I understand that estimation bias occurs when there is simultaneity between the dependent and independent variable, but I’m not sure if I understand completely how to identify it. I have been reading Introductory Econometrics (Wooldridge) where there are interesting examples of how to identify and deal with simultaneous equation models when the market clears or when a company has a monopolistic position.
Since I cannot find an example of simultaneity when an individual is trying to maximize profit, I proposed the following problem, but I’m not sure if my analysis is right.
The problem
Suppose I have a house that I rent by the week, and I want to find the price that maximizes my profit. In the last ten years, I have been tracking the price I was asking every week and whether the house was rented i.e. I have a record of 52x10 weeks. With this information, I can run the following simple model to estimate the probability my house will be rented at a price and then use this information to maximize my profit:
$$rented_i = \beta_0 + price_i \beta_1 + dm_2 m_{i2} +\dots + dm_{12} m_{i12} +u_t$$
where
$i \in [1\dots 520]:$ weekly observation for the last ten years
$rented_i:$ 0 = not rented, and 1 = rented
$price_i:$ asking price
$m_{i2} \dots m_{i12}: $ month of the year dummy
The questions
a) If I randomly set the price, I understand that $price$ variable is exogenous as I don’t adjust it based on previous occupancies. Is this correct?
b) If set the price based on previous occupancies, would I still be able to consider price exogenous? If I run OLS, I’m incurring in simultaneity bias as $price$ -> $rent$ -> $price$. Is this correct?
c) If in (b), $price$ is endogenous, what can be a useful IV? Is it valid to use the rent price of my friend’s house as an instrumental variable for the $price$ variable?