A dimension reduction factor approach for multivariate time series with long-memory: a robust alternative method

Abstract

This paper studies factor modeling for a vector of time series with long-memory properties to investigate how outliers affect the identification of the number of factors and also proposes a robust method to reduce their impact. The number of factors is estimated using an eigenvalue analysis for a non-negative definite matrix introduced by Lam et al. (2011). Two estimators are proposed; the first is based on the classical sample covariance function, and the second uses a robust covariance function estimate. In both cases, it is shown that the eigenvalues estimates have similar convergence rates. Empirical simulations support both estimators for multivariate stationary long-memory time series and show that the robust method is preferable when the data is contaminated with additive outliers. Time series of daily log returns are used as an example of application. In addition to abrupt observations, exchange rates exhibit non-stationarity properties with long memory parameters greater than one. Then we use semi-parametric long memory estimators to estimate the fractional parameters of the series. The number of factors was estimated using the classical and robust approaches. Due to the influence of the abrupt observations, these tools suggested a different number of factors to model the data. The robust method suggested two factors, while the classical approach indicated only one factor.

6 Appendix: Technical lemmas

Lemmas 1 and 2 were stated and proved in Reisen et al. (2019) but are recalled here for the reader convenience.

Lemma 1

Let $\widehat{\varvec{A}}_n$ be a sequence of $p\times p$ symmetric matrices and $\varvec{A}$ a $p\times p$ symmetric matrix such that $u_n(\widehat{\varvec{A}}_n-\varvec{A})=O_p(1)$, where $u_n$ is a sequence of positive numbers tending to infinity as n tends to infinity, then

$$\begin{aligned} u_n \sup _{1\le j\le p} |\lambda _j(\widehat{\varvec{A}})-\lambda _j(\varvec{A})|=O_p(1),\; \text { as } n\rightarrow \infty , \end{aligned}$$

where $(\lambda _j(\widehat{\varvec{A}}))_{1\le j\le p}$ and $(\lambda _j(\varvec{A}))_{1\le j\le p}$ are the eigenvalues of $\widehat{\varvec{A}}_n$ and A, respectively.

Lemma 2

Let $\widehat{\varvec{A}}_n(h)$ be a sequence of $p\times p$ symmetric matrices and A(h) a $p\times p$ symmetric matrix such that $u_n(\widehat{\varvec{A}}_n(h)-\varvec{A}(h))=O_p(1)$, for each fixed $h\in \{1,\ldots ,h_{max}\}$, where $u_n$ is a sequence of positive numbers tending to infinity as n tends to infinity, then

$$\begin{aligned} u_n \left( \sum _{h=1}^{h_{max}}\widehat{\varvec{A}}_n(h)\widehat{\varvec{A}}_n(h)'-\sum _{h=1}^{h_{max}}\varvec{A}(h)\varvec{A}(h)'\right) =O_p(1), \end{aligned}$$

as n tends to infinity.

Lemma 3

Let h be a non negative integer and i and j two integers in $\{1,\ldots ,k\}$. Assume that (2) holds, then the autocovariance estimator ${\widehat{\gamma }}^{Q}_{i,j}(h)$ defined in (10) satisfies the following limit theorems as n tends to infinity.

• (i) If, for all i in $\{1,\ldots ,k\}$, $D_i>1/2$,

  $$\begin{aligned} \sqrt{n}({\widehat{\gamma }}^{Q}_{i,j}(h)-\gamma _{ij}(h)){\mathop {\longrightarrow }\limits ^{d}}{\mathcal {N}}(0,{\widetilde{\sigma }}_{i,j}^2(h)),\text { as } n\rightarrow \infty , \end{aligned}$$

  where

  $$\begin{aligned} {\widetilde{\sigma }}_{i,j}^2(h)=[\psi (Y_{i,1},Y_{j,1+h})^2]+2\sum _{k\ge 1}{\mathbb {E}}[\psi (Y_{i,1},Y_{j,1+h})\psi (Y_{i,k+1},Y_{j,k+1+h})], \end{aligned}$$

  where $\psi$ is

  $$\begin{aligned} \psi (x,y)=\frac{1}{2}\left( \gamma _{i,i}(0)+\gamma _{j,j}(0)+2\gamma _{i,j}(h)\right) \nonumber \\ \text {IF}\left( \frac{x+y}{\sqrt{\gamma _{i,i}(0)+\gamma _{j,j}(0)+2\gamma _{i,j}(h)}},Q,\Phi \right) \nonumber \\ -\frac{1}{2}\left( \gamma _{i,i}(0)+\gamma _{j,j}(0)-2\gamma _{i,j}(h)\right) \nonumber \\ \text {IF}\left( \frac{x-y}{\sqrt{\gamma _{i,i}(0)+\gamma _{j,j}(0)-2\gamma _{i,j}(h)}},Q,\Phi \right) , \end{aligned}$$

  (12)

  and $\text {IF}$ is defined in Lévy-Leduc et al. (2011b, Equation (20)).

