# Interpreting regression coefficient - what units are fractions?

I am regressing a growth measure (in fraction form, between 0 and 1) on another fraction that lies between 0 and 1 (let's call the variable share"). The regression is performed using OLS. What is the interpretation of my regression coefficient, b?

My best guess: a one unit change in the share changes growth by b units.

But what is a one unit change in the share? Is it the same as a 1% change? Moreover, does the same interpretation of a unit hold for the left-hand-side variable? That is, am I right to interpret the coefficient as: "for a 1% change in share, growth changes by b%"?

On a broader note, what unit measurement do we use for fractions?

EDIT: I'll now be more precise about my lack of understanding, as prompted by @NickCox. I have results from a number of regressions and I am struggling to understand the output. The key right-hand-side variable is a fraction between 0 and 1 (there are other controls in every regression). The left-hand-side varies: in some regressions it is a fraction between 0 and 1, in some it is a change (first-difference) in logs of a variable, and sometimes it is a dummy or indicator variable. I know the standard deviations of each variable, and I'd like to be able to interpret all of my coefficients on the key dependent variable in terms of a one standard deviation change in variable and in terms of a one unit change.

That is, I would like help interpreting the coefficient b -- in unit changes and standard deviations changes -- in the regression of a:

• Fraction on a fraction (and other controls)
• Change in log on a fraction (and other controls)
• Binary variable on a fraction (and other controls).

Note for the final model, I am using a linear-probability model (I'm confident of my choice of this model for other reasons, plus it is consistent with the literature).

Finally, a lot of my problems have come from reading published papers and trying to back out their logic for interpreting coefficients. There seems to be a lack of consistency. For example, I am studying a paper where the regression is a binary dependent variable regressed on a change (first-difference) in logs of a variable (estimated by OLS). The authors have interpreted the coefficient (-0.21) as: "a one standard deviation increase in the change in logs leads to a 6.3% percentage point decrease in the probability of an event"; a result that seems to come from dividing the coefficient by 1 over the standard deviation of the change in logs (1/0.3=3.33). Is this correct?

• If the units on both variables are the same, they wash out in forming the slope coefficient. The coefficient itself is unit-free. (On a smaller detail, why say "reduces" and "increases" in different places here?) – Nick Cox Jun 8 '15 at 16:23
• You're right, Nick, it's very sloppy. I will edit my question to be more precise. If my coefficient is unit free, do I report it based on standard deviations of the independent variable? I'm searching for a way to provide a so-called "economic" interpretation of my coefficient estimate. And thanks for the help, @NickCox. – singlepeaked Jun 8 '15 at 18:19
• Just to follow up, I think I have the answer and I would really appreciate it if you could check my logic, @NickCox: First, I find the standard deviation of my right-hand-side variable (X). Let's say it is 0.1. Second, I need my slope coefficient. Let's say $b=5$. Third, I compare a unit change to the SD: in this case, a 1 unit change in X is a change of 10 standard deviations. Finally, I can interpret my slope coefficient as: a change of 10 standard deviations of X will result in a 5 standard deviation change in the outcome, Y. – singlepeaked Jun 8 '15 at 18:36
• If the slope is 5, and the SD of $X$ is 0.1, then a change of 1 SD = 0.1 in $X$ is a change of 0.5 in $Y$; or similarly a change of 10 SD = 1 in $X$ is a change of 5 in $Y$. You need to know the SD of $Y$ to say more. – Nick Cox Jun 8 '15 at 20:20
• Thanks, again @NickCox: If I knew that the SD of Y is 0.5, where does that get me? Just to point out to anyone who comes across this post, I worked out my previous (but incorrect) attempt from this blog-post: davegiles.blogspot.dk/2015/04/…. The post is very informative, the misunderstanding is obviously all mine. – singlepeaked Jun 8 '15 at 21:26