Robust test for structural instability in dynamic factor models

Abstract

In this paper, we consider a robust test for structural breaks in dynamic factor models. The proposed framework considers structural changes when the underlying high-dimensional time series is contaminated by outlying observations, which are often observed in many real applications such as fMRI, economics and finance. We propose a test based on the robust estimation of a vector autoregressive model for principal component factors using minimum density power divergence. The simulations study shows excellent finite sample performance, higher powers while achieving good sizes in all cases considered. Our method is illustrated to the resting state fMRI series to detect brain connectivity changes.

Appendix

Proof of Lemma 1

Under (A1), there exist a constant mean vector $\mu$ and a covariance matrix $\varGamma _0$ of $Y_t$ such that $\mu =E(Y_t)$ and $\varGamma _0=E(Y_tY_t^T)-\mu \mu ^T$ for all t, which implies $E(Y_{t,i}^2)<\infty$ for all t and $i=1,\ldots ,r$. Hence, by Lemma S1 of the Supplementary Material and Cauchy–Schwartz inequality, we obtain

$$\begin{aligned} E\left( \sup _{\theta \in \varTheta }\left| \frac{\partial ^2h_{\alpha ,t}(\theta )}{\partial \theta _i\partial \theta _j}\right| \right) <\infty \end{aligned}$$

and

$$\begin{aligned} E\left( \sup _{\theta \in \varTheta }\left| \frac{\partial h_{\alpha ,t}(\theta )}{\partial \theta _i}\frac{\partial h_{\alpha ,t}(\theta )}{\partial \theta _j}\right| \right) \le E\left( \sup _{\theta \in \varTheta }\left| \frac{\partial h_{\alpha ,t}(\theta )}{\partial \theta _i}\right| \sup _{\theta \in \varTheta }\left| \frac{\partial h_{\alpha ,t}(\theta )}{\partial \theta _j}\right| \right) <\infty \end{aligned}$$

for $i,j=1,\ldots ,\eta$. Therefore, the lemma is validated. $\square$

Proof of Lemma 2

Due to the first part of Lemma 1, $J_{\alpha }$ is finite. Note that

$$\begin{aligned} E\left( \frac{\partial ^2h_{\alpha ,t}(\theta _0)}{\partial \theta \partial \theta ^T}\right)= & {} E\left\{ E\left( \frac{\partial ^2h_{\alpha ,t}(\theta _0)}{\partial \theta \partial \theta ^T}\Big |Y_{t-p:t-1}\right) \right\} \\= & {} (1+\alpha )E\left\{ \int f_{\theta _0}^{\alpha -1}(y|Y_{t-p:t-1})\frac{\partial f_{\theta _0}(y|Y_{t-p:t-1})}{\partial \theta }\frac{\partial f_{\theta _0}(y|Y_{t-p:t-1})}{\partial \theta ^T}\mathrm{d}y\right\} . \end{aligned}$$

Letting A be a $r(r+1)/2\times r(r+1)/2$ diagonal matrix with $A_{ii}=1$ if $i=(m-1)(r+1-m/2)+1$ for $m=1,\ldots ,r$ and 2 otherwise, we can write that for $\nu =(\nu _1^T,\nu _2^T,\nu _3^T)^T\in {\mathbb {R}}^r\times {\mathbb {R}}^{r^2p} \times {\mathbb {R}}^{r(r+1)/2}$,

$$\begin{aligned}&\nu ^T\frac{\partial f_{\theta _0}(Y_t|Y_{t-p:t-1})}{\partial \theta }\\&\quad =\nu _1^T\frac{\partial f_{\theta _0}(Y_t|Y_{t-p:t-1})}{\partial c}+\nu _2^T\frac{\partial f_{\theta _0}(Y_t|Y_{t-p:t-1})}{\partial vec(A_1,\ldots ,A_p)}+\nu _3^T\frac{\partial f_{\theta _0}(Y_t|Y_{t-p:t-1})}{\partial vech(\varSigma )}\\&\quad =f_{\theta _0}(Y_t|Y_{t-p:t-1})\bigg [\nu _1^T\varSigma _0^{-1}\epsilon _t(\theta _0)+\nu _2^T\left\{ (Y_{t-1}^T,Y_{t-2}^T,\ldots ,Y_{t-p}^T)^T\otimes \varSigma _0^{-1}\epsilon _t(\theta _0)\right\} \\&\qquad -\frac{1}{2}\nu _3^T A vech\left\{ \varSigma _0^{-1}-\varSigma _0^{-1}\epsilon _t(\theta _0)\epsilon _t(\theta _0)^T\varSigma _0^{-1}\right\} \bigg ]\\&\quad :=f_{\theta _0}(Y_t|Y_{t-p:t-1})M_t(\nu ,\theta _0), \end{aligned}$$

