A novel copula-based approach for parametric estimation of univariate time series through its covariance decay

Abstract

In this note we develop a new technique for parameter estimation of univariate time series by means of a parametric copula approach. The proposed methodology is based on a relationship between a process’ covariance decay and parametric bivariate copulas associated to lagged variables. This relationship provides a way for estimating parameters that are identifiable through the process’ covariance decay, such as in long range dependent processes. We provide a rigorous asymptotic theory for the proposed estimator. We also present a Monte Carlo simulation study to asses the finite sample performance of the proposed estimator.

Appendix 1: Mathematical proofs

Proof of Theorem 2.1:

We present the proof for the case where $\varvec{a}\notin \textrm{int}(\Theta )$. The other cases are dealt analogously. Let $\{\varvec{\alpha }_m\}_{m\in \mathbb {N}^*}$ be an arbitrary sequence of parameters in D such that $\varvec{\alpha }_{m}\rightarrow \varvec{a}$ (assuming the adequate lateral limit when necessary, allowing for s coordinates to remain fixed). Applying a second order Taylor expansion in $\varvec{\theta }_n$ around $\varvec{a}$, apart from an $o(\Vert \varvec{\theta }_n -\varvec{a}\Vert ^2)$ remainder, we have

$$\begin{aligned} C_{\varvec{\theta }_n}(u,v)&=\lim _{m\rightarrow \infty }\!C_{\varvec{\alpha }_m}\!(u,v)\!+\!\lim _{m\rightarrow \infty }\!\!d_{\varvec{\theta }}C_{\varvec{\alpha }_m}\!(u,v)(\varvec{\theta }_n \!-\!\varvec{a})\!+\!\frac{1}{2}(\varvec{\theta }_n \!-\!\varvec{a})'\lim _{m\rightarrow \infty }\!\!d^2_{\varvec{\theta }}C_{\varvec{\alpha }_m}\!(u,v)(\varvec{\theta }_n \!-\!\varvec{a})\nonumber \\&=uv+\sum _{i=1}^k\lim _{m\rightarrow \infty }\bigg [\frac{\partial C_{\varvec{\theta }}(u,v)}{\partial \theta _i}\bigg |_{\varvec{\theta }=\varvec{\alpha }_m}\bigg ] (\theta _n^{(i)}-a_i)+ \nonumber \\&\quad +\frac{1}{2}\sum _{i,j=1}^k\lim _{m\rightarrow \infty }\!\bigg [\frac{\partial ^2 C_{\varvec{\theta }}(u,v)}{\partial \theta _i\partial \theta _j}\bigg |_{\varvec{\theta }=\varvec{\alpha }_m} \bigg ](\theta _n^{(i)}-a_i)(\theta _n^{(j)}-a_j). \end{aligned}$$

(7)

Let $\{X_n\}_{n\in \mathbb {N}}$ and $\{F_n\}_{n\in \mathbb {N}}$ be as in the enunciate. Hoeffding’s lemma combined with (7) yields

$$\begin{aligned}&\mathop {\mathrm {\textrm{Cov}}}\limits (X_0,X_n)=\sum _{i=1}^k\left[ \iint _{I^2}\frac{1}{l_0(u)l_n(v)}\lim _{m\rightarrow \infty }\frac{\partial C_{\varvec{\theta }}(u,v)}{\partial \theta _i}\bigg |_{\varvec{\theta }=\varvec{\alpha }_m}dudv\right] (\theta _n^{(i)}-a_i)+\nonumber \\&\qquad +\frac{1}{2}\sum _{i,j=1}^k\bigg [\iint _{I^2}\frac{1}{l_0(u)l_n(v)}\lim _{m\rightarrow \infty }\frac{\partial ^2 C_{\varvec{\theta }}(u,v)}{\partial \theta _i\partial \theta _j}\bigg |_{\varvec{\theta }=\varvec{\alpha }_m} dudv\bigg ]\times (\theta _n^{(i)}-a_i)(\theta _n^{(j)}-a_j)\nonumber \\&\quad =\sum _{i=1}^kK_1^{(i)}(n)(\theta _n^{(i)}-a_i)+\frac{1}{2}\sum _{i,j=1}^kK_2^{(i,j)}(n)(\theta _n^{(i)}-a_i)(\theta _n^{(j)}-a_j) \sim R(n)+o\big (R(n)\big )\sim R(n), \end{aligned}$$

by the hypothesis on $K_1^{(i)}$ and $K_2^{(i,j)}$. $\blacksquare$

Proof of Lemma 4.1:

