Fine structure of spectral properties for random correlation matrices: An application to financial markets

We study some properties of eigenvalue spectra of financial correlation matrices. In particular, we investigate the nature of the large eigenvalue bulks which are observed empirically, and which have often been regarded as a consequence of the supposedly large amount of noise contained in financial data. We challenge this common knowledge by acting on the empirical correlation matrices of two data sets with a filtering procedure which highlights some of the cluster structure they contain, and we analyze the consequences of such filtering on eigenvalue spectra. We show that empirically observed eigenvalue bulks emerge as superpositions of smaller structures, which in turn emerge as a consequence of cross correlations between stocks. We interpret and corroborate these findings in terms of factor models, and we compare empirical spectra to those predicted by random matrix theory for such models.

Figure 1

(a) Empirical eigenvalue density of the covariance matrix for $T=3400$ daily returns of $N=396$ assets belonging the S&P500 Index over the years $1996-2009$. (b) The same density as in (a) without the largest eigenvalue. (c) Eigenvalue density for $T=1423$ daily returns of $N=243$ assets belonging to the FTSE$350$ Index over the years $2005-2010$. (d) The same density as in (c) without the largest eigenvalue. All figures were produced with standardized data. In (b) and (d) the Marčenko-Pastur distributions for the corresponding values of $q=N/T$ are also plotted.

Figure 2

(a) Eigenvalue density for $100$ simulations of the factor model described in (18), (19), and (20) with $N=500$, $T=2000$ ($q=0.25$). One cluster made of $N_1=\mathrm{N̄}=100$ variables, correlated via a coefficient ${\gamma }_1=0.7$, is present. A common mode is introduced via a coefficient ${\gamma }_N=0.3$. As expected from Eq. (25), two eigenvalue bulks are clearly visible. The mean value in the bulk on the left is $0.04$, while Eq. (25) would predict zero as a consequence of the strong correlation limit (${\gamma }_k\to 1$) approximation. On the other hand, the mean value in the bulk on the right is $0.85$, in remarkable agreement with the predicted value $(1-{\gamma }_N){}^2/[(1-{\gamma }_N){}^2+{\gamma }_N^2]=0.84$. For the sake of readability, the “large” eigenvalues are not shown. (b) By setting ${\gamma }_1=0.4$, the two bulks in (a) merge into a single one. Such a structure, despite emerging as a consequence of (weak) correlations, is very well fitted by a Marčenko-Pastur distribution [see Eq. (16)] with $q=0.29$ and $\sigma =0.88$ (solid line).

Figure 3

(a) Eigenvalue spectrum of the correlation matrix for the factor model yielding the spectrum in (29) for $N=500$, $T=2000$, $\mathrm{N̄}=100$, and $\rho =0.84$. This model yields two degenerate eigenvalues: ${\Lambda }_1=1-\rho =0.16$ and ${\Lambda }_2=1$ (see also Fig. 2). The histogram is the result of $100$ Monte Carlo simulations of such model, while the solid line represents the density obtained from the solution of Eq. (43) (solved as in[29]). (b) Eigenvalue spectrum for the same model with $\rho =0.65$, that is, for ${\Lambda }_1=0.35$. It can be clearly seen that the two separated bulks shown in (a) start to merge as a consequence of the smaller correlations (smaller value of $\rho$). Again, the solid line represents the density obtained from Eq. (43). (c) Posing $\rho =0.30$ the two eigenvalue bulks merge completely into one single structure. In analogy to Fig. 2b, such a structure is apparently well fitted by a Marčenko-Pastur distribution with $q=0.26$ and $\sigma =0.97$, plotted as a solid line. On this scale, the Marčenko distributions would be barely distinguishable from the density obtained from Eq. (43) with ${\Lambda }_1=1-\rho =0.7$ and ${\Lambda }_2=1$. (d) Comparison between two such densities (the dark line represents the Marčenko-Pastur distribution, while the grey line is the solution of Eq. (43)) in correspondence of their peak, where they differ the most. Despite the quite small deviation between the two, a Kolmogorov-Smirnov (KS) performed on the data gave the following results. The critical values for different significance levels $\alpha$, are given by ${\mathrm{V}}_{\mathrm{KS}}^{*}(\alpha =0.10)=5.5\times {10}^{-3}$, ${\mathrm{V}}_{\mathrm{KS}}^{*}(\alpha =0.05)=6.1\times {10}^{-3}$ and ${\mathrm{V}}_{\mathrm{KS}}^{*}(\alpha =0.01)=7.3\times {10}^{-3}$. Under a null hypothesis of data distributed according to the Marčenko-Pastur distribution, the value of the KS statistic was ${\mathrm{S}}_{\mathrm{KS}}=7.9\times {10}^{-3}$, allowing for the rejection of the null hypothesis for all the significance levels considered. On the other hand, under the null assumption of data distributed according to the density obtained from Eq. (43), we obtained ${\mathrm{S}}_{\mathrm{KS}}=2.3\times {10}^{-3}$, thus preventing from rejecting the null hypothesis. Clearly the large statistics in this example plays a relevant role in helping the KS test to “distinguish” the two densities. Smaller data samples would prevent the Marčenko-Pastur distribution from being rejected. Indeed, when performing a KS test on a smaller version ($N=250$, $T=1000$, $\mathrm{N̄}=50$, $\rho =0.30$) of the system in (d), we obtained ${\mathrm{S}}_{\mathrm{KS}}=3.0\times {10}^{-3}$ with the density given by Eq. (43) and ${\mathrm{S}}_{\mathrm{KS}}=5.0\times {10}^{-3}$ for the Marčenko-Pastur distribution, the critical value being ${\mathrm{V}}_{\mathrm{KS}}^{*}(\alpha =0.05)=8.6\times {10}^{-3}$. Thus this example shows how rescaled and smaller system dimensions might prevent from discriminating the two densities.

