From an investigation of the residuals of fitting a linear regression by ordinary least squares it is clear that the spread of the residuals increases as the predicted values increase. Observed values of the dependent variable range from 0 to 100. Determine which of the following statements is/are true with regard to transforming the dependent variable to make the variance of the residuals more constant.

• Taking the logarithm of one plus the value of the dependent variable may make the variance of the residuals more constant.
• A square root transformation may make the variance of the residuals more constant.
• A logit transformation may make the variance of the residuals more constant.

• None
• I and II only
• I and III only
• II and III only
• The correct answer is not given by (A), (B), (C), or (D).

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Key: B

Adding a constant to the dependent variable avoids the problem of the logarithm of zero being negative infinity. In general, a log transformation may make the variance constant. Hence I is true.

Power transformations with the power less than one, such as the square root transformation, may make the variance constant. Hence II is true.

A logit transformation requires that the variable take on values between 0 and 1 and hence cannot be used here.

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.