Estimation of long-range dependence in gappy Gaussian time series

Abstract

Knowledge of the long-range dependence (LRD) parameter is critical to studies of self-similar behavior. However, statistical estimation of the LRD parameter becomes difficult when the observed data are masked by short-range dependence and other noises or are gappy in nature (i.e., some values are missing in an otherwise regular sampling). Currently there is a lack of theory for spectral- and wavelet-based estimators of the LRD parameter for gappy data. To address this, we estimate the LRD parameter for gappy Gaussian semiparametric time series based upon undecimated wavelet variances. We develop estimation methods by using novel estimators of the wavelet variances, providing asymptotic theory for the joint distribution of the wavelet variances and our estimator of the LRD parameter. We introduce sandwich estimators to compute standard errors for our estimates. We demonstrate the efficacy of our methods using Monte Carlo simulations and provide guidance on practical issues such as how to select the range of wavelet scales. We demonstrate the methodology using two applications: one for gappy Arctic sea-ice draft data and another for gap-free and gappy daily average temperature data collected at 17 locations in south central Sweden.

Proofs

We first need the following lemmas.

Proposition 1

Let $U_{l,l',t}$ and $V_{l,l',t}$ be stationary processes that are independent of each other for any choice of k, $k'$, l and $l'$, and let

$$\begin{aligned} U_{l,l',t}= & {} \theta _{l,l'} + \int _{-1/2}^{1/2} e^{i 2\pi ft} d{{\mathcal {U}}}_{l,l'}(f),\\ V_{l,l',t}= & {} \omega _{l,l'} + \int _{-1/2}^{1/2} e^{i 2\pi ft} d{{\mathcal {V}}}_{l,l'}(f) \end{aligned}$$

be their respective spectral representations. For any $k,k',l$ and $l'$, let $S_{k,k',l,l'}$ and $G_{k,k',l,l'}$ denote the respective cross-spectrum between $U_{k,k',t}$ and $U_{l,l',t}$ and between $V_{k,k',t}$ and $V_{l,l',t}$. For fixed real numbers $\{ a_{l,l'} \}$ define

$$\begin{aligned} Q_t = \sum _{l,l'} a_{l,l'} ( U_{l,l',t} V_{l,l',t} - \theta _{l,l'} \omega _{l,l'}), \end{aligned}$$

Then, $Q_t$ is a second-order stationary process whose spectral density function is given by

$$\begin{aligned} S_Q(f)\equiv & {} \sum _{k,k'} \sum _{l,l'} a_{k,k'} a_{l,l'} \Big [ \theta _{k,k'} \theta _{l,l'} G_{k,k',l,l'}(f) \nonumber \\&+ \,\omega _{k,k'} \omega _{l,l'} S_{k,k',l,l'}(f) + S*G_{k,k',l,l'}(f) \Big ], \end{aligned}$$

(21)

where $S*G_{k,k',l,l'}(f) \equiv \int _{-1/2}^{1/2} G_{k,k',l,l'}(f-f') S_{k,k',l,l'}(f')\,\text {d}f'.$

Proof

The proof is given in the preprint that accompanies Mondal and Percival (2010). $\square$

Proposition 2

Let $X_t$ be a real-valued Gaussian stationary process with zero mean, and SDF $S_X$ that satisfies

$$\begin{aligned} \int _{-1/2}^{1/2} \sin ^4( \pi f) S^2_X(f) \, \text {d}f < \infty . \end{aligned}$$

Let $\eta _t$ be a binary-valued strictly stationary process that is independent of $X_t$ and satisfies assumptions stated in Mondal and Percival (2010). Using $Y_{j,t}$ as defined by Eq. (9), then $Q_t$, defined by

$$\begin{aligned} Q_t = \sum _{j=J_0}^J \gamma _j \Bigl ( Y_{j,t} - \nu _j^2 \Bigr ), \end{aligned}$$

(22)

is a second-order stationary process whose SDF at zero is a strictly positive finite number.

