Query with many date ranges
As a minimum, you should use OR instead of AND, because it is impossible for a date to be simultaneously in February AND January.

SELECT sum(sell) AS sell
  FROM stat 
  LEFT JOIN stat_key ON stat.stat_key = stat_key.id 
 WHERE stat.date BETWEEN '2017-02-16' AND '2017-02-26'
    OR stat.date BETWEEN '2017-01-16' AND '2017-01-26'
    OR stat.date BETWEEN '2016-12-16' AND '2017-12-26';


Now that I see your expected output, I know that you want something like this:

WITH ct(d) AS ( 
  SELECT d::date 
  FROM generate_series('2016-11-26', '2017-02-26', INTERVAL '1 month') AS gs(d)
) 
SELECT COALESCE(SUM(s.sell), 0) AS sell, 
       ct.d AS dt  
    FROM  ct 
    LEFT JOIN stat s ON s.date BETWEEN ct.d + '-10 days' AND ct.d 
    GROUP BY ct.d;


Unfortunately, your expected output cannot be generated by the sample input, so I had to test it with my data. 

Notes: 


generate_series works with timestamps: you can convert them to dates in the main SELECT or the CTE. Other approaches are possible.
It is possible to avoid the CTE (i.e. the WITH at the beginning) and join directly with SELECT generate_series(.... However, I find it more readable this way, given that in this way the report dates are near the top and that is what you will change most often.
LEFT JOIN and COALESCE() let you display results (0) for ranges that have no values at all in the data table.