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Outline the ways of working with heteroskedastic data.

FRM® Part I · Quantitative Analysis · Regression Diagnostics · FRM1.QA.9.b

The idea

Once heteroskedasticity is found there are three standard responses. The first keeps the OLS coefficients and replaces the standard errors with heteroskedasticity-robust (White) standard errors, which are valid whatever the form of the variance. The second is to transform the data so the variance becomes constant, for example taking logs or scaling a loss by firm size when dispersion grows with size. The third is weighted least squares (generalised least squares in this setting). Suppose the error variance is σ² × x². Divide every term, including the intercept, by x, which is the same as weighting each observation by the inverse of its error variance. Robust standard errors change inference only. Weighted least squares changes the coefficients too and is more efficient if the variance model is right. Exam trap: weights go to the less noisy observations.

A practice question

From the question bank, with the reason each wrong option is wrong.

Which approach corrects the standard errors for heteroskedasticity while leaving the OLS coefficient estimates unchanged?

  1. Weighted least squares with weights inversely proportional to the error variance.Why not: WLS re-estimates the model with new weights, so the coefficients change; it is not only a correction to the standard errors.
  2. Adding a lagged dependent variable as an extra regressor.Why not: A lagged dependent variable is a response to serial dependence and changes the model; it does not target non-constant error variance.
  3. Dropping the intercept from the regression.Why not: Removing the intercept changes all the estimates and can bias them, while leaving the variance problem unsolved.
  4. Heteroskedasticity-robust (White) standard errors. - correct

Robust standard errors keep the OLS point estimates and replace the variance formula with one that remains valid when the error variance is not constant. Only inference changes.

A harder one

In a regression of daily loss (USD) on firm size, the dispersion of the residuals grows roughly in proportion to firm size. Which remedy addresses this most directly?

  1. Drop the largest firms, treating them as outliers.Why not: The high variance is systematic rather than erroneous; discarding the large firms removes valid data without modelling the structure.
  2. First-difference the data.Why not: Differencing addresses non-stationarity or serial correlation and does not remove variance that scales with size.
  3. Re-specify the model with loss as a percentage of firm size (or in logs) so the dispersion is stabilised. - correct
  4. Do nothing, because the coefficients remain unbiased and inference is unaffected.Why not: Unbiasedness does not make the standard errors valid, so t-statistics and confidence intervals are still unreliable.

When the scale of the errors rises with the size of the unit, scaling the dependent variable by size or taking logs usually restores roughly constant variance; robust standard errors or WLS are alternatives.

The hardest tier

A regression gives a slope of 0.80 with a conventional standard error of 0.30. A Breusch-Pagan test rejects homoskedasticity, and the White robust standard error for the same slope is 0.45. Using a 5% two-sided critical value of 1.96, which result is correct?

  1. The slope becomes 1.20; the t-statistic is 2.67, so it is significant at 5%.Why not: Scales the point estimate by the ratio of standard errors; robust standard errors never change the coefficient.
  2. The slope stays 0.80; the robust t-statistic is 1.78, so it is not significant at 5%. - correct
  3. The slope stays 0.80; the t-statistic is 2.67, so it is significant at 5%.Why not: Keeps the conventional standard error that heteroskedasticity has made unreliable.
  4. The slope stays 0.80; the t-statistic is 0.56, so it is not significant at 5%.Why not: Inverts the t-statistic as standard error divided by coefficient; the result is right in direction only by luck.

Robust standard errors leave the estimate at 0.80. The t-statistic is 0.8/0.45 = 1.78, below 1.96, so after the correction the slope is not significant, whereas the conventional 0.8/0.3 = 2.67 overstated significance.

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