Proof

Note that $Q_t$ in (22) can be written as

$$\begin{aligned} \sum _{l,l' =0}^{L_J-1} a_{l,l'} ( U_{l,l',t} V_{l,l',t} - \theta _{l,l'} \omega _{l,l'}). \end{aligned}$$

where $U_{l,l',t} = 1/2 (X_{t-l} - X_{t-l'})^2$, $V_{l,l',t} = \eta _{t-l} \eta _{t-l'}$, $a_{l,l'} = \sum _{j=J_0}^J \gamma _j h_{j,l} h_{j,l'} \pi _{l-l'}$, $\theta _{l,l'} = \gamma _{X, l-l'}$, and $\omega _{l,l'} = \pi ^{-1}_{l-l'}$. Next, following the argument given in Mondal and Percival (2010), we can show that the bivariate process $\mathbf{U}_t = \left[ 1/2 (X_{t-k} - X_{t-k'})^2, 1/2 (X_{t-l} - X_{t-l'})^2 \right] ^T$, for any choice of $k,k',l$ and $l'$, has a spectral matrix ${{\mathcal {S}}}_\mathbf{U}$ that is continuous. In addition, if $S_{k,k',l,l'}$ is the $(k,k',l,l')$ component of ${{\mathcal {S}}}_\mathbf{U}$, then

$$\begin{aligned} \sum _{k,k'} \sum _{l,l'} h_{j,k} h_{j,k'} h_{j,l} h_{j,l'} S_{k,k',l,l'}(0) > 0. \end{aligned}$$

Now by Lemma 1, the SDF of $Q_t$ is given by the right hand side of Eq. (21), where $G_{k,k',l,l'}$ is the cross-spectrum between $\eta _{t-k} \eta _{t-k'}$ and $\eta _{t-l} \eta _{t-l'}$. Since $a_{l,l'} \omega _{l,l'} = \sum _{j=J_0}^J \gamma _j h_{j,l}h_{j,l'}$,

$$\begin{aligned}&\sum _{k,k'} \sum _{l,l'}a_{k,k'} a_{l,l'}\omega _{k,k'} \omega _{l,l'} S_{k,k',l,l'}(0) \nonumber \\&\quad =\! \sum _{j=J_0}^J \sum _{j'=J_0}^J \sum _{k,k'} \sum _{l,l'} \gamma _j \gamma _{j'} h_{j,k} h_{j,k'} h_{j',l} h_{j',l'} S_{k,k',l,l'}(0)\nonumber \\ \end{aligned}$$

(23)

and after a few lines of algebra, reduces to

$$\begin{aligned} 2 \int _{-1/2}^{1/2} \left( \sum _{j=J_0}^J \gamma _j | H_j(f)|^2 \right) ^2 S_X^2(f) \, \text {d}f \end{aligned}$$

where $| H_j(f)|^2$ is the squared gain function associated with the level j wavelet filter $h_{j,l}$. This integral is a strictly positive finite number. Now $\sum _{k,k'} \sum _{l,l'}a_{k,k'} a_{l,l'} \theta _{k,k'} \theta _{l,l'} G_{k,k',l,l'}(0)$ and $\sum _{k,k'} \sum _{l,l'}a_{k,k'} a_{l,l'} S*G_{k,k',l,l'}(0)$ are nonnegative and finite because $G_{k,k',l,l'}$ and $S_{k,k',l,l'}$ are entries of spectral density matrices. Finally, $\sum _{k,k'} \sum _{l,l'}a_{k,k'} a_{l,l'} \theta _{k,k'} \theta _{l,l'} G_{k,k',l,l'}(0)$ is also nonnegative and finite. This completes the proof. $\square$

Before we prove Theorem 1, the following lemmas are consequences of Brillinger (1981, p. 21) and are proved in Mondal and Percival (2010).

