1). $$

Summarizing, we obtain result (i), that is

$$ [D_{n}(J)]^{-1/2}\left\{ 2\pi n\sum _{j=J}^{\hat{J}}\sum _{k=1}^{2^{j}}\hat{\alpha }_{jk}^{2}-[C_n(\hat{J})-C_n(J)]\right\} =o_{P}(1). $$

We now show (ii), that is $D_{n}(\hat{J})/D_n(J)=1+o_{P}(1)$. Using the fact that $2^{\hat{J}}/2^J = 1 + o_P(2^{-J/2})$ and $J \rightarrow \infty$ such that $2^J/n \rightarrow 0$, one shows easily that

$$ \frac{D_{n}(\hat{J})}{D_n(J)} = \frac{2^{\hat{J}+1} - 1}{2^{J+1} - 1} \rightarrow ^p 1. $$

This shows (ii). This completes the proof of Theorem 5. $\Box$

Proof

(Theorem 6) We first show $Q(\hat{f},f)=Q(\tilde{f},f)+o_{P}(2^{J}/T+2^{-2qJ})$. Write

$$\begin{aligned} Q(\hat{f},f)-Q(\tilde{f},f)= & {} Q(\hat{f},\tilde{f})+2\int _{-\pi }^{\pi }[\hat{f}(\omega )-\tilde{f}(\omega )][\tilde{f}(\omega )-f(\omega )]d\omega , \nonumber \\= & {} \hat{Q}_{1}+2\hat{Q}_{2}. \end{aligned}$$

(50)

For the first term in (50), by Parseval’s identity, Proposition 4 (which can be shown to continues to hold given Assumptions 2–3 and 7–10; See Remark 1), and $D_{n}(J)\propto O(2^{J+1})$, we have

$$\begin{aligned} \hat{Q}_{1}=\sum _{j=0}^{J}\sum _{k=1}^{2^{j}}(\hat{\alpha }_{jk}-\tilde{\alpha }_{jk})^{2}=O_{P}[n^{-1}+2^{J}n^{-2}]=o_{P}(2^{J}/n), \end{aligned}$$

(51)

as $n\rightarrow \infty$. For the second term, we have $\hat{Q}_{2}=o_{P}(2^{J}/n+2^{-2qJ})$ by the Cauchy-Schwarz inequality, (51) and the fact that $Q(\tilde{f},f)=O_{P}(2^{J}/n+2^{-2qJ})$, which follows by Markov’s inequality and $EQ(\tilde{f},f)=O(2^{J}/n+2^{-2qJ})$. The latter is to be shown below.

To compute $E[Q(\tilde{f},f)],$ we write

$$\begin{aligned} E[Q(\tilde{f},f)]= E[Q(\tilde{f},E\tilde{f})] + Q[E(\tilde{f}),f]. \end{aligned}$$

(52)

We first consider the second term in (52). Put $B(\omega ) \equiv \sum _{j=J+1}^{\infty }\sum _{k=1}^{2^{j}}\alpha _{jk}\varPsi _{jk}(\omega )$. Then

$$\begin{aligned} Q[E(\tilde{f}),f]= \int _{-\pi }^{\pi }B^{2}(\omega )d\omega +\sum _{j=0}^{J}\sum _{k=1}^{2^{j}} (E\tilde{\alpha }_{jk}-\alpha _{jk})^{2}. \end{aligned}$$

(53)

We evaluate directly $\int _{-\pi }^{\pi }B^{2}(\omega )d\omega$. Using the orthonormality of the wavelet basis, we have that

$$ \int _{-\pi }^{\pi }B^{2}(\omega )d\omega = \sum _{j=J+1}^{\infty }\sum _{k=1}^{2^j}\alpha _{jk}^2. $$

