Interpretation of the magnitude of regression coefficients

[Francesco Franzoni]

Dear Statalisters,

I am running a regression like this:
Y = a + b1*X1 + b2*X2 + e

Note that X1 and X2 are measured in the same units, but they have very different standard deviations.

In comparing the magnitude of the effects of X1 and X2 on Y, should I just compare the estimated b1 and b2, or should I consider the fact that the two variables have very different standard deviations and possibly standardize them?

Thank you

[Maarten Buis]

That is a substantive question. It is best to discuss this concretely instead of abstractly. So, given that you gave us no context, I will add my own (if that does not work for you, you can add your own context and we can discuss that). Lets say that the unit of analysis are heterosexual households with 2 working adults, i.e. a male and a female. X1 is the occupational status of the male, and X2 the occupational status of the female, measured in some standard scale, e.g. ISEI (https://doi.org/10.1016/0049-089X(92)90017-B). The variance of X1 is in all likelihood much higher than X2: Women are much less likely to reach very high occupation, but they are also less likely to end up in very low status occupation. Instead, women tend to lump together in occupations like teacher, nurse, secretary, which are neither very higher nor very low status. Should we standardize or not? If we were to standardize we would argue that a unit change in occupational status is more "valuable" to women than for men. In many cases I could imagine that would confound the actual mechanism we are trying to study if we are interested in male/female differences, and would thus be a very bad idea. But maybe there are cases where it would make sense.

In general, I think standardization is a bad idea with the occasional exception (though I have not come across a really convincing case), so it should come as no surprise that I would argue against it. However, you should make up your own mind. What is your analysis trying to say, and what steps would help or hinder that?

[Francesco Franzoni]

Dear Maarten,

thank you for taking the time. Your explanation is clear and makes sense. In my context, however, the two variables, although measured in the same units, seem to have a different scale. (I could tell you that the X variables measure ownership shares in a company and Y measures stock price volatility...but I guess it would not help much).
Here are the distributions of X1 and X2 (i.e. holding1 and holding2):

Variable | Obs Mean Std. Dev. Min Max
-------------+---------------------------------------------------------
holding1 | 589,752 .0378966 .041858 0 .5514704


Variable | Obs Mean Std. Dev. Min Max
-------------+---------------------------------------------------------
holding2 | 802,214 .1002854 .1125305 0 2.08562



So, X2 seems to exist in a 'bigger universe' than X1. I am not sure that comparing a one-unit change across the two variables makes much sense.
Thank you.

[Maarten Buis]

It would seem like a well defined unit, so I would keep it as is. At least you need a very strong substantive argument before throwing away the substantial unit. Just looking at variances is not a substantive argument.

[Francesco Franzoni]

I get it. Thanks a lot.

[Phil Bromiley]

Let me reiterate something Maarten said more directly - you cannot meaningfully just compare the two parameters unless a one unit change in each of the rhs variables has the same meaning in both variables. If the standard deviations are different, then it is likely that what a one unit change means is different in the two x's. For example, if the two measures of shares were common and preferred, they are different. A related practice is to look at the change in predicted value for a one standard deviation change in a variable - I'm not sure if this has the same problems as standardizing.

Maarten has good reasons to avoid standardization, but he probably does a sophisticated analysis of what the parameters mean. That's what you probably need to do.