• (ii) If, there exists $i_0$ in $\{1,\ldots ,k\}$ such that $D_{i_0}<1/2$,

  $$\begin{aligned} n^{D_{i_0}\wedge D_j}({\widehat{\gamma }}^{Q}_{i_0,j}(h)-\gamma _{i_0,j}(h))=O_P(1),\text { as } n\rightarrow \infty . \end{aligned}$$

Proof of Lemma 3

Observe that the autocovariance $\gamma _{i,j}^{(+)}(\ell )$ of the process $(Y_{i,t}+Y_{j,t+h})_{t\ge 1}$ is equal to

$$\begin{aligned} \gamma _{i,j}^{(+)}(\ell ){} & {} ={{\,\textrm{Cov}\,}}(Y_{i,t}+Y_{j,t+h},Y_{i,t+\ell }+Y_{j,t+h+\ell })\\{} & {} =\gamma _{i,i}(\ell )+\gamma _{i,j}(h+\ell )+\gamma _{i,j}(\ell -h)+\gamma _{j,j}(\ell ). \end{aligned}$$

By (2) and by using a Taylor expansion, $\gamma _{i,j}^{(+)}(\ell )$ is proportional to $\ell ^{D_i\wedge D_j}$. Hence, the process $(Y_{i,t}+Y_{j,t+h})_{t\ge 1}$ satisfies (Lévy-Leduc et al. 2011b, Assumption A2) with $D=D_i\wedge D_j$. Since the autocovariance $\gamma _{i,j}^{(-)}(\ell )$ of the process $(Y_{i,t}-Y_{j,t+h})_{t\ge 1}$ is equal to

$$\begin{aligned} \gamma _{i,j}^{(-)}(\ell ){} & {} ={{\,\textrm{Cov}\,}}(Y_{i,t}-Y_{j,t+h},Y_{i,t+\ell }-Y_{j,t+h+\ell })\\{} & {} =\gamma _{i,i}(\ell )-\gamma _{i,j}(h+\ell )-\gamma _{i,j}(\ell -h)+\gamma _{j,j}(\ell ), \end{aligned}$$

by following the same lines, the process $(Y_{i,t}-Y_{j,t+h})_{t\ge 1}$ also satisfies (Lévy-Leduc et al. 2011b, Assumption A2) with $D=D_i\wedge D_j$. In the case (i), the proof follows the same lines as the ones of the proof of (i) in Lévy-Leduc et al. (2011b, Theorem 4). In the case (ii), by applying the Delta method to Lévy-Leduc et al. (2011b, Equation (74)), we get

$$\begin{aligned} n^{D_{i_0}\wedge D_j}\left( Q_{n-h}(Y_{i_0,1:n-h}+Y_{j,h+1:n})^2-{{\,\textrm{Var}\,}}(Y_{i_0,t}+Y_{j,t+h})\right) =O_p(1). \end{aligned}$$

Similarly, we have that

$$\begin{aligned} n^{D_{i_0}\wedge D_j}\left( Q_{n-h}(Y_{i_0,1:n-h}-Y_{j,h+1:n})^2-{{\,\textrm{Var}\,}}(Y_{i_0,t}-Y_{j,t+h})\right) =O_p(1), \end{aligned}$$

which gives the result. $\square$

Lemma 4

Let h be a non negative integer and i and j two integers in $\{1,\ldots ,k\}$. Assume that (2) holds, then the autocovariance estimator ${\widehat{\gamma }}_{i,j}(h)$ defined in (8) satisfies the following limit theorems as n tends to infinity.

• (i) If, for all i in $\{1,\ldots ,k\}$, $D_i>1/2$,

  $$\begin{aligned} \sqrt{n}({\widehat{\gamma }}_{i,j}(h)-\gamma _{ij}(h)){\mathop {\longrightarrow }\limits ^{d}}{\mathcal {N}}(0,{\check{\sigma }}_{i,j}^2(h)),\text { as } n\rightarrow \infty , \end{aligned}$$

  where

  $$\begin{aligned} {\check{\sigma }}_{i,j}^2(h)&={\mathbb {E}}\left[ \left( Y_{i,1}Y_{j,h+1}-\gamma _{i,j}(h)\right) ^2\right] \\&\quad +2\sum _{k\ge 1}{\mathbb {E}}\left[ \left( Y_{i,1}Y_{j,h+1}-\gamma _{i,j}(h)\right) \left( Y_{i,1+k}Y_{j,1+h+k}-\gamma _{i,j}(h)\right) \right] . \end{aligned}$$

• (ii) If, there exists $i_0$ in $\{1,\ldots ,k\}$ such that $D_{i_0}<1/2$,

  $$\begin{aligned} n^{D_{i_0}\wedge D_j}({\widehat{\gamma }}_{i_0,j}(h)-\gamma _{i_0,j}(h))=O_P(1),\text { as } n\rightarrow \infty . \end{aligned}$$

Proof of Lemma 4

• (i) Note that

  $$\begin{aligned} {\widehat{\gamma }}_{i,j}(h)=\frac{1}{n}\sum _{t=1}^{n-h} Y_{i,t}Y_{j,t+h} -{\bar{Y}}_i{\bar{Y}}_j. \end{aligned}$$

