BUG: cov buggy when having NaT in column

[phofl]

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• I have confirmed this bug exists on the latest version of pandas.

• I have confirmed this bug exists on the main branch of pandas.

Reproducible Example

df = DataFrame({
    "a": [1, 2, 3],
    # "b": [pd.Timestamp("1970-01-01 00:00:00.000000001"), pd.Timestamp("1970-01-01 00:00:00.000000002"), pd.NaT],
    "b": [1, 2, np.nan],
    "c": [1, 2, 3],
})
df.cov()

Issue Description

If I understand this correctly, both calculations should be equivalent. The np.nan case returns

     a    b    c
a  1.0  0.5  1.0
b  0.5  0.5  0.5
c  1.0  0.5  1.0

while the NaT case returns

              a             b             c
a  1.000000e+00 -4.611686e+18  1.000000e+00
b -4.611686e+18  2.835686e+37 -4.611686e+18
c  1.000000e+00 -4.611686e+18  1.000000e+00

Both are equivalent without any missing values

Expected Behavior

Same result for both

Installed Versions

main

[jbrockmendel]

1. I think in .cov we do a self.to_numpy(dtype=np.float64).
   a) i think that should raise on a dt64 column
   b) the NaT gets converted to -9.22337204e+18 where it should be np.nan
2. .cov definitely shouldn't work with a dt64 column (as df["b"].var() correctly raises); .corr should be OK

[phofl]

Raising sounds fine.

Timedelta has the same problem.

corr has the same problem for Timestamp and Timedelta, so this needs fixing.

[fbourgey]

@phofl @jbrockmendel
I am thinking of working on this. To sum up, we need to

• raise an error with df.cov() when there is a dt64 column? I don't get why df.corr() should be OK.

[jbrockmendel]

I don't get why df.corr() should be OK

bc .corr() has units of td64 while .cov() has units of td64^2 which isn't a thing.