Figure 4

Distribution of the largest eigenvalue ${\lambda }_{\mathrm{MAX}}$ from $5000$ Monte Carlo simulations of the factor model yielding the spectrum in (29) with $\rho =0.85$, $N=500$, and $\mathrm{N̄}=100$. The distribution is well fitted by a normal distribution (dark line) with expected value $m=84.79$, very close to the theoretical value predicted by Eq. (29): $\mathrm{N̄}\rho +(1-\rho )=85.15$. Three different statistical tests (Jarque-Bera, Lilliefors, and Kolmogorov-Smirnov)[32] were performed, assuming a null hypothesis of normally distributed data. In the following we report the different critical values (${\mathrm{V}}^{*}$) obtained for the different tests and for different significance levels $\alpha$. Also, we report the statistic values (S), which, if smaller than the critical values, prevent the null hypothesis from being rejected. Jarque-Bera test: ${\mathrm{S}}_{\mathrm{JB}}=3.114$, ${\mathrm{V}}_{\mathrm{JB}}^{*}(\alpha =0.10)=4.605$, ${\mathrm{V}}_{\mathrm{JB}}^{*}(\alpha =0.05)=5.992$, ${\mathrm{V}}_{\mathrm{JB}}^{*}(\alpha =0.01)=9.210$. Lilliefors test: ${\mathrm{S}}_{\mathrm{L}}=0.78\times {10}^{-2}$, ${\mathrm{V}}_{\mathrm{L}}^{*}(\alpha =0.10)=1.14\times {10}^{-2}$, ${\mathrm{V}}_{\mathrm{L}}^{*}(\alpha =0.05)=1.25\times {10}^{-2}$, and ${\mathrm{V}}_{\mathrm{L}}^{*}(\alpha =0.01)=1.56\times {10}^{-2}$. Kolmogorov-Smirnov test: ${\mathrm{S}}_{\mathrm{KS}}=0.78\times {10}^{-2}$, ${\mathrm{V}}_{\mathrm{KS}}^{*}(\alpha =0.10)=1.73\times {10}^{-2}$, ${\mathrm{V}}_{\mathrm{KS}}^{*}(\alpha =0.05)=1.92\times {10}^{-2}$, ${\mathrm{V}}_{\mathrm{KS}}^{*}(\alpha =0.01)=2.30\times {10}^{-2}$. All of the previous results prevent from rejecting the hypothesis of normally distributed data. We also generated a TW distribution with the algorithm described in[33], fitted it (grey line) to the data, and performed a KS test on it (the KS being the only non-Gaussian test among the ones mentioned previously) obtaining ${\mathrm{S}}_{\mathrm{KS}}=2.16\times {10}^{-2}$, which allows to reject the null hypothesis of data distributed according to a TW distribution for $\alpha =0.05,0.10$.

Figure 5

Graphical representation of correlation matrices. (a) $40\times 40$ correlation matrix for the selected returns belonging to the S&P500 Index. (b) $28\times 28$ correlation matrix for the selected returns belonging to the FTSE350 Index. In (a) and (b) the stocks have been sorted in order to highlight the cluster structure. (c) and (d) Model correlation matrices corresponding to the cases shown in (a) and (b), respectively. In all plots, white diagonal blocks correspond to ones, while black ones correspond to zeros. Gray shadings are intermediate values. As already explained in the main text, the gray shadings in (c) and (d) correspond to $\rho =0.712$ and $\rho =0.707$, respectively. The presence of unresolved structures in (a) and, less evident, in (b), suggests that the model matrix in (44), depicted in (c) and (d), does not fully capture all empirical features.

Figure 6

(a) and (b) Eigenvalue densities of the correlation matrices represented in Figs. 5a and 5b. In both cases one can clearly distinguish two well separated bulks, while the largest eigenvalues have not been plotted for better visualization (see main text for further explanation). (c) and (d) Comparison between the theoretically expected spectra derived via Eq. (43) (solid lines) and the empirical ones. The latter have been modified with respect to (a) and (b) according to the following approach. A bootstrap random sampling (100 iterations) has been performed on the weakly correlated subsets of stocks, picking 30 stocks out of 33 in the S&P500 Index case and 18 out of 21 in the FTSE350 Index case. The presence of undetected structures [see Figs. 5a and 5b and main text] leads to a poor agreement between data and theory. The values and weights of the eigenvalues used to plot the theoretical density obtained from Eq. (38) are detailed in (49). (e) and (f) As in (c) and (d) but with weakly correlated data reshuffled before bootstrap, leading to a much better agreement between data and theory. The reference eigenvalue for the bulks on the right is now assumed to be equal to one (see main text).