Weights

[mattwigway]

Add support for weights - this will likely be important when performing scenario analysis as it affects how much demand there is for different types of housing units.

[mattwigway]

One concern is that the equilibration process is based on the assumption that maximum likelihood estimates should perfectly predict market shares, but that's no longer true, I think, with weights, unless I weight the likelihood function too.

Or maybe I should just use a bootstrap sample from the PUMS.

[mattwigway]

Tra (2007) appears to have used the weights - see page 104. But that doesn't mean it was right.

[mattwigway]

I think we actually can use weights. We just don't use them during the estimation process. Ben-Akiva and Lerman (1985, left the book at home so I don't have the page number but will add later) demonstrate that if the weights are conditional on the choices, all parameters except the constants are recovered correctly (I wonder if this extends to the Box-Cox parameter for the outside good). The constants can be recovered using an additional step where you subtract ln (unweighted proportion of choice in sample) / (weighted proportion of choice). This is equivalent to the step we use to find the ASCs in the estimation (eq. 16 of Bayer), because the unweighted proportion of the choice in the sample is the sum of the predicted probabilities after the ASCs have been found during estimation.

I think considering our sample to be choice-based wouldn't be the worst thing in the world, since many of the weighting variables are probably correlated with geography.

So when we find the "full" ASCs, we just need to find weighted full ASCs, and similarly when clearing the market.