How to interpret an index of values between -2.5 and +2.5 (an independent variable) in a regression?

Question

I am in the proccess of writing my Master's Thesis and I'm performing a multivariate regression (OLS). One of my independent variables is Chinn-Ito Index (financial openness index) which takes values from -2.5 to +2.5. The coefficient is .261. And the dependent variable is in log.

Normally, we would interpret this coefficient as: -> one unit increase in Financial Openness Index increases the dependent variable by 26.1%.

I'm worried this interval (-2.5 and +2.5) might cause some extreme value in my coefficient as other control variables are in different units.

I am not sure what is 1 unit increase in this case? Does this mean that an increase from, for instance, -2 to -1 increases the dependent variable by 26.1%? Seems kind of odd to me.

Should I maybe normalize the index somehow? I have seen a paper when they normalized this index between 0 and 1. If this is the solution, how to do it in Stata? And if normalizing is appropriate in this case, how to interpret a coefficient?

Thank you in advance for taking the time to answer my questions.

Victoria

Edit:

I am trying to get some results on what explaines the crypto adoption. Logcrypto is the log transformation of the crypto index, and the rest are independent variables.

Answer 1

Look at this post to see how to normalize between 0, 1 with Stata. And this post for the general idea.

The coefficient for f_opennavg is $-0.261$, so assuming your model looks like $log(y)=\beta_0 + \beta_1x + u$, the interpretation would be: "An increase in $x$ by one unit is associated with a change in $y$ by $100* \beta_1$% approximately, on average, all other things equal."

So when f_opennavg changes from $1$ to $2$, you would expect that this leads to a change of $-26.1$% in whatever is your $y$. Keep in mind that the model is linear which implies that a change from -2 to -1 has the same marginal effect as a change from -1 to 0 etc.

This is not neccessarily a reasonable assumption, i.e. in cases where you have "bunching" around -1 and 1 (or so) and few "extreme" observations at the top/bottom end of the scale (e.g. North Korea would be extremely "closed" and at the same time is very different to most other countries). You could try to use a dummy to differentiate "extreme" cases from others (or more technically speaking: to "control for the effect of extreme values in openness").

You could also use a GAM model (as described here) to see if the effect of f_opennavg on $y$ can be assumed to be linear. There seem to be GAM models for Stata as well.

When you change the scale of $x$ (e.g. by squeezing it into the interval 0,1), the OLS coefficient will be "scaled" as well (but will not change its fundamental meaning). So putting $x$ on a scale between 0 and 1 will not change the overall result.

Data:

  y  x
1 1  0
2 5  0
3 4  5
4 3  5
5 6  5
6 8 10

Regression:

Coefficients:
            Estimate Std. Error t value Pr(>|t|)
(Intercept)   2.5882     1.2255   2.112    0.102
x             0.4588     0.2269   2.022    0.113

Data with $x$ in range 0,1:

  y   x
1 1 0.0
2 5 0.0
3 4 0.5
4 3 0.5
5 6 0.5
6 8 1.0

Regression:

Coefficients:
            Estimate Std. Error t value Pr(>|t|)
(Intercept)    2.588      1.225   2.112    0.102
x              4.588      2.269   2.022    0.113

So the coefficient for $x$ is simply "sclaed" $\beta*10$ (the former max value in my data)

R-Code:

df = data.frame(y=c(1,5,4,3,6,8), x=c(0,0,5,5,5,10))
summary(lm(y~x,df))
xs = (df$x- min(df$x)) / (max(df$x)-min(df$x))
dfs = data.frame(y=df$y, x=xs)
summary(lm(y~x,dfs))