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I've been looking for prior research by economists that predict the number of house listings in a particular city (or even, say, the probability that an individual in the city will list his house that year).

So, for example, the former would be something like:

$NumListings=1.5*GDP+ 2*income+3*interestRate+unemployment+.3*population$

and the second could be a logit regression like

$ProbListing=\frac{(GDP+1.1*income)}{population}$

Is there any existing literature on this? The only things I have been able to find is literature on people using the number of listings to predict house prices-- I want to know if anyone has used exogeneous data like unemployment, income, consumption, gdp, interest rate, unsold listings, etc.. to predict the number of listings in the future.

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It looks like what you're really after is determinants of housing mobility or migration. You get that, you can then just apply it to a specific city and aggregate to predict. One of the earliest and most seminal works in this field was Rossi's Why Families Move, originally published in '56 and redone in '80. The book is here and a review is here.

Another giant paper in the literature is Massey "Social Structure, Household Strategies, and the Cumulative Causation of Migration". It is also a little older (1990), but brings together a lot of the previous literature (great paper for lit review) and formalizes the decision making process of migration.

Newer papers in the literature look more at specific issues related to relocation, like eviction, fertility and housing decisions, and connection with life-course events.

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