A multi-stage emulsion processing aid polymer comprising one or more functionalized ethylenically unsaturated monomer into the emulsion polymerization reactor, wherein the functionality is selected from the group consisting of β-keto esters, β-keto amides, β-diketones, cyanoacetic esters, malonates, nitroalkanes, β-nitro esters, sulfonazides, thiols, thiol-s-triazines, and amine, where the functionality is incorporated into polymers by polymerizing, ethylenically unsaturated monomers containing these functionalities or by post functionalization of a polymer with additional reactions after polymerization in one of the first or second stages. Foamable halogenated polymers comprising the multi-stage emulsion processing aid polymer is also provided. Also provided are methods for making the multi-stage emulsion processing aid polymer and foamable halogenated polymers.