BUG: Implement interpolating NaT values in datetime series

Closes pandas-dev#11701

doc/source/whatsnew/v0.21.0.txt

@@ -163,6 +163,7 @@ Other Enhancements
 - :func:`Categorical.rename_categories` now accepts a dict-like argument as `new_categories` and only updates the categories found in that dict. (:issue:`17336`)
 - :func:`read_excel` raises ``ImportError`` with a better message if ``xlrd`` is not installed. (:issue:`17613`)
 - :meth:`DataFrame.assign` will preserve the original order of ``**kwargs`` for Python 3.6+ users instead of sorting the column names
+- Implement interpolating ``NaT`` values in ``datetime`` series (:issue:`11701`)
 
 
 .. _whatsnew_0210.api_breaking:

pandas/core/series.py

@@ -19,6 +19,7 @@
     is_integer, is_integer_dtype,
     is_float_dtype,
     is_extension_type, is_datetimetz,
+    is_datetime64_dtype,
     is_datetime64tz_dtype,
     is_timedelta64_dtype,
     is_list_like,
@@ -34,8 +35,10 @@
     maybe_upcast, infer_dtype_from_scalar,
     maybe_convert_platform,
     maybe_cast_to_datetime, maybe_castable)
-from pandas.core.dtypes.missing import isna, notna, remove_na_arraylike
-
+from pandas.core.dtypes.missing import (isna, notna,
+                                        remove_na_arraylike,
+                                        isnull)
+from pandas.core.tools.datetimes import to_datetime
 from pandas.core.common import (is_bool_indexer,
                                 _default_index,
                                 _asarray_tuplesafe,
@@ -2734,6 +2737,14 @@ def from_csv(cls, path, sep=',', parse_dates=True, header=None,
 
         return result
 
+    def interpolate(self, *args, **kwargs):
+        if is_datetime64_dtype(self) and self.isnull().any():
+            s2 = self.astype('i8').astype('f8')
+            s2[self.isnull()] = np.nan
+            return to_datetime(s2.interpolate(*args, **kwargs))            
+        else:
+            return super(Series, self).interpolate(*args, **kwargs)        
+
     def to_csv(self, path=None, index=True, sep=",", na_rep='',
                float_format=None, header=False, index_label=None,
                mode='w', encoding=None, date_format=None, decimal='.'):

pandas/tests/series/test_missing.py

@@ -1218,3 +1218,13 @@ def test_series_interpolate_intraday(self):
         result = ts.reindex(new_index).interpolate(method='time')
 
         tm.assert_numpy_array_equal(result.values, exp.values)
+
+    def test_series_interpolate_nat(self):
+        # GH 11701
+        expected = pd.Series(pd.date_range('2015-01-01', '2015-01-30'))
+        result = expected.copy()
+        result[[3, 4, 5, 13, 14, 15]] = pd.NaT
+        result = result.interpolate()
+        print(result)
+        print(expected)
+        tm.assert_series_equal(result, expected)