A Statistical Prism
Light is a mixture of colors. A prism reveals the bouquet of hidden colors. The prism's power of spectral decomposition has led to discoveries in fields ranging from molecular biology to astrophysics. The prism is a metaphor for the cooperative resolution process.
Variance decomposition is a statistical procedure that takes as its starting point another statistical model. For example, variance decomposition can be based on linear regression, which is widely used to model the behavior of a variable as a result of the behavior of several explanatory factors. One use of variance decomposition is to attribute the explained variance of a statistical model to its factors. This may provide a better understanding of the relative importance of the factors.
A regression coefficient, or beta, of a factor is an estimate of how much the dependent variable being modeled changes given a change in the factor. The statistical significance of the factor is an indicator of how accurately it is measured. One of the key determinants of the statistical significance of a factor is its marginal contribution to variance.
Variance decomposition is another approach to determining the relative importance of factors in a statistical model. Marginal contributions to explained variance may add up to much more or much less than the total variance of the model. One interpretation of this fact is that there may sometimes be a certain type of inconsistency in conventional statistical significance testing. A statistical significance test implicitly assumes that other factors in that model but are not tested are themselves significant.
A new current of thought in statistics explicitly sees techniques such as variance decomposition as providing information not available from the model itself or from conventional statistical tests. Variance decomposition is not simply an academic issue because of the wide use of statistical models in business, financial and public policy decisions.
Prism Analytics's variance decomposition method, proportional marginal variance decomposition (PMVD), consistently attributes variance shares to individual model factors. PMVD is consistent, both in the sense that all joint contributions to variance are treated in a uniform manner, and in the sense that attributions are consistent with statistical significance testing. PMVD is the only available variance decomposition technique that is consistent with statistical theory.
PMVD is a general procedure with wide potential application. Our first application is to returns-based analysis, an area of financial econometrics. The principal anticipated use is the analysis of hedge fund fund performance, however, returns based tools using PMVD can also be used in the analysis of mutual funds and other financial assets. This application was chosen because of the expertise of the principals and because of need for better methods to analyze available hedge fund performance data.
Prism’s patented Proportional Marginal Variance Decomposition (PMVD) technique is the best available method for decomposing variance and determining the relative importance of model factors. It is a statistical prism. It is an improvement on existing statistical techniques because it provides a more consistent way to assess the relative importance of factors in a statistical analysis.