## Reference Materials

## Model Selection

The Advanced Parameters tab in the Edit Analysis Parameters dialog allows the user to set a number of parameters that modify the way the calculations are made.

StyleADVISOR allows the user to modify the mathematical calculation used to analyze a manager's style. This is done on the Advanced Parameter tab in the Edit Analysis Parameters dialog under Model Selection. Changing the model will affect all style weights and style bases, i.e., it will affect the Manager Style and Asset Allocation graphs, as well as everything that involves a Style Benchmark.

When the Standard Model is chosen, StyleADVISOR will perform the classical style analysis as set forth by William F. Sharpe. This means that StyleADVISOR will determine the style weights (i.e., the asset allocation) so as to minimize the variance of excess return. Looking at the formula for the standard R2 as explained earlier, it is easy to see that minimizing the variance of excess return is equivalent to maximizing the standard R2.

The four other models all have one thing in common: They modify the calculation of the style weights so that there is a penalty for having a larger number of non-zero weights. In other words, there is a reward for dropping indices from the analysis.

To see an example, open the workbook Models.zsa. The page named "Standard Model" shows a standard style analysis of the Acorn fund. Here, StyleADVISOR has determined the weights for the four indices so as to maximize the standard R2. In the left hand style table, we have chosen "Standard" as our definition of R2. Hence, the value 88.87 that is shown in the left hand style table is in fact the value that StyleADVISOR has maximized.

Now contrast this with the results on the page named "Adjusted R2 Model." Here, we have chosen the second one of the five models. The weights for the four indices are different from the first page. As is to be expected, the standard R2 as shown in the left hand style table is lower than before: it has dropped slightly to 88.86. That is because only the weights of the standard style analysis will give the maximal standard R2. All other weights result in a lower standard R2. However, since we have chosen a non-standard model, StyleADVISOR has given a reward for dropping one index, namely the Russell 1000 Value index. That reward has outweighed the small decrease of the R2.

The right hand style tables on the two pages of Models.zsa will be explained below.

The overall structure of the calculation is the same for all four non-standard models. StyleADVISOR will first determine all possible combinations of dropping one or more of the chosen indices. In other words, it will look at all possible subsets of the set of chosen indices. For each of these subsets, it will perform a regular style analysis, and it will calculate a "utility function" which rewards high standard R2 and penalizes for using a large number of indices. The final result is obtained by picking the subset for which the utility function has the optimal value. (This optimum can be the maximal or minimal value depending on the nature of the utility function). Finally, the indices that were dropped are assigned zero weights.

What distinguishes the four non-standard models is the way in which the reward for dropping indices is determined. In other words, different models use different ways to calculate the utility function as explained in the previous paragraph. It is important to understand that there are no compelling mathematical reasons to prefer one model over the other. The criteria employed by the different utility functions are contentious even among mathematicians. The choice of a particular model is therefore highly discretionary.

For a deeper mathematical discussion of the four models, we refer the reader to Judge et al., Sections 20.4.1-20.4.3.

**References**

Judge, George J., R. Carter Hill, William E. Griffiths, Helmut Luetkepohl and Tsoung-Chang Lee. Introduction to the Theorie and Practice of Econometrics, Second Edition, John Wiley and Sons, New York, 1988. Sections 20.4.1-20.4.3