Hello everyone,

I am analyzing a panel data set with 55 countries. My dependent variable is firm equity issuance (aggregated at the country level) and my independent variable is aggregate stock market liquidity. I initially ran a panel regression with fixed effects as below,

xtreg equity liquiditity $controls i.year, fe vce(robust) // (1st regression)

However, since the scales of the two variables are different, the coefficients are not naturally interpretable. I recently saw in a paper that demeaning and standardizing variables allows meaningful interpretation of the coefficients. To try this, I ran the following command which creates a new set of standardized variables with the prefix "c_".

by country: center equity liquiditity , standardize

reg c_equity c_liquidity $controls i.year, vce(robust) // (2nd regression)

My questions are;
01) Is it Ok to standardize only the dependent and independent variable? Do I need to standardize the control variables in my model when running the 2nd regression?
02) When running the regression with demeaned and standardized variables, is the code stated above correct; is it correct to use "reg" instead of "xtreg"?
03) Even though I did not standardize the control variables, the coefficients and the p values of those control variables in the 2nd regression are different from the 1st regression. Is that to be expected?

Any help is much appreciated. Thank you.