I have panel data of various countries where exchange rate is one of the variables. I have an exchange rate denominated in local currencies only. My question is that Do I need to transform local currency to dollar denominated currency?
I would do so. When regressing some FE models, your dependent and independent variables have the same scale across countries. Otherwise, interpretation is more difficult, as estimates also contain exchange rate responses.
you may have to consider the dollar fluctuations per year for each country depending on average published value per dollar domiciled in each country. Alternatively, instead of creating too many assumptions with dollar, each country's local currency may be explored and then sight a particular year (or more) in a discussion with a variant in dollar value in all countries to buttress a point. Good luck with your research
It is recommended to use uniform scale of measurement of variables pertaining to each cross section of panel data to ease interpretations & analysis. The difference in scale of measurement make interpretations difficult.
If you are using the log of the exchange rate in your analysis it is not important which one you use as
log(US dollar price of one unit of domestic currency) = - Log (domestic currency price of one US dollar)
If you are considering comparing volume series (real GDP) at constant prices you must convert them at base year (or quarter) exchange rates. Otherwise, you will be contaminating your common currency volume data with changes due to changing exchange rates.