Replacing $\alpha _{jk}=\sum _{h=-\infty }^{\infty }R_e(h)\hat{\psi }_{jk}(2 \pi h)$ and since $\hat{\psi }_{jk}(2 \pi h)=e^{-i 2\pi h k /2^j} 2^{-j/2}\hat{\psi }(2\pi h/2^j)$,

$$\begin{aligned} \sum _{j=J+1}^{\infty } \sum _{k=1}^{2^j} \alpha _{jk}^2= & {} \sum _{j=J+1}^{\infty } \sum _{h=-\infty }^{\infty }\sum _{m=-\infty }^{\infty } R_e(h)R_e(m) \{ 2^{-j} \sum _{k=1}^{2^j} e^{i 2\pi (m-h) k/2^j} \} \hat{\psi }(2\pi h/2^j)\hat{\psi }^{*}(2\pi m/2^j), \\= & {} \sum _{j=J+1}^{\infty }\sum _{h=-\infty }^{\infty }\sum _{m=-\infty }^{\infty } c_j(h,m)R_e(h)R_e(m)\hat{\psi }(2\pi h/2^j)\hat{\psi }^{*}(2\pi m/2^j), \end{aligned}$$

where $c_j(h,m) = 2^{-j} \sum _{k=1}^{2^j} e^{i 2\pi (m-h) k/2^j}$ is as in (49). By a change of variables,

$$\begin{aligned} \sum _{j=J+1}^{\infty } \sum _{k=1}^{2^j} \alpha _{jk}^2 = \sum _{j=J+1}^{\infty } \sum _{h=-\infty }^{\infty } \sum _{r=-\infty }^{\infty } R_e(h)R_e(h+2^j r) \hat{\psi }(2\pi h/2^j)\hat{\psi }^{*}(2\pi h/2^j + 2\pi r). \end{aligned}$$

(54)

We evaluate separately the case corresponding to $r=0$ and $r \ne 0$ in (54).

$$\begin{aligned}&\sum _{j=J+1}^{\infty }\sum _{h=-\infty }^{\infty } R_e^2(h) \hat{\psi }(2\pi h/2^j)\hat{\psi }^{*}(2\pi h/2^j) \\= & {} \sum _{j=J+1}^{\infty }\sum _{h=-\infty }^{\infty } R_e^2(h) |\hat{\psi }(2\pi h/2^j)|^2, \\= & {} \sum _{j=J+1}^{\infty }\sum _{h=-\infty }^{\infty } R_e^2(h)|2\pi h/2^j|^{2q} \frac{|\hat{\psi }(2\pi h/2^j)|^2}{|2\pi h/2^j|^{2q}}, \\= & {} \lim _{z \rightarrow 0} \frac{|\hat{\psi }(z)|^2}{|z|^{2q}}[1+o(1)](2\pi )^{2q} \sum _{j=J+1}^{\infty }\sum _{h=-\infty }^{\infty }|h|^{2q}R_e^2(h)(2^{-2q})^j, \\= & {} \lim _{z \rightarrow 0} \frac{|\hat{\psi }(z)|^2}{|z|^{2q}}[1+o(1)] (2\pi )^{2q} \sum _{j=J+1}^{\infty } (2^{-2q})^j \sum _{h=-\infty }^{\infty } |h|^{2q} R_e^2(h), \\= & {} (2\pi )^{2q+1} \lim _{z \rightarrow 0} \frac{|\hat{\psi }(z)|^2}{|z|^{2q}} [1+o(1)] \frac{2^{-2q(J+1)}}{1-2^{-2q}} \int _{-\pi }^{\pi } [f_e^{(q)}(\omega )]^2 d\omega , \end{aligned}$$

where $f^{(q)}_e(\cdot )$ is defined in Sect. 4.3 and o(1) is uniform in $\omega \in [-\pi ,\pi ].$ It follows that

$$\begin{aligned} \int _{-\pi }^{\pi } B^{2}(\omega )d\omega =2^{-2q(J+1)}\vartheta _{q}\int _{-\pi }^{\pi }\left[ f_e^{(q)}(\omega )\right] ^{2}d\omega +o(2^{-2qJ}). \end{aligned}$$

(55)

It may be show that the term corresponding to $r\ne 0$ is $o(2^{-2qJ})$.

