A residual-based multivariate constant correlation test

Abstract

We propose a new multivariate constant correlation test based on residuals. This test takes into account the whole correlation matrix instead of the considering merely marginal correlations between bivariate data series. In financial markets, it is unrealistic to assume that the marginal variances are constant. This motivates us to develop a constant correlation test which allows for non-constant marginal variances in multivariate time series. However, when the assumption of constant marginal variances is relaxed, it can be shown that the residual effect leads to nonstandard limit distributions of the test statistics based on residual terms. The critical values of the test statistics are not directly available and we use a bootstrap approximation to obtain the corresponding critical values for the test. We also derive the limit distribution of the test statistics based on residuals under the null hypothesis. Monte Carlo simulations show that the test has appealing size and power properties in finite samples. We also apply our test to the stock returns in Euro Stoxx 50 and integrate the test into a binary segmentation algorithm to detect multiple break points.

Appendices

Proof

1.1 Proof of Proposition 1

To prove the convergence of the random vector $\frac{1}{\sqrt{n}}\sum _{t=1}^{[ns]} vech( \hat{\varvec{Z}}_{t} \hat{\varvec{Z}}_{t}')$, we need to prove each element in this $\frac{d(d-1)}{2}$ dimensional vector has such convergence, for $1\le i<j\le d$:

$$\begin{aligned} \frac{1}{\sqrt{n}}\sum _{t=1}^{[ns]} \hat{Z}_{t,i}\hat{Z}_{t,j}= & {} \frac{1}{\sqrt{n}}\sum _{t=1}^{[ns]} Z_{t,i}Z_{t,j} + \frac{1}{\sqrt{n}} \sum _{t=1}^{[ns]} \frac{\partial z_{i}z_{j}}{\partial \varvec{z}}\Big |_{\varvec{z} = \varvec{Z}_t^*} \frac{\partial \varvec{Z}_t^*}{\partial \varvec{\theta }}\Big |_{\varvec{\theta }=\varvec{\theta }_{\lambda _0}^*} (\hat{\varvec{\theta }}_{\lambda _0}-\varvec{\theta }_{\lambda _0}) \\= & {} \frac{1}{\sqrt{n}}\sum _{t=1}^{[ns]} Z_{t,i}Z_{t,j} + \frac{1}{\sqrt{n}} \sum _{t=1}^{[ns]} \frac{\partial z_{i}z_{j}}{\partial \varvec{z}}\Big |_{\varvec{z} = \varvec{Z}_t} \frac{\partial \varvec{Z}_t}{\partial \varvec{\theta }}\Big |_{\varvec{\theta }=\varvec{\theta }_{\lambda _0}} (\hat{\varvec{\theta }}_{\lambda _0} -\varvec{\theta }_{\lambda _0})\\&+ \,\frac{1}{\sqrt{n}}\sum _{t=1}^{[ns]} \Bigg (\frac{\partial z_{i}z_{j}}{\partial \varvec{z}}\Big |_{\varvec{z} = \varvec{Z}_t^*} \frac{\partial \varvec{Z}_t^*}{\partial \varvec{\theta }}\Big |_{\varvec{\theta }=\varvec{\theta }_{\lambda _0}^*}-\frac{\partial z_{i}z_{j}}{\partial \varvec{z}}\Big |_{\varvec{z} = \varvec{Z}_t} \frac{\partial \varvec{Z}_t}{\partial \varvec{\theta }}\Big |_{\varvec{\theta }=\varvec{\theta }_{\lambda _0}}\Bigg ) \\&(\hat{\varvec{\theta }}_{\lambda _0}-\varvec{\theta }_{\lambda _0}) \end{aligned}$$

Note that $\varvec{\theta }_{\lambda _0}$ is true parameter vector with length 4d, the estimator $\varvec{\theta }_{\lambda _0}^*$ is the convex combination of $\varvec{\theta }_{\lambda _0}$ and $\hat{\varvec{\theta }}_{\lambda _0}$ such that it lies in the neighborhood $\Phi _n$ defined in Assumption 2 (as $\hat{\varvec{\theta }}_{\lambda _0}-\varvec{\theta }_{\lambda _0} = O_p(1/\sqrt{n})$, so $\varvec{\theta }_{\lambda _0}^*$ lies in $\sqrt{n}$-neighborhood of $\varvec{\theta }_{\lambda _0}$ hence in $\Phi _n$). The first equal sign is given by expansion with mean value theorem around $\varvec{\theta }_{\lambda _0}$, and the third term in the second line vanishes asymptotically as