where $\epsilon _t(\theta _0)=Y_t-c_0-A_{01}Y_{t-1}-\cdots -A_{0p}Y_{t-p} \sim N(0,\varSigma _0)$ and the symbol $\otimes$ stand for Kronecker product. Thus, we have

$$\begin{aligned} \nu ^T(-J_{\alpha })\nu & = (1+\alpha )E\left\{ \int f_{\theta _0}^{\alpha -1}(y|Y_{t-p:t-1})\left( \nu ^T\frac{\partial f_{\theta _0}(y|Y_{t-p:t-1})}{\partial \theta }\right) ^2\mathrm{d}y\right\} \\& = (1+\alpha )E\left\{ f_{\theta _0}^{\alpha -2}(Y_t|Y_{t-p:t-1})\left( \nu ^T\frac{\partial f_{\theta _0}(Y_t|Y_{t-p:t-1})}{\partial \theta }\right) ^2\right\} \\ & = (1+\alpha )E\left\{ f_{\theta _0}^{\alpha }(Y_t|Y_{t-p:t-1})M_t(\nu ,\theta _0)^2\right\} \ge 0. \end{aligned}$$

Since $M_t(\nu ,\theta _0)$ has a continuous distribution induced from Gaussian random variables, one can see that the equality holds only for $\nu =0$. Hence, $J_{\alpha }$ is a non-singular matrix. The non-singularity of $K_\alpha$ can be shown by similar arguments, so we omit the proof for $K_\alpha$. $\square$

Proof of Lemma 3

Let $\{ Y_t | t\in {\mathbb {Z}}\}$ be the strictly stationary and ergodic solution to VAR(p) model (3). Recall that $\{h_{\alpha ,t}(\theta ) | t=1,\ldots ,n\}$ is calculated from the observations $Y_1,\ldots ,Y_n$ and some initial values for $Y_{-p},\ldots , Y_0$. To verify the lemma, we introduce stationary version of $\{h_{\alpha ,t}(\theta ) | t=1,\ldots ,n\}$. Let $\{h^o_{\alpha ,t}(\theta ) | t\in {\mathbb {Z}}\}$ and $\{H^o_{\alpha ,t}(\theta ) | t\in {\mathbb {Z}}\}$ be the counterparts of $\{h_{\alpha ,t}(\theta )\}$ and $\{H_{\alpha ,t}(\theta )\}$, respectively, obtained by using the solution $\{ Y_t | t\in {\mathbb {Z}}\}$. Then, $\{\partial h^o_{\alpha ,t}(\theta _0)/\partial \theta \}$ is a strictly stationary and ergodic process. Further, noting that

$$\begin{aligned}&E\left( f_{\theta _0}^{\alpha -1}(Y_t|Y_{t-p:t-1})\frac{\partial f_{\theta _0}(Y_t|Y_{t-p:t-1})}{\partial \theta }\Big |{\mathcal {F}}_{t-1}\right) \\&\quad =\int f_{\theta _0}^{\alpha }(y|Y_{t-p:t-1})\frac{\partial f_{\theta _0}(y|Y_{t-p:t-1})}{\partial \theta }\mathrm{d}y, \end{aligned}$$

where ${\mathcal {F}}_{t-1}$ be the sigma field generated by $\{ Y_{t-1}, Y_{t-2},\ldots \}$, we have

