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FRM Part I · FRM Exam Part I · Regression Diagnostics

A model of bank trading losses on market volatility shows residual variance proportional to the square of the regressor: Var(e_i) = sigma^2 x X_i^2, with X_i > 0. The original model is Y_i = b0 + b1 X_i + e_i. Which weighted least squares transformation restores homoskedasticity, and what is the interpretation of the intercept in the transformed regression?

Divide every term by X, because the error standard deviation is proportional to X. The transformed model Y/X = b1 + b0(1/X) + e/X has constant error variance. The original intercept b0 becomes the coefficient on 1/X, while the original slope b1 becomes the constant term.

  1. ADivide every term by X_i; the regression is Y/X = b1 + b0(1/X) + u, so b0 becomes the slope on 1/XCorrect
  2. BDivide every term by X_i^2; the intercept b0 remains the constant term
  3. CMultiply every term by X_i; the intercept b0 becomes the slope on X
  4. DTake the square root of Y only; the intercept is unchanged

Explanation

If Var(e)=sigma^2 X^2, then sd(e)=sigma X, so dividing by X gives error e/X with constant variance sigma^2. The transformed model is Y/X = b1 + b0(1/X) + e/X, so b1 becomes the constant and b0 the slope on 1/X. Dividing by X^2 would leave error variance sigma^2/X^2, which is not constant.

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