$$\begin{aligned}&\left\| \frac{1}{\sqrt{n}}\sum _{t=1}^{[ns]} \Big (\frac{\partial z_{i}z_{j}}{\partial \varvec{z}}\Big |_{\varvec{z} = \varvec{Z}_t^*} \frac{\partial \varvec{Z}_t^*}{\partial \varvec{\theta }}\Big |_{\varvec{\theta }=\varvec{\theta }_{\lambda _0}^*}-\frac{\partial z_{i}z_{j}}{\partial \varvec{z}}\Big |_{\varvec{z} = \varvec{Z}_t} \frac{\partial \varvec{Z}_t}{\partial \varvec{\theta }}\Big |_{\varvec{\theta }=\varvec{\theta }_{\lambda _0}} \Big ) (\hat{\varvec{\theta }}_{\lambda _0}-\varvec{\theta }_{\lambda _0}) \right\| \\&\quad \le \Vert \sqrt{n}(\hat{\varvec{\theta }}_{\lambda _0} -\varvec{\theta }_{\lambda _0}) \left\| \sup _{\varvec{\theta }_{\lambda _0}^*,t=1,\ldots ,n} \Vert \frac{\partial z_{i}z_{j}}{\partial \varvec{z}}\Big |_{\varvec{z} = \varvec{Z}_t^*} \frac{\partial \varvec{Z}_t^*}{\partial \varvec{\theta }}\Big |_{\varvec{\theta }=\varvec{\theta }_{\lambda _0}^*}-\frac{\partial z_{i}z_{j}}{\partial \varvec{z}}\Big |_{\varvec{z} = \varvec{Z}_t} \frac{\partial \varvec{Z}_t}{\partial \varvec{\theta }}\Big |_{\varvec{\theta }=\varvec{\theta }_{\lambda _0}}\right\| \\&\quad \rightarrow _p 0 \end{aligned}$$

The last line is guaranteed by Assumption 2. By applying Lemma 18.7 in Davidson (1994), p. 285, for $1\le i <j \le d$, we have

$$\begin{aligned} \frac{1}{\sqrt{n}}\sum _{t=1}^{[ns]} \Big (\frac{\partial z_{i}z_{j}}{\partial \varvec{z}}\Big |_{\varvec{z} = \varvec{Z}_t^*} \frac{\partial \varvec{Z}_t^*}{\partial \varvec{\theta }}\Big |_{\varvec{\theta }=\varvec{\theta }_{\lambda _0}^*}-\frac{\partial z_{i}z_{j}}{\partial \varvec{z}}\Big |_{\varvec{z} = \varvec{Z}_t} \frac{\partial \varvec{Z}_t}{\partial \varvec{\theta }}\Big |_{\varvec{\theta }=\varvec{\theta }_{\lambda _0}} \Big ) (\hat{\varvec{\theta }}_{\lambda _0}-\varvec{\theta }_{\lambda _0}) \rightarrow _p 0 \end{aligned}$$

Note that, from Assumption 1, we have

$$\begin{aligned} \sqrt{n}(\hat{\varvec{\theta }}_{\lambda _0}-\varvec{\theta }_{\lambda _0}) \rightarrow _d \Sigma _{\lambda _0}^{1/2}\varvec{\Theta }_{\lambda _0}(1) \end{aligned}$$

In addition, once Assumption 1 is valid, it holds that the sample (cross-) moments of $\varvec{Z}_{t}$ converges to its theoretical counterparts, respectively, in the subsamples separated by the change point $\lambda _0$. As a consequence, the asymptotic property of the essential part of residual effect follows

$$\begin{aligned} \frac{1}{n}\sum _{t=1}^{[ns]}\frac{\partial vech(\varvec{z}\varvec{z}')}{\partial \varvec{z}}\Big |_{\varvec{z} = \varvec{Z}_t} \frac{\partial \varvec{Z}_t}{\partial \varvec{\theta }}\Big |_{\varvec{\theta }=\varvec{\theta }_{\lambda _0}} \Rightarrow \varvec{\tau }_{\lambda _0}(s) \end{aligned}$$

where the asymptotic residual effect part is

$$\begin{aligned} \varvec{\tau }_{\lambda _0}(s) = \begin{pmatrix} \varvec{\tau }_{\varvec{\theta }_{0,1}}(s) \bar{D}_{t,\lambda _0} + \varvec{\tau }_{\varvec{\theta }_{0,1}}(\lambda _0) D_{t,\lambda _0}&(\varvec{\tau }_{\varvec{\theta }_{0,2}}(s)-\varvec{\tau }_{\varvec{\theta }_{0,2}}(\lambda _0))D_{t,\lambda _0} \end{pmatrix} \end{aligned}$$

Following the elementwise convergence derived above, together with continuous mapping theorem and Assumptions 1–2, we have