  By Theorem 5.1 of Taqqu (1975),

  $$\begin{aligned} {\bar{Y}}_i=O_P(n^{-D_i/2}) \text { and } {\bar{Y}}_j=O_P(n^{-D_j/2}). \end{aligned}$$

  Let ${\mathcal {Y}}_t=(Y_{i,t},Y_{j,t+h})$ and $f:(x,y)\mapsto xy$ then, by Theorem 4 of Arcones (1994), we get that

  $$\begin{aligned} \frac{1}{\sqrt{n}}\sum _{t=1}^{n-h} \left( f({\mathcal {Y}}_t)-{\mathbb {E}}(f({\mathcal {Y}}_t))\right) =\frac{1}{\sqrt{n}}\sum _{t=1}^{n-h} \left( Y_{i,t}Y_{j,t+h}-\gamma _{ij}(h)\right) {\mathop {\longrightarrow }\limits ^{d}}{\mathcal {N}}(0,{\check{\sigma }}_{i,j}^2(h)) \end{aligned}$$

  (13)

  since f is of Hermite rank 2, $r^{(1,2)}(h)={\mathbb {E}}[Y_{i,t} Y_{j,t+h}]=\gamma _{i,j}(h)$, $r^{(1,1)}(h)={\mathbb {E}}[Y_{i,t} Y_{i,t+h}]=\gamma _{i,i}(h)$, $r^{(2,2)}(h)={\mathbb {E}}[Y_{j,t} Y_{j,t+h}]=\gamma _{j,j}(h)$ and $D_i>1/2$, for all i. In (13),

  $$\begin{aligned} {\check{\sigma }}_{i,j}^2(h)={\mathbb {E}}\left[ \left( Y_{i,1}Y_{j,h+1}-\gamma _{i,j}(h)\right) ^2\right] \\+2\sum _{k\ge 1}{\mathbb {E}}\left[ \left( Y_{i,1}Y_{j,h+1}-\gamma _{i,j}(h)\right) \left( Y_{i,1+k}Y_{j,1+h+k}-\gamma _{i,j}(h)\right) \right] . \end{aligned}$$

• (ii) Observe that

  $$\begin{aligned} {\widehat{\gamma }}_{i,j}(h)=\frac{1}{4}\left( {{\widehat{\sigma }}_{n-h,Y_{i,1:n-h} +Y_{j,h+1:n}}}^2-{{\widehat{\sigma }}_{n-h,Y_{i,1:n-h}-Y_{j,h+1:n}}}^2\right) \end{aligned}$$

  (14)

  where

  $$\begin{aligned} {{\widehat{\sigma }}_{n-h,Y_{i,1:n-h}+Y_{j,h+1:n}}}^2=:{\widehat{\sigma }}_{+,i,j}^2 =\frac{1}{n}\sum _{t=1}^{n-h}(Y_{i,t}+Y_{j,t+h})^2-({\bar{Y}}_i+{\bar{Y}}_j)^2 \\ \text { with } {\bar{Y}}_i=\frac{1}{n}\sum _{t=1}^nY_{i,t} \text { and } {\bar{Y}}_j=\frac{1}{n}\sum _{t=1}^nY_{j,t} \end{aligned}$$

  and

  $$\begin{aligned} {{\widehat{\sigma }}_{n-h,Y_{i,1:n-h}-Y_{j,h+1:n}}}^2=:{\widehat{\sigma }}_{-,i,j}^2 =\frac{1}{n}\sum _{t=1}^{n-h}(Y_{i,t}-Y_{j,t+h})^2-({\bar{Y}}_i-{\bar{Y}}_j)^2. \end{aligned}$$

  By using the same arguments as those used in the proof of Lemma 3, we get that $(Y_{i,t}+Y_{j,t+h})_{t\ge 1}$ and $(Y_{i,t}-Y_{j,t+h})_{t\ge 1}$ satisfy (Lévy-Leduc et al. 2011b, Assumption A2) with $D=D_i\wedge D_j$. Since $D_{i_0}<1/2$, $D_{i_0}\wedge D_j<1/2$ for all j in $\{1,\ldots ,k\}$. Thus, $(Y_{i_0,t}+Y_{j,t+h})_{t\ge 1}$ and $(Y_{i_0,t}-Y_{j,t+h})_{t\ge 1}$ satisfy Assumption (A2) with $D<1/2$. Hence, by Lévy-Leduc et al. (2011b, Proposition 3(b)) and the Delta method,

  $$\begin{aligned} n^{D_{i_0}\wedge D_j}\left( {\widehat{\sigma }}_{+,i_0,j}^2-{{\,\textrm{Var}\,}}(Y_{i_0,t}+Y_{j,t+h})\right) =O_p(1),\text { as } n\rightarrow \infty \end{aligned}$$

  and

  $$\begin{aligned} n^{D_{i_0}\wedge D_j}\left( {\widehat{\sigma }}_{-,i_0,j}^2-{{\,\textrm{Var}\,}}(Y_{i_0,t}-Y_{j,t+h})\right) =O_p(1),\text { as } n\rightarrow \infty , \end{aligned}$$

  which concludes the proof by (14).