# 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!

• If you use the logarithm of the rates, you're already modeling a log-normal distribution. If the logarithm by itself is not enough to make the variable normally distributed, try to remove autocorrelation and heteroskedasticity and plug into OLS the residuals. If even this is not enough, try to partially differentiate the time series: while the difference order grows, you'll make the series more stationary... Above some threshold, say $d=0.4$, the distribution will exhibit two first moments quite strong and the third and fourth one hopefully less pronounced. – Lisa Ann Jun 1 at 6:18

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.