The Agri-Food & Veterinary Authority (AVA) has stated on their website:
PETS DO NOT CAUSE ASTHMA.
Asthma is a genetically inherited condition. Attacks can be triggered by house dust, dust mites, pollen, lint, stress, cold water and animal hair/saliva. Animal hair/saliva are not considered to be major triggers. If you suspect that pet fur triggers your attack, you should get a physician to conduct proper allergy tests to confirm this before you make a decision to remove the animal.