Visualising Dependence
- Peter Urbani
- Jan 15, 2013
- 2 min read
Since the beginning of the Global Financial Crisis, average Equity market correlations have almost doubled from around 0.22 to 0.39. Current 0.26.
This month we look at the application of graph theory to attempting to better visualise these connections.
Our ultimate aim is as always to better understand the underlying dynamics and dependencies between assets in order to be better able to diversify our portfolios against the next crisis.
In particular we examine the use of Partial Correlation to help deepen our understanding of the co-relationships that exist and also to help filter out extraneous or spurious correlations.
The two pictures below show the dependencies between actual markets ( Left ) and theoretical random matrix theory with the same number of connections ( Right ). As you can see the actual markets display considerable more structure and in particular many more zones of concentration than the hypothetical random walk model does.
The Partial Correlation Matrix is calculated from the inverse matrix of either the standard Pearson product moment Correlation Matrix or the Variance-Co-Variance Matrix preferably after robustifying it through some or other form of shrinkage towards the global mean.
In its univariate form it can be thought of as the correlation of the residuals of X and Y after subtracting the returns of a third variable Z, from both. A Dependency matrix is then constructed by deducting the Partial Correlation from the Pearson Correlation to help identify those components with the highest level of interaction. It is hoped that the resulting matrix is more representative of the ‘true’ dependencies.
The Dependency Matrix is then converted to an adjacency matrix, typically after further threshold filtering, and then to a Directed Adjacency Graph. The Shrinkage helps ensure the positive definiteness of the matrix prior to inversion and the filtering ensures the sparseness of the resultant matrix which is desirable to avoid the ‘bowl of spaghetti’ affect.
Once in the form of either an adjacency list or Incidence Matrix the information can be converted into Graph vertices and nodes and presented in various ways.
The above charts represent the inter-linkages between the 30 stocks in the Dow Jones Industrial Index for the period during the GFC and post the bottom of the market in Feb 2009. A toy version of the Partial Correlation method can be downloaded here, but for further efforts I would recommend the use of more powerful packages such as NodeXL , Gephi and R.
References and further Reading
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