A few variables expressed in units of a standard normal distribution in a regression

Question

I am using three of the Worldwide Governance Indicators (WGI) estimates. They represent broad dimensions of governance for countries. All indicators range from -2.5 to 2.5 approximately. They are expressed in units of a standard normal distribution.

For your information: here is the link to the website: https://info.worldbank.org/governance/wgi/Home/Documents#doc-intro

I want to use these indicators in a panel regression as control variables. All my other variables are not expressed in units of a standard normal distribution. Is it a problem? Do I need to convert the governance indicators (-2.5, 2.5) to another scale (0, 1)? Will the 'econometrics' make sense?

Answer 1

In general, there is no expectation that right hand side variable in a regression are in the same units. We often have combinations of variables like years of education, price, regional dummies, etc. that are all included together. So in principle, there’s no reason why you should have to convert your controls to some other scale.

From a numerical standpoint, it can be useful to normalize variables to maintain numerical stability. The eigenvalues of the covariance matrix can be a little funky if the relative scales are way different (for example, one variable ranging between 1-4 and another variable ranging between 1-4 billion). It can also be useful to normalize variables to help with interpretation of the coefficient values (easier to compare 1 and 2 than it is to compare 1 and 0.000000002).

From your example, though, if you’re just considering adjusting from something that ranges on the order of -2 to 2 to something that ranges from 0 to 1, neither of these reasons would probably hold. As such, based on the info you’ve given, I would probably just leave the RHS variables as they are.

Answer 2

The interpretation of the coefficient $\beta$ on the standard normal explanatory variable is that a one standard deviation increase in that variable will lead to a $\beta$ increase in the dependent variable. Some examples: If the dependent variable was GDP, then $\beta$ would be the amount of an increase in GDP associated with a one standard deviation increase in the governance score. Or, if the dependent variable was log GDP, then $100\cdot\beta$ would approximately represent the percent increase in GDP associated with a one standard deviation increase in the governance score. So, this interpretation is straightforward, and converting the variable to a range between 0 and 1 will make the interpretation more difficult and unintuitive.

Furthermore, a normally-distributed variable actually ranges from $-\infty$ to $\infty$, mathematically speaking. Even though 98.8% of the values lie between 2.5 standard deviations from the mean, the variable can still take on larger values, moreso in large samples. So, mathematically speaking, the transformation $f(x) = (x + 2.5)/5$ will not actually keep all the values between 0 and 1.

It is perfectly fine to have different units in your regression—the only difference with using different units is how you interpret the coefficients. You may even want to normalize the dependent variable (by dividing it by its standard deviation), so that the interpretation becomes "A one standard deviation increase in X increases Y by $\beta$ standard deviations."