st: 3 levels of variation, 1 fully nested in the 2nd--which Fixed Effects are needed?

[Jen Zhen <jenzhen99@gmail.com>]

Dear listers,

I would like to explain per-unit price P_ijt contracted between buyer
i and seller j in month t.

There are multiple buyers per seller, but also multiple sellers per
buyer, so i and j are not nested in each other.
By contrast, each buyer i shows up only in one month t, but each month
t contains multiple buyers and the number of buyers varies from month
to month. So if I'm using the terminology correctly, then dimension i
is nested in dimension t.

I explicitly include in my regression seller fixed effects beta_j.
Since there are too many buyers to explicitly estimate a set of buyer
fixed effects, I do this implicitly by substracting from each P_ijt a
buyer-specific mean, i.e. my outcome variable is (P_ijt - pi_i).

Now I am wondering whether I do still need month fixed effects
gamma_t. I think that implicitly including buyer fixed effects (5 for
the first month, 10 for the second, 3 for the third, etc) should make
month fixed effects superfluous, but I am not entirely sure.

Would anyone know?

Thank you so much and kind regards,
JZ
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