Under Framework A, there exists $M>0$ such that $\Big |\lim _{\theta \rightarrow a}\frac{\partial C_{\theta }(u,v)}{\partial \theta }\Big |\le M$ for all $u,v\in I$, since it is a continuous function defined on a compact set. Now, since

$$\begin{aligned}&|{\hat{K}}_1-K_1|\nonumber \\&\quad \le \iint _{I^2}\bigg |\frac{1}{{\hat{F}}_n'\big ({\hat{F}}_n^{(-1)}(u)\big ){\hat{F}}_n'\big ({\hat{F}}_n^{(-1)}(v)\big )}-\frac{1}{F'\big (F^{(-1)}(u)\big )F'\big ( F^{(-1)}(v)\big )}\bigg |\bigg |\lim _{\theta \rightarrow a}\frac{\partial C_{\theta }(u,v)}{\partial \theta }\bigg |\,dudv,\nonumber \\&\quad \le M \iint _{I^2}\bigg |\frac{1}{{\hat{F}}_n'\big ({\hat{F}}_n^{(-1)}(u)\big ){\hat{F}}_n'\big ({\hat{F}}_n^{(-1)}(v)\big )}-\frac{1}{F'\big (F^{(-1)}(u)\big )F'\big ( F^{(-1)}(v)\big )}\bigg |\,dudv, \end{aligned}$$

(8)

given $\varepsilon >0$, it follows from (8) that

$$\begin{aligned}&\mathbb {P}\big (|{\hat{K}}_1-K_1|>\varepsilon \big )\le \\&\quad \le \mathbb {P}\bigg (\iint _{I^2}\bigg |\frac{1}{{\hat{F}}_n'\big ({\hat{F}}_n^{(-1)}(u)\big ){\hat{F}}_n'\big ({\hat{F}}_n^{(-1)}(v)\big )}-\frac{1}{F'\big (F^{(-1)}(u)\big )F'\big (F^{(-1)}(v)\big )}\bigg |\,dudv>\frac{\varepsilon }{M}\bigg )\rightarrow 0, \end{aligned}$$

as $n\rightarrow \infty$, by condition (a). Hence ${\hat{K}}_1\overset{\mathbb {P}}{\longrightarrow }K_1$ as desired. To complete the proof, observe that (b) $\Rightarrow$ (a) trivially. $\blacksquare$

Proof of Theorem 4.1:

Under A0, ${\hat{K}}_1 \overset{\mathbb {P}}{\longrightarrow }\ K_1$, while under A1, ${\hat{\theta }}_k(n)-a\overset{\mathbb {P}}{\longrightarrow }\theta _k^0-a\in \mathbb {R}$, for all $s \le k\le m$, as n tends to infinity, so that $\widehat{\varvec{L}}_{s,m}(n)\overset{\mathbb {P}}{\longrightarrow }\varvec{R}_{s,m}({\varvec{\eta }}_0)$. Now, by assumption A2 ${\mathscr {D}}$ is equivalent to the usual metric in ${\mathbb {R}^{m-s+1}}$ and since $({\mathbb {R}^{m-s+1}},{\mathscr {D}})$ is a complete metric space, it follows that,

$$\begin{aligned} {\mathscr {D}}\big (\widehat{\varvec{L}}_{s,m}(n),\varvec{R}_{s,m}({\varvec{\eta }}_0)\big )\overset{\mathbb {P}}{\longrightarrow }0, \end{aligned}$$

(9)

as n tends to infinity. Let $\hat{\varvec{\eta }}_{s,m}(n)$ be as in (3) and notice that, for sufficiently large n,

$$\begin{aligned} {\mathscr {D}}\big ({\varvec{R}}_{s,m}(\varvec{\eta }_0),{\varvec{R}}_{s,m}(\hat{\varvec{\eta }}_{s,m}(n))\big )&<{\mathscr {D}}\big ({\varvec{R}}_{s,m}(\varvec{\eta }_0),\widehat{\varvec{L}}_{s,m}(n)\big )+{\mathscr {D}}\big (\widehat{\varvec{L}}_{s,m}(n),{\varvec{R}}_{s,m}(\hat{\varvec{\eta }}_{s,m}(n))\big )\nonumber \\&<\varepsilon +{\mathscr {D}}\big (\widehat{\varvec{L}}_{s,m}(n),{\varvec{R}}_{s,m}(\hat{\varvec{\eta }}_{s,m}(n))\big ), \end{aligned}$$

(10)

hence $\lim _{n\rightarrow \infty }{\mathscr {D}}\big ({\varvec{R}}_{s,m}(\varvec{\eta }_0),{\varvec{R}}_{s,m}(\hat{\varvec{\eta }}_{s,m}(n))\big )\le \lim _{n\rightarrow \infty }{\mathscr {D}}\big (\widehat{\varvec{L}}_{s,m}(n),{\varvec{R}}_{s,m}(\hat{\varvec{\eta }}_{s,m}(n))\big )$. By the definition of $\hat{\varvec{\eta }}_{s,m}(n)$, given $\delta >0$, ${\mathscr {D}}\big (\widehat{\varvec{L}}_{s,m}(n),{\varvec{R}}_{s,m}(\hat{\varvec{\eta }}_{s,m}(n))\big )\le {\mathscr {D}}\big (\widehat{\varvec{L}}_{s,m}(n),{\varvec{R}}_{s,m}({\varvec{\eta }})\big )$, for all ${\varvec{\eta }}\in \overline{B_\delta (\hat{\varvec{\eta }}_{s,m}(n))}$, the closed ball in $\mathbb {R}^p$ with radius $\delta$ centered at $\hat{\varvec{\eta }}_{s,m}(n)$. Now, by (9), it follows that for sufficiently large n, ${\varvec{\eta }}_0\in \overline{B_\delta (\hat{\varvec{\eta }}_{s,m}(n))}$, so that

