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Why OLS Does Not Require Normally Distributed Predictors

Article Quant Q&A · Author: Emre
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Summary

The document addresses whether interest-rate predictors must be transformed to normal distributions before they can be used in an ordinary least squares regression. Its central point is that normality of the explanatory variables is not an OLS requirement. The distributional assumption more relevant to classical inference concerns the regression errors; normal errors support stronger finite-sample statistical conclusions, but nonnormal predictors alone do not invalidate the estimator.

It also cautions that least squares depends on suitable moment conditions, and that heavy-tailed or Cauchy-like data can make the method problematic. The response mentions equity returns and other quantities as possible examples where ordinary least squares may fail. It gives no transformation recipe or empirical analysis of the Euro-area interest-rate data, and its broad claim about estimator properties depends on the relevant assumptions, including finite moments and the regression specification.

Key ideas

  • OLS does not require explanatory variables to be normally distributed.
  • Normality of regression errors supports stronger statistical inference, but is distinct from predictor normality.
  • The distribution of the data matters when required moments do not exist.
  • Heavy-tailed or Cauchy-like observations can undermine least squares methods.
  • Transforming interest rates solely to make their marginal distribution normal is not necessary for OLS.

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StatisticsFixed income
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# Transforming non-normally distributed interest rates for OLS regression


# Transforming non-normally distributed interest rates for OLS regression












I am studying the effects of short- and long-term interest rates on bank risk-taking in the Euro zone countries. To analyse the effects, I will use, amongst other, an OLS regression. However I have the problem that the interest rates are non-normally distributed, so I need to transform them in order to use them in a regression. Additionally, my data includes negative interest rates that have been in the Euro zone during the past couple of years. Please see the image that I attached for the histogram of the short-term interest rates over the years 2004-2017 of the Euro zone countries. The interest rates are likely distributed this way, due to that the values do not change significantly from year-to-year (usually a low rate is followed by low rate), and because the European Central Bank sets a monetary policy rate, which guides these countries, and thus leads to similar interest rates. Other variables that I have included in my regression are for example bank capitalization and bank size, which are normally distributed.

Using transformations such as logarithm, square root, etc. is not very useful in this case. However, is it possible to transform this variable in such a way that it will follow a normal distribution? Is a log-normal distribution applicable? Thank you very much for your help!

## Answer by Dave Harris (score 3)

https://quant.stackexchange.com/a/45887

You do not need to transform the variables into normally distributed data in order to use them in a regression. That is not a requirement of ordinary least squares.

If the error terms are normally distributed, then there are stronger interpretive statements that you could make, but if it is not true it does not make the OLS estimator any less the minimum variance unbiased estimator. Amongst other things, if you could assume normality then the MVUE would also be the most efficient estimator which it often is not. The distribution of the data only impacts least squares regression if it lacks a first or second moment. For example, returns on equity securities lack a first moment and so all forms of least squares regression will generate an incorrect solution.

The first to note this was Augustin Cauchy in 1851 in a battle with Bienayme' over the properties of regression estimators.

Least squares can be problematic if your data is drawn from a Pareto distribution or a variant of the Cauchy distribution. Equity returns, accounting ratios and the distribution of certain types of prices or values such as oil fields will mandate alternative methods.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.