No, at least not directly.
But look on it as a mechanical problem. Long hair is easier to be lost than a short one and if a person with tendency of losing hair takes his hat on and off, say sixty times a day as a habit and the hat pulls the hair, it might contribute to hair loss.
But that is surely not significant. Hair loss of men is caused mainly with DHT hormone which is a modified testosterone hormone mostly produced in damaged prostate. Note that the term "damage" is very relative, but the bigger the damage is the greater the level of DHT will be.
DHT stands for dihydrotestosterone.
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