$$\begin{aligned}&E\left( \frac{\partial h^o_{\alpha ,t}(\theta _0)}{\partial \theta }\Big |{\mathcal {F}}_{t-1}\right) =(1+\alpha )E\left( \int f_{\theta _0}^{\alpha }(y|Y_{t-p:t-1})\frac{\partial f_{\theta _0}(y|Y_{t-p:t-1})}{\partial \theta }\mathrm{d}y \right. \\&\qquad \left. -f_{\theta _0}^{\alpha -1}(Y_t|Y_{t-p:t-1})\frac{\partial f_{\theta _0}(Y_t|Y_{t-p:t-1})}{\partial \theta }\Big |{\mathcal {F}}_{t-1}\right) =0. \end{aligned}$$

Hence, it follows from the functional limit theorem for martingales (cf. Section 18 in Billingsley 1999) that

$$\begin{aligned} \frac{[ns]}{\sqrt{n}}\frac{\partial H^o_{\alpha ,[ns]}(\theta _0)}{\partial \theta } =\frac{1}{\sqrt{n}}\sum _{t=1}^{[ns]} \frac{\partial h^o_{\alpha ,t}(\theta _0)}{\partial \theta }{\mathop {\longrightarrow }\limits ^{w}} K_\alpha ^{1/2}\,B_\eta (s)~~in~~{\mathbb {D}}([0,1],{\mathbb {R}}^\eta ). \end{aligned}$$

Since $h^o_{\alpha ,t}(\theta _0)=h_{\alpha ,t}(\theta _0)$ for $t\ge p+1$, one can also see that

$$\begin{aligned}&\sup _{0\le s\le 1} \frac{[ns]}{\sqrt{n}}\left\| \frac{\partial H^o_{\alpha ,[ns]}(\theta _0)}{\partial \theta }- \frac{\partial H_{\alpha ,[ns]}(\theta _0)}{\partial \theta }\right\| \\&\quad \le \frac{1}{\sqrt{n}}\sum _{t=1}^{p}\left\| \frac{\partial h^o_{\alpha ,t}(\theta _0)}{\partial \theta }-\frac{\partial h_{\alpha ,t}(\theta _0)}{\partial \theta } \right\| =o(1)\quad a.s., \end{aligned}$$

which establishes the lemma. $\square$

Proof of Lemma 4

By Lemma 1, we have

$$\begin{aligned} E \left( \sup _{\theta \in \varTheta } \left\| \frac{\partial ^2 h_{\alpha ,t}(\theta )}{\partial \theta \partial \theta ^T }- \frac{\partial ^2 h_{\alpha ,t}(\theta _0)}{\partial \theta \partial \theta ^T }\right\| \right) <\infty . \end{aligned}$$

Since $\partial ^2 h_{\alpha ,t}(\theta )/\partial \theta \partial \theta ^T$ is continuous in $\theta$, for any $\epsilon >0$, one can take a neighborhood $N_\epsilon (\theta _0)$ such that

$$\begin{aligned} E \left( \sup _{\theta \in N_\epsilon (\theta _0)} \left\| \frac{\partial ^2 h_{\alpha ,t}(\theta )}{\partial \theta \partial \theta ^T }- \frac{\partial ^2 h_{\alpha ,t}(\theta _0)}{\partial \theta \partial \theta ^T }\right\| \right) <\epsilon \end{aligned}$$

(12)

by decreasing the neighborhood to the singleton $\theta _0$. Noting that $\hat{\theta }_{\alpha ,n}$ converges almost surely to $\theta _0$, we have, for sufficiently large n,