Proposition 3

Assume that $X_t$ satisfies the conditions stated in Theorem 1. Let $U_{p,t} = -\,1/2 (X_{t-l}- X_{t-l'})^2$ and $E U_{p,t}=\theta _p$, in which $p=(l,l')$. Then, for $n \ge 3$ and fixed $p_1, \ldots , p_n$,

$$\begin{aligned} \sum _{t_1,\ldots t_n} | \text{ cum }(U_{p_1,t_1} -\theta _{p_1}, \ldots , U_{p_n,t_n}-\theta _{p_n}) | = o(M^{n/2}), \end{aligned}$$

where each $t_i$ ranges from 0 to $M-1$.

Proposition 4

Let $U_{p,t}$ be either as in Lemma 3, and assume $\kappa _n(p_1, \ldots , p_n, t_1, \ldots , t_n) = \text{ cum }(U_{p_1, t_1}-\theta _{p_1}, \ldots , U_{p_n,t_n}-\theta _{p_n}).$ Define for $i = 1,2,\ldots , n-1$

$$\begin{aligned}&\kappa _n(p_1, \ldots , p_n, t_1, \ldots , t_i)\\= & {} \sum _{t_{i+1}, \ldots , t_n} M^{-1/2 (n-i-1)} \kappa _n(p_1, \ldots , p_n, t_1, \ldots , t_n), \end{aligned}$$

where the summation in $t_j$ ranges from 0 to $M-1$. Then $\kappa _n(p_1, \ldots , p_n, t_1, \ldots , t_i)$ is bounded and satisfies

$$\begin{aligned} {} \sum _{t_1, \ldots , t_i} \kappa _n(p_1, \ldots , p_n, t_1, \ldots , t_i) = o\left( M^{1/2(i+1)}\right) \end{aligned}$$

(24)

for $i =1,2,\ldots ,n$.

Proof

(Proof of Theorem 1) First we will apply the Cramer–Wold theorem to derive the asymptotic normality of the time averages of the multivariate process $(Y_{J_0,t}, Y_{J_0+1,t}, \ldots , Y_{J,t} )^T$. Thus, as in Lemma 2, let $\gamma _j$, $j=J_0, \ldots , J$ be real constants and take the univariate time series formed by the linear combination, i.e.,

$$\begin{aligned} Q_t= & {} \sum _{j=J_0}^J \gamma _j \Bigl ( Y_{j,t} - \nu _j^2 \Bigr )\\= & {} \sum _{l,l' =0}^{L_J-1} a_{l,l'} ( U_{l,l',t} V_{l,l',t} - \theta _{l,l'} \omega _{l,l'}), \end{aligned}$$

where $U_{l,l',t} = 1/2 (X_{t-l} - X_{t-l'})^2$, $V_{l,l',t} = \eta _{t-l} \eta _{t-l'}$, $a_{l,l'} = \sum _{j=J_0}^J \gamma _j h_{j,l} h_{j,l'} \pi _{l-l'}$, $\theta _{l,l'} = \gamma _{X, l-l'}$, and $\omega _{l,l'} = \pi ^{-1}_{l-l'}$. We first prove a central limit theorem for $R = M^{-1/2} \sum _{t=0}^{M-1} Q_t$. Using Zurbenko (1986, p. 2), to write the log of the characteristic function of R as

$$\begin{aligned} \log F(z) = \sum _{n=1}^\infty \frac{i^n z^n}{n!} \sum _{t_1,\ldots ,t_n} \frac{B_n (t_1, \ldots , t_n)}{M^{n/2}}, \end{aligned}$$

where $B_n$ is the nth-order cumulant of $Q_t$, and each $t_i$ ranges from 0 to $M-1$. Since $Q_t$ is centered, $B_1 (t_1)=0$. By Lemma 2, $Q_t$ is stationary, the autocovariances $s_{Q,\tau }$ of $Q_t$ are absolutely summable and $M^{-1} \sum _{t_1}\sum _{t_2} B_2 (t_1,t_2) \rightarrow \sum _\tau s_{Q,\tau } = S_Q (0) > 0$. In order to prove the CLT for R, it suffices to show that $\sum _{t_1,\ldots ,t_n} M^{-n/2} B_n (t_1, \ldots , t_n) \rightarrow 0$ for $n=3,4,\ldots$.

