Dear all
In one of my study, I am using, Economic Policy Uncertainty Index (epu) as my independent variable. I am using firm-level data(panel), and for each firm-year, I have corresponding economic policy uncertainty index sourced from http://www.policyuncertainty.com/all_country_data.html. Thus for all firms in the year 2010, I have epu=105. My doubt is related to the usage of EPU as an independent variable. My dependent variable is the investment and I want to check the relation between firm-level investment and epu. I referred some papers and in some of them, they directly used epu as an independent variable. I am doubtful whether we can use an index directly as a variable of interest. Hence I transformed epu into logarithmic differences and used it as my independent variable. I ran the following commands

Code:
gen delta_log_epu= log(epu)-l.log(epu)
xtreg investment delta_log_epu other_controls i.year, fe vce(robust)
Question 1: Does my transformation of epu makes sense? Is it logically and empirically correct?
Question 2: Should I use delta_log_epu or lag of delta_log_epu?
Question 3: Can I use firm-fixed effects in this setting?

Can someone help me here?
Thanks in advance