Fixed effects IV regressions for demand elasticity estimation

[Tomas FNog]

I am trying to estimate demand elasticity using Hausman type instruments. Hausman type instruments have been used in the literature to identify demand elasticities (example of studies using Hausman type instruments: Hausman (1996) and Nevo (2001)).

The key idea is to use prices in other markets to identify the demand.

I have a dataset the looks like the following (the actual dataset is much bigger this is a minimal working example).

There is one firm/retailer that sells multiple products (identified through Product_id), in multiple dates (Date_id). Then I also have price and quantity of each product (Price and Quantity) and then the price and quantity of the same product in another market (Price_other and Quantity_other). Quantity and Price data are in logs.

.

Product_id	Date_id	Price	Quantity	Price_other	Quantity_other
1	1	1.0	7	1	5
1	2	1.0	8	3	7
1	3	1.0	7	1	5
1	4	2.0	10	1	7
1	5	2.0	7	2	10
1	6	2.0	8	2	8
2	3	3.0	8	2	7
2	4	4.0	8	2	5
2	5	4.0	8	3	7
2	6	4.0	9	2	6
2	7	4.0	5	4	8
3	2	2.0	9	2	7
3	3	2.0	10	1	5
3	4	2.0	5	4	8
3	5	2.0	5	4	10
3	6	3.0	5	4	9
3	7	3.0	6	2	7
3	8	3.0	5	4	8
3	9	1.0	7	1	10

.

I am trying to estimate an average demand elasticity for for this retailer using the price in other markets as instruments. To do so, I am running the following in stata:

xtset Product_id Date_id
xtivreg Quantity (Price = Price_other), fe

Is this doing what is supposed to do? I.e. estimating an average elasticity for the retailer using prices in other markets as instruments (and including fixed effects at the product level)?

Thanks,
Tom