For the second term in (53), we have

$$\begin{aligned} \sum _{j=0}^{J}\sum _{k=1}^{2^{j}} (E\tilde{\alpha }_{jk}-\alpha _{jk})^{2}= & {} \sum _{j=0}^{J}\sum _{k=1}^{2^{j}} \left[ n^{-1}\sum _{h=1-n}^{n-1}|h|R_e(h)\hat{\psi }_{jk} (2\pi h)+\sum _{|h|\ge n}R_e(h)\hat{\psi }_{jk}(2\pi h)\right] ^{2}, \nonumber \\\le & {} 4Cn^{-2}\sum _{h=1}^{n-1}\sum _{m=1}^{n-1}|h m R_e(h)R_e(m) b_{J}(h,m)|, \nonumber \\= & {} O[(J+1)/n^{2}), \end{aligned}$$

(56)

given Lemma 3(vii) and $\sum _{h=-\infty }^{\infty }|hR_e(h)|\le C$ as implied by Assumption 10.

Finally, we consider the variance factor in (52). We write

$$\begin{aligned} E[Q(\tilde{f},E\tilde{f})]= & {} \sum _{h=1-n}^{n-1} \sum _{m=1-n}^{n-1}b_{J}(h,m) {\text {cov}}[\tilde{R}_e(h),\tilde{R}_e(m)], \\= & {} \sum _{h=1-n}^{n-1} \sum _{m=1-n}^{n-1}b_{J}(h,m)n^{-1}\sum _{l} \left[ 1-\frac{\eta (l)+m}{n}\right] \\&\times \left[ R_e(l)R_e(l+m-h)+R_e(l+m)R_e(l-h)+ \kappa (l,h,m-h)\right] , \\\equiv & {} V_{1n}+V_{2n}+V_{3n},\text { say,} \end{aligned}$$

where the function $\eta (l)$ satisfies

$$ \eta (l)\equiv \left\{ \begin{array}{ll} l, &{} \text {if }l>0, \\ 0, &{} \text {if }h-m\le l\le 0, \\ -l+h-m, &{} \text {if }-(n-h)+1\le l\le h-m. \end{array} \right. $$

For more details see [61, p. 326]. Given Assumption 9 and Lemma 3(vii), we have $|V_{2n}|\le C(J+1)n^{-1}$ and $|V_{3n}|\le C(J+1)n^{-1}$. For the first term $V_{1n}$, we can write

$$\begin{aligned} V_{1n}= & {} \sum _{h=1-n}^{n-1} b_{J}(h,h)n^{-1}\sum _{l=-\infty }^{\infty }(1-|l|/n)R_e^{2}(l)+\sum _{h}\sum _{|r|=1}^{n-1}b_{J}(h,h+r)n^{-1}\sum _{l=-\infty }^{\infty }R_e(l)R_e(l+r), \\= & {} n^{-1}(2^{J+1}-1) \sum _{h=-\infty }^{\infty }R_e^{2}(h)+ O[(J+1)/n], \end{aligned}$$

where we have used Lemma 3(v) for the first term, which corresponds to $h=m$; the second term corresponds to $h\ne m$ and it is $O[(J+1)/T]$ given $\sum _{h=-\infty }^{\infty }|R(h)|\le C$ and Lemma 3(v). It follows that as $J\rightarrow \infty$

$$\begin{aligned} E[Q(\tilde{f},E\tilde{f})]= \frac{2^{J+1}}{n} \int _{-\pi }^{\pi }f_e^{2}(\omega )d\omega +o(2^{J}/n). \end{aligned}$$

(57)

Collecting (55)–(57) and $J\rightarrow \infty$, we obtain

$$ E[Q(\hat{f},f)] = \frac{2^{J+1}}{n}\int _{-\pi }^{\pi }f_e^{2}(\omega )d\omega +2^{-2qJ}\vartheta _{q} \int _{-\pi }^{\pi }[f_e^{(q)}(\omega )]^{2}d\omega +o(2^{J}/n+2^{-2qJ}). $$

This shows the Theorem. $\Box$

Proof

(Corollary 2) The result follows immediately from Theorem 6 because Assumption 9 implies $2^{\hat{J}}/2^{J}-1=o_{P}(T^{-1/2(2q+1)})=o_{P}(2^{-J/2})$, where the nonstochastic finest scale J is given by $2^{J+1}\equiv \max \{[2\alpha \vartheta _{q}\zeta _{0}(q)T]^{1/(2q+1)},0\}.$ The latter satisfies the conditions of Theorem 6. $\Box$