BUG: Override mi-columns in to_csv if requested (pandas-dev#18110)

Previously, MultiIndex columns weren't being
overwritten when header was passed in for to_csv.

Closes pandas-devgh-5539

Author: gfyoung

doc/source/whatsnew/v0.21.1.txt

@@ -79,6 +79,7 @@ I/O
 
 - Bug in class:`~pandas.io.stata.StataReader` not converting date/time columns with display formatting addressed (:issue:`17990`). Previously columns with display formatting were normally left as ordinal numbers and not converted to datetime objects.
 - Bug in :func:`read_csv` when reading a compressed UTF-16 encoded file (:issue:`18071`)
+- Bug in :meth:`DataFrame.to_csv` when the table had ``MultiIndex`` columns, and a list of strings was passed in for ``header`` (:issue:`5539`)
 
 Plotting
 ^^^^^^^^

pandas/io/formats/format.py

@@ -1695,7 +1695,7 @@ def _save_header(self):
             else:
                 encoded_labels = []
 
-        if not has_mi_columns:
+        if not has_mi_columns or has_aliases:
             encoded_labels += list(write_cols)
             writer.writerow(encoded_labels)
         else:

pandas/tests/frame/test_to_csv.py

@@ -1203,3 +1203,16 @@ def test_period_index_date_overflow(self):
 
         expected = ',0\n1990-01-01,4\n,5\n3005-01-01,6\n'
         assert result == expected
+
+    def test_multi_index_header(self):
+        # see gh-5539
+        columns = pd.MultiIndex.from_tuples([("a", 1), ("a", 2),
+                                             ("b", 1), ("b", 2)])
+        df = pd.DataFrame([[1, 2, 3, 4], [5, 6, 7, 8]])
+        df.columns = columns
+
+        header = ["a", "b", "c", "d"]
+        result = df.to_csv(header=header)
+
+        expected = ",a,b,c,d\n0,1,2,3,4\n1,5,6,7,8\n"
+        assert result == expected