$$\begin{aligned} \begin{array}{ll} &{}\frac{1}{\sqrt{n}}\displaystyle \sum \limits _{t=1}^{[ns]} (vech(\hat{\varvec{Z}}_{t}\hat{\varvec{Z}}_{t}')- \mathbb {E}(vech(\varvec{Z}_{t}\varvec{Z}_{t}'))) \\ &{}\quad = \frac{1}{\sqrt{n}}\displaystyle \sum \limits _{t=1}^{[ns]} (vech(\varvec{Z}_{t}\varvec{Z}_{t}')- \mathbb {E}(vech(\varvec{Z}_{t}\varvec{Z}_{t}'))) + \frac{1}{\sqrt{n}} \displaystyle \sum \limits _{t=1}^{[ns]} \frac{\partial vech(\varvec{z}\varvec{z}')}{\partial \varvec{z}}\Big |_{\varvec{z} = \varvec{Z}_t} \frac{\partial \varvec{Z}_t}{\partial \varvec{\theta }}\Big |_{\varvec{\theta }=\varvec{\theta }_{\lambda _0}} (\hat{\varvec{\theta }}_{\lambda _0}-\varvec{\theta }_{\lambda _0})\\ &{}\qquad + \frac{1}{\sqrt{n}}\displaystyle \sum \limits _{t=1}^{[ns]} \Bigg (\frac{\partial vech(\varvec{z}\varvec{z}')}{\partial \varvec{z}}\Big |_{\varvec{z} = \varvec{Z}_t^*} \frac{\partial \varvec{Z}_t^*}{\partial \varvec{\theta }}\Big |_{\varvec{\theta }=\varvec{\theta }_{\lambda _0}^*}-\frac{\partial vech(\varvec{z}\varvec{z}')}{\partial \varvec{z}}\Big |_{\varvec{z} = \varvec{Z}_t} \frac{\partial \varvec{Z}_t}{\partial \varvec{\theta }}\Big |_{\varvec{\theta }=\varvec{\theta }_{\lambda _0}}\Bigg ) (\hat{\varvec{\theta }}_{\lambda _0}-\varvec{\theta }_{\lambda _0}) \\ &{}\quad \Rightarrow \Omega ^{1/2}\varvec{\Gamma }(s) + \varvec{\tau }_{\lambda _0}(s)\Sigma _{\lambda _0}^{1/2}\varvec{\Theta }_{\lambda _0}(1) \end{array} \end{aligned}$$

1.2 Proof of Proposition 2

Following Proposition 1, we have

$$\begin{aligned} \begin{array}{cl} &{} \frac{j}{\sqrt{n}}(\hat{\varvec{S}}_j-\hat{\varvec{S}}_n) \\ &{}\quad = \frac{j}{\sqrt{n}} \left( \frac{1}{j} \displaystyle \sum \limits _{t=1}^j vech( \hat{\varvec{Z}}_{t} \hat{\varvec{Z}}_{t}' ) - \frac{1}{n}\displaystyle \sum \limits _{t=1}^n vech(\hat{\varvec{Z}}_{t} \hat{\varvec{Z}}_{t}')\right) \\ &{}\quad = \frac{1}{\sqrt{n}} \displaystyle \sum \limits _{t=1}^j [ vech( \hat{\varvec{Z}}_{t} \hat{\varvec{Z}}_{t}' ) -\mathbb {E}(vech(\varvec{Z}_{t} \varvec{Z}_{t}')) ] - \frac{j}{n} \frac{1}{\sqrt{n}}\displaystyle \sum \limits _{t=1}^n [vech( \hat{\varvec{Z}}_{t} \hat{\varvec{Z}}_{t}' ) - \\ &{} \mathbb {E}(vech(\varvec{Z}_{t} \varvec{Z}_{t}')) ] \\ &{}\quad \Rightarrow \Omega ^{1/2}\varvec{\Gamma }(s) + \varvec{\tau }_{\lambda _0}(s)\Sigma _{\lambda _0}^{1/2}\varvec{\Theta }_{\lambda _0}(1) - \frac{j}{n} [\Omega ^{1/2}\varvec{\Gamma }(1)+\varvec{\tau }_{\lambda _0}(1)\Sigma _{\lambda _0}^{1/2}\varvec{\Theta }_{\lambda _0}(1)] \\ &{}\quad :=\Omega ^{1/2} (\hat{\varvec{\Gamma }}(s) - s\hat{\varvec{\Gamma }}(1)) \end{array} \end{aligned}$$

where $\hat{\varvec{\Gamma }}(s) = \varvec{\Gamma }(s) + \Omega ^{-1/2}\varvec{\tau }_{\lambda _0}(s)\Sigma _{\lambda _0}^{1/2}\varvec{\Theta }_{\lambda _0}(1)$ and $j=[ns]$. The conclusion of this proposition follows with the continuous mapping theorem together with Assumption 3.

Tables

See Tables 3, 4 and 5.

Table 3 Empirical size of the multivariate constant correlation test for serially independent random variables with 1000 Monte Carlo simulations

Table 4 Empirical size of the multivariate constant correlation test for serially dependent random variables with 1000 Monte Carlo simulations

Table 5 Empirical power of the multivariate constant correlation test for both serially independent and serially dependent random variables with 1000 Monte Carlo simulations

Figures

See Figs. 8 and 9.

Fig. 8

ACF plots of cross products of residuals $\hat{Z}_{t,i}\hat{Z}_{t,j}, \forall i\ne j$ for stocks, and A, B, D, E, ING, INT, L, S are abbreviations for ARCELOR, BASF, DAIMLER, ENEL, INGGROEP, INTESA, LVMH, SANOFI, respectively

Fig. 9

Histogram of asymptotic distribution of test statistics $Q_n$ approximated by block bootstrap procedure with $B=10{,}999$ bootstrap replications and block length $T^{1/3}$ (the blue vertical line indicates the 0.95 quantile) (color figure online)