Let's suppose a regression between earnings and age (and suppose I do not know the distribution of earnings). Would it be possible for the residuals to be normally distributed?I am thinking it would not be possible since earnings only takes on positive values and since the support of the normal is from − ∞ to ∞, it would not be normal. However, since residuals are errors, they can be both positive and negative, so I am starting to question my hypothesis here.

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Let&#039;s suppose a regression between earnings and age (and suppose I do not know the distribution of earnings). Would it be possible for the residuals to be normally distributed?I am thinking it would not be possible since earnings only takes on positive values and since the support of the normal is from   −  ∞ to   ∞, it would not be normal. However, since residuals are errors, they can be both positive and negative, so I am starting to question my hypothesis here.

Tanya Anthony · Accepted Answer

If earnings are always positive then no, the residuals cannot be normally distributed, even though many may be negative: the magnitude of the negative residuals are bounded by the highest predicted earnings on the regression line.That may not be the major issue: more important might be issues such as the skewness of earnings distributions at any age, or a non-linear relationship between earnings and age .

Isaac Barry · Accepted Answer

You can always skew-zero transform a   y-variable (earnings) if transforming skewed   x-variables do not result in normally-distributed residuals. van Der Waerden scores would do a good job here, so to begin:1. Determine percentile values,   p  c      t    i  , of each   y-value based on rank position,   R  (      y    i    ), after an ascending sort.2. Obtain the van der Waerden scores by plugging in the percentile values into the inverse CDF, i.e.,       Z    i    =      Φ          −      1        (  p  c      t    i    )3. Then regress   Z on age, providing age is not skewed too much.By definition, van der Waerden scores are mean-zero standard normal distributed,   N  (  0  ,  1  ), so the residuals should now be normally distributed.To interpret the coefficient on age, just deconvolve.

Let's suppose I do a regression between earnings and age (and suppose I do not know the distribution of earnings). Would it be possible for the residuals to be normally distributed?

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