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FRM Part I · FRM Exam Part I · Regression Diagnostics

A researcher suspects the error variance in a regression is proportional to the square of an explanatory variable x, that is Var(e_i) = sigma^2 x_i^2. To obtain efficient estimates using weighted least squares for the model y_i = a + b x_i + e_i, which transformation is appropriate?

Divide every term by x, giving y/x = a(1/x) + b + e/x. The transformed error e/x has variance sigma squared times x squared divided by x squared, which equals sigma squared, a constant. This restores homoskedasticity, so OLS on the transformed model is efficient.

  1. AMultiply every term by x_i^2
  2. BDivide every term by x_i^2
  3. CDivide every term by x_i, estimating y/x = a(1/x) + b + e/xCorrect
  4. DTake the square root of y_i only and keep the right side unchanged

Explanation

Dividing by x_i gives error e_i/x_i with variance sigma^2 x_i^2/x_i^2 = sigma^2, which is constant. Dividing by x_i^2 would give error variance sigma^2/x_i^2, not constant. The transformed model has the intercept a on the 1/x term and b as the constant.

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