$$\begin{aligned}&\max _{ 1\le k\le n}\frac{k}{n}\left\| \frac{\partial ^2 H_{\alpha ,k}({\bar{\theta }}_{\alpha ,n,k})}{\partial \theta \partial \theta ^T }+J_\alpha \right\| \\&\quad \le \max _{ 1\le k\le n} \frac{k}{n}\left\| \frac{\partial ^2 H_{\alpha ,k}({\bar{\theta }}_{\alpha ,n,k})}{\partial \theta \partial \theta ^T }-\frac{\partial ^2 H_{\alpha ,k}(\theta _0)}{\partial \theta \partial \theta ^T }\right\| + \max _{ 1\le k\le n} \frac{k}{n}\left\| \frac{\partial ^2 H_{\alpha ,k}(\theta _0)}{\partial \theta \partial \theta ^T }+J_\alpha \right\| \\&\quad \le \frac{1}{n} \sum _{t=1}^n \sup _{\theta \in N_{\epsilon }(\theta _0)} \left\| \frac{\partial ^2 h_{\alpha ,t}(\theta )}{\partial \theta \partial \theta ^T }-\frac{\partial ^2 h_{\alpha ,t}(\theta _0)}{\partial \theta \partial \theta ^T }\right\| + \max _{ 1\le k\le n} \frac{k}{n}\left\| \frac{\partial ^2 H_{\alpha ,k}(\theta _0)}{\partial \theta \partial \theta ^T }+J_\alpha \right\| \\&\quad :=I_n +II_n\quad a.s. \end{aligned}$$

Using the ergodic theorem and (12), we can see that

$$\begin{aligned} \lim _{n\rightarrow \infty }I_n = E \left( \sup _{\theta \in N_\epsilon (\theta _0)} \left\| \frac{\partial ^2 h_{\alpha ,t}(\theta )}{\partial \theta \partial \theta ^T }- \frac{\partial ^2 h_{\alpha ,t}(\theta _0)}{\partial \theta \partial \theta ^T }\right\| \right) < \epsilon \quad a.s. \end{aligned}$$

Also, from the fact that $\Vert \partial ^2 H_{\alpha ,n}(\theta _0)/\partial \theta \partial \theta ^T +J_\alpha \Vert = o(1)$ a.s., we have

$$\begin{aligned}&\max _{1\le k \le \sqrt{n}} \frac{k}{n}\left\| \frac{\partial ^2 H_{\alpha ,k}(\theta _0)}{\partial \theta \partial \theta ^T }+J_\alpha \right\| \le \frac{1}{\sqrt{n}} \sup _k \left\| \frac{\partial ^2 H_{\alpha ,k}(\theta _0)}{\partial \theta \partial \theta ^T }+J_\alpha \right\| =o(1)\quad a.s. \end{aligned}$$

and

$$\begin{aligned}&\max _{\sqrt{n} < k \le n} \left\| \frac{\partial ^2 H_{\alpha ,k}(\theta _0)}{\partial \theta \partial \theta ^T }+J_\alpha \right\| =o(1)\quad a.s., \end{aligned}$$

which yield $II_n=o(1)$ a.s. The lemma is therefore obtained. $\square$

Proof of Lemma 5

By the second result in Lemma 1 and the continuity of $\partial h_{\alpha ,t}(\theta )/\partial \theta$ in $\theta$, we can also take a neighborhood $N_\epsilon (\theta _0)$ such that

$$\begin{aligned}&\lim _{n\rightarrow \infty }\frac{1}{n}\sum _{t=1}^n\sup _{\theta \in N_\epsilon (\theta _0)}\left\| \frac{\partial h_{\alpha ,t}(\theta )}{\partial \theta }\frac{\partial h_{\alpha ,t}(\theta )}{\partial \theta ^T}- \frac{\partial h_{\alpha ,t}(\theta _0)}{\partial \theta }\frac{\partial h_{\alpha ,t}(\theta _0)}{\partial \theta ^T}\right\| \nonumber \\&\quad = E\left( \sup _{\theta \in N_\epsilon (\theta _0)}\left\| \frac{\partial h_{\alpha ,t}(\theta )}{\partial \theta }\frac{\partial h_{\alpha ,t}(\theta )}{\partial \theta ^T}- \frac{\partial h_{\alpha ,t}(\theta _0)}{\partial \theta }\frac{\partial h_{\alpha ,t}(\theta _0)}{\partial \theta ^T}\right\| \right) < \epsilon ~~ a.s.~ \end{aligned}$$

(13)

Since ${\hat{\theta }}_n \overset{a.s.}{\longrightarrow } \theta _0$, one can prove the lemma by using (13) and the ergodic theorem. $\square$