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Are there any data sources or variables that measure how protectionist a country is? If so, what are they?

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Looked a bit further, I think I possibly found what you're looking for:

The Overall Trade Restrictiveness Index (OTRI) summarizes the trade policy stance of a country by calculating the uniform tariff that will keep its overall imports at the current level when the country in fact has different tariffs for different goods.

The data: http://siteresources.worldbank.org/INTRES/Resources/469232-1107449512766/OTRI2009.xlsx

Background: http://econ.worldbank.org/WBSITE/EXTERNAL/EXTDEC/EXTRESEARCH/0,,contentMDK:22574446~pagePK:64214825~piPK:64214943~theSitePK:469382,00.html

hope this helps, best regards

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Another useful World Bank data set on protectionism is the Temporary Trade Barriers Database (TTBD)

The TTBD website hosts newly collected, freely available, and detailed data on more than thirty different national governments' use of policies such as antidumping (AD), global safeguards (SG), China-specific transitional safeguard (CSG) measures, and countervailing duties (CVD).

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