Now using Brillinger (1981, p. 19), we break up the nth-order cumulant as follows:

$$\begin{aligned}&B_n (t_1, \ldots , t_n)\\&\quad = \sum _{p_1} \cdots \sum _{p_n} a_{p_1} \cdots a_{p_n} \text{ cum }( U_{p_1,t_1} V_{p_1,t_1}-\theta _{p_1} \omega _{p_1}, \ldots ,\\&\qquad U_{p_n,t_n} V_{p_n,t_n}-\theta _{p_n} \omega _{p_n} ). \end{aligned}$$

But, applying the argument given in Mondal and Percival (2010), it follows that $\text{ cum }( U_{p_1,t_1} V_{p_1,t_1}-\theta _{p_1} \omega _{p_1}, \ldots , U_{p_n,t_n} V_{p_n,t_n}-\theta _{p_n} \omega _{p_n} ) = o(M^{n/2}).$

Note that $\widehat{\nu }^2_j - \nu ^2_j$ is the average of $Q_t$ over $L_j-1 \le t\le N-1$ with $\eta _{l-l'}$ replaced by its consistent estimate ${\hat{\eta }}_{l,l'}$. Then, Slutsky’s theorem is invoked to complete the proof that $\widehat{\nu }^2_j, j=J_0, \ldots J$ is jointly asymptotically normal. $\square$

Proof

(Proof of Theorem 2) Note that $\varvec{a}$ depends on J. We have $\varvec{a}^T \varvec{1} = 0$ and $\varvec{a}^T \varvec{w} = 1.$ Then, since $K = J - J_0 + 1$ is finite and $\varvec{a}$ is normalized, as $J \rightarrow \infty$, $\varvec{a}^T = O(\varvec{1}^T)$; that is, for each i, $a_i = O(1)$.

Letting $\varvec{\xi } = (\log \nu _j^2: j=J_0,\ldots ,J)$, we have

$$\begin{aligned}&N_J^{1/2} ( \widehat{\delta } - \delta ) / \sqrt{\kappa }\\&\quad = N_J^{1/2} ( 1/2 + \varvec{a}^T \varvec{z} - \delta ) / \sqrt{\kappa }\\&\quad = N_J^{1/2} \varvec{a}^T ( \varvec{z} - \varvec{\xi } ) / \sqrt{\kappa } + N_J^{1/2} ( \varvec{a}^T \varvec{\xi } - (\delta - 1/2)) / \sqrt{\kappa }\\&\quad = N_J^{1/2} \varvec{a}^T ( \varvec{z} - \varvec{\xi } ) / \sqrt{\kappa } +\\&N_J^{1/2} \varvec{a}^T \left\{ \varvec{\xi } - \varvec{w} (\delta - 1/2) - \varvec{1} \log (S_0(0) K_{\delta }) \right\} / \sqrt{\kappa }. \end{aligned}$$

An immediate consequence of Eqs. (2) and (13) is that the second term in the above equation goes to zero. The asymptotic normality of the first term holds by Theorem 1 and (14) in conjunction with the applications of the delta method and Slustky’s theorem.

To see that condition (14) holds in the gap-free case, note that $\sigma _{j,j}$ is equal to

$$\begin{aligned}&2 \int _{-1/2}^{1/2} | H_j(f)|^4 S_X^2(f) \, \text {d}f\\&\quad = 2 S_0^2(0) K_{1, \delta } 2^{j (4\delta -1)} + \text{ a } \text{ small } \text{ term }, \end{aligned}$$

where $K_{1,\delta } = \int |f|^{-4 \delta } | \varPsi (f)|^4 \, \text {d}f.$ This implies that

$$\begin{aligned} \sigma _{j,j}/\nu _j^4= & {} c_1 2^{j} + \text{ a } \text{ small } \text{ term }, \end{aligned}$$

for some constant $c_1$. $\square$