$$\begin{aligned} {{\mathscr {D}}\big (\widehat{\varvec{L}}_{s,m}(n),{\varvec{R}_{s,m}}(\hat{\varvec{\eta }}_{s,m}(n))\big )\le {\mathscr {D}}\big (\widehat{\varvec{L}}_{s,m}(n),{\varvec{R}_{s,m}}({\varvec{\eta }}_0)\big )\overset{\mathbb {P}}{\longrightarrow }0.} \end{aligned}$$

Now, since ${\mathscr {D}}$ is a metric in $\mathbb {R}^{m-s+1}$, by the continuity of R and by the identifiability of ${\varvec{\eta }}_0$, it follows that $\mathbb {P}\big (\Vert \hat{\varvec{\eta }}_{s,m}(n)-{\varvec{\eta }}_0\Vert <\varepsilon \big )\longrightarrow 1$, as n tends to infinity. $\blacksquare$

Proof of Theorem 4.2:

Without loss of generality, we shall assume that $a=0$. Let $s_0=\max \{k_0,k_1\}$ and $\Omega =\Omega _0\cap \Omega _1$ in A3 and A4 and let $s>s_0$. Under the hypothesis, upon defining

$$\begin{aligned} {S_{s,m}(\varvec{\eta };n)=\sum _{k=0}^{m-s}\big (\widehat{\theta }_{s+k}(n)-L(s+k,\varvec{\eta })\big )^2,} \end{aligned}$$

as n tends to infinity, with probability 1

$$\begin{aligned} {\varvec{0}=\frac{\partial S_{s,m}(\varvec{\eta };n)}{\partial \varvec{\eta }} \bigg |_{{\widehat{\varvec{\eta }}}}=\frac{\partial S_{s,m}(\varvec{\eta };n)}{\partial \varvec{\eta }}\bigg |_{\varvec{\eta }_0} + \bigg (\frac{\partial ^2 S_{s,m}(\varvec{\eta };n)}{\partial \varvec{\eta }\partial \varvec{\eta }'}\bigg |_{\overline{\varvec{\eta }}}\bigg )(\widehat{\varvec{\eta }}-\varvec{\eta }_0),} \end{aligned}$$

for some $\overline{\varvec{\eta }}\in \Omega$ such that $\Vert \overline{\varvec{\eta }}-\varvec{\eta }_0\Vert \le \Vert \widehat{\varvec{\eta }}-\varvec{\eta }_0\Vert$. In order to prove the result, it suffices to show that

$$\begin{aligned} b_n\bigg (\frac{\partial S_{s,m}(\varvec{\eta };n)}{\partial \varvec{\eta }}\bigg |_{\varvec{\eta }_0}\bigg ){\mathop {\longrightarrow }\limits ^{d}}-2\sum _{k=0}^{m-s}\varvec{a}_{s+k} Z_{s+k} \quad \text{ and } \quad \frac{\partial ^2 S_{s,m}(\varvec{\eta };n)}{\partial \varvec{\eta }\partial \varvec{\eta }'}\bigg |_{\overline{\varvec{\eta }}}{\mathop {\longrightarrow }\limits ^{\mathbb {P}}}2\sum _{k=0}^{m-s} \varvec{a}_{s+k}\varvec{a}_{s+k}^\prime \end{aligned}$$

(11)

and to observe that the right hand side in the second relation in (11) is positive definite by A3. On one hand, by A3, we can write

$$\begin{aligned} {\frac{\partial S_{s,m}(\varvec{\eta };n)}{\partial \varvec{\eta }} = -2\sum _{k=0}^{m-s}\varvec{a}_{s+k}\big (\widehat{\theta }_{s+k}(n)-L(s+k,\varvec{\eta })\big ),} \end{aligned}$$

so that the first equation in (11) follows from A4 by multiplying both sides by $b_{n}$ and taking the limit as n tends to infinity. On the other hand, by A3 and A4, for $\varvec{\eta }\in \Omega$,

$$\begin{aligned} \frac{\partial ^2 S_{s,m}(\varvec{\eta };n)}{\partial \varvec{\eta }\partial \varvec{\eta }'}&=-2\sum _{k=0}^{m-s}\bigg \{\frac{\partial ^2 L(s+k,\varvec{\eta })}{\partial \varvec{\eta }\partial \varvec{\eta }'}\big (\widehat{\theta }_{s+k}(n)-L(s+k,\varvec{\eta })\big )-\varvec{a}_{s+k}\varvec{a}_{s+k}^\prime \bigg \}\nonumber \\&=2\sum _{k=0}^{m-s} \varvec{a}_{s+k}\varvec{a}_{s+k}^\prime +o_P(1), \end{aligned}$$

and the proof is complete.$\blacksquare$