A priori it seems astoundingly unlikely to me that rent prices are causing people to become homeless at the margins. There's a huge gap in means between someone who just barely can't afford rent in SF and someone who lives on the street in SF. The former will move somewhere cheaper rather than living on the street. Almost certainly this effect is mediated by the fact that homeless people choose to move to cities which have high rent (because those cities tend to have better weather, better social services, etc.).