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I want to get the raw number of units of housing stock that there is in a particular city in month X of year Y (even yearly data would be fine). This place shows yearly stock from 2010 onwards. Here shows data for every 10 years. This link

www(dot)washingtoncitypaper.com/blogs/housingcomplex/2014/10/24/d-c-s-housing-stock-in-charts/ 

makes me think there should be more yearly data since this has 2013 data.

Is there anywhere I can find a more comprehensive data set?

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The US Census collects housing data in multiple instruments. The one I'm most familiar with is the American Community Survey (ACS). ACS data is released in 1-year and 5-year tabulations (and until 2013, in a 3-year tabulation). In order to mitigate sampling error, the 1-year release only covers geographies with population >= 65,000, which means that it only has "city-level" data (Census summary level "place" or 160) for about 2% of US cities. At the metro level, the 1-year sample has a bit over half of the "core based statistical areas" (CBSA, a catch all for metropolitan and micropolitan areas). So if you can deal with those constraints, you could consult table B25001 for all CBSA in the US, ACS2014-1 year.

There is another product with which I'm less familiar, the American Housing Survey. The AHS is conducted every two years and includes data down to the metropolitan level. Here is the download page for the 2013 AHS metropolitan summary tables

Base housing units are also counted for the decennial census, of course, but since timely data is part of your request, that's just a footnote.

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