Postgres indexing for timestamp range does not work - postgresql

Question

I have a following table with 1.000.000 rows
create table event
(
id serial
constraint event_pk
primary key,
type text not null,
start_date timestamp not null,
end_date timestamp not null,
val text
);
and I need to execute a following SQL query
EXPLAIN (analyse, buffers, format text)
SELECT *
from event
WHERE end_date >= '2010-01-12T18:00:00'::timestamp
AND start_date <= '2010-01-13T00:00:00'::timestamp;
Please note that end_date in being compared with date which is eariler than that for start_date
The question is what index should I create for such query?
I've tried following one:
create index my_index
on event (end_date, start_date desc);
But it doesn't work, I can see that sequential search is being used
Seq Scan on event (cost=0.00..53040.01 rows=1967249 width=57) (actual time=0.142..149.163 rows=1971694 loops=1)
Filter: ((end_date >= '2010-01-12 18:00:00'::timestamp without time zone) AND (start_date <= '2010-01-13 00:00:00'::timestamp without time zone))
Rows Removed by Filter: 28307
Buffers: shared hit=15762 read=7278
Planning:
Buffers: shared hit=4
Planning Time: 0.127 ms
Execution Time: 201.610 ms
I can not understand why my index is not working, because if we just try following index and query:
create index simple
on event (start_date, end_date DESC);
SELECT *
from event
WHERE event.start_date >= '2010-01-12T18:00:00'::timestamp
AND event.end_date <= '2011-01-13T00:00:00'::timestamp;
indexing works just fine
Bitmap Heap Scan on event (cost=466.91..23418.68 rows=18035 width=57) (actual time=1.954..8.551 rows=15944 loops=1)
Recheck Cond: ((start_date >= '2010-01-12 00:00:00'::timestamp without time zone) AND (end_date <= '2011-01-13 00:00:00'::timestamp without time zone))
Heap Blocks: exact=7694
Buffers: shared hit=7734 read=26
-> Bitmap Index Scan on simple (cost=0.00..462.40 rows=18035 width=0) (actual time=1.314..1.314 rows=15944 loops=1)
Index Cond: ((start_date >= '2010-01-12 00:00:00'::timestamp without time zone) AND (end_date <= '2011-01-13 00:00:00'::timestamp without time zone))
Buffers: shared hit=55 read=11
Planning:
Buffers: shared hit=8
Planning Time: 0.133 ms
Execution Time: 9.025 ms
This query does not do what I need, but I'm just wondering now why this index works for such query, but my index above does not work for query which I need

Answer 1

The answer is right there in the execution plan:
Seq Scan on event (...) (actual ... rows=1971694 ...)
Filter: ((end_date >= '2010-01-12 18:00:00'::timestamp without time zone) AND (start_date <= '2010-01-13 00:00:00'::timestamp without time zone))
Rows Removed by Filter: 28307
the query returns almost two million rows, and the filter only removes 30000. It is more efficient to use a sequential scan than an index scan in that case.