Building and Testing a Multiple Linear Regression Model 87
regression at first. Then we consider all variables for exclusion on a step-
wise removal basis.
For each independent variable, we compute
Fn=−()SSE SSE SSEkk−− 11 /()k ×−()k (4.6)
to find the ones where F is insignificant. The one that yields the least signifi-
cant value F is discarded. We proceed stepwise by alternatively considering all
remaining variables for exclusion and, likewise, compute the F-test statistic given
by equation (4.7) for the new change in the coefficient of partial determination.
In general, at each step i, we compute
Fn=−()SSE SSEki−−ki+− 1 /()SSEki×−()ki+−^1 (4.7)
to evaluate the coefficient of partial determination lost due to discarding the
ith independent variable.^6 If no variable with an insignificant F-test statistic
can be found, we terminate the elimination process.
Standard Stepwise Regression Method
The standard stepwise regression method involves introducing indepen-
dent variables based on significance and explanatory power and possibly
eliminating some that have been included at previous steps. The reason for
elimination of any such independent variables is that they have now become
insignificant after the new independent variables have entered the model.
Therefore, we check the significance of all coefficient statistics according to
equation (3.16) in Chapter 3. This methodology provides a good means for
eliminating the influence from possible multicollinearity discussed earlier.
Application of the Stepwise Regression Method In the previous chapter, we
used an illustration to show how multiple linear regression analysis can
be used for hedge fund style analysis. We first explained the use of the
Sharpe benchmark for this purpose and then explained using an illustra-
tion by Dor and Jagannathan the issues with using the Sharpe benchmark
for hedge fund style analysis.^7 We will continue with that illustration here
(^6) The SSEk−i+ 1 is the sum of square residuals before independent variable i is dis-
carded. After the ith independent variable has been removed, the sum of square
residuals of the regression with the remaining k − i variables is given by SSEk−i.
(^7) Arik Ben Dor and Ravi Jagannathan, “Style Analysis: Asset Allocation and Perfor-
mance Evaluation,” in The Handbook of Equity Style Management, 3rd ed., ed. T.
Daniel Coggin and Frank J. Fabozzi (Hoboken, NJ: John Wiley & Sons, 2003).