I am working on a panel dataset for a project which comprises 5 survey rounds. I have test score data for 4 of these rounds (2-5), so I am omitting the first round and running a regression using panel random effects on the remaining rounds.

My main regression is: Yit= B0 + B1Zi + B2ROUND + BiZi*ROUNDi (1)

Where Round is a set of time dummies with round 2 as the baseline (and round 1 omitted)- 3, 4, and 5. Zi is a continuous variable. The interpretation of B1 is the expected increase in Y due to a unit increase in Zi in Round 2, and the interpretation of the interaction coefficients seems to be the 'additional' effect of a unit increase in Zi in Rounds 3-5, c.f. the base round of round 2, on Y.

This interpretation, whilst making sense, is somewhat harder to interpret from regression tables alone. I was hoping to revise my regression such that, the interpretation on a coefficient would be instead 'The effect of increasing Zi on Yit in round 3' as opposed to the 'additional effect of round 3 vs round 2 '.

I have thus created the following regression:

Yit = B0 + B1Round + BiZi*Roundi (2)

Where Round, as before, is a set of time dummies with round 2 as the baseline.

My question is, I think that the interpretation of Bi in this case is slightly different in equation 2 vs equation 1. From what I understand, the coefficient on Zi*Round3 in equation (2) would be the effect of a unit increase in Zi on Yit if the round is equal to 3, as opposed to the additional effect. Could someone confirm if this intuition is correct?

I have also run these two separate regressions in STATA and the results seem to indicate that my intuition is correct, however I am a little confused on why the interpretation of the equations changes between (1) and (2). If someone could provide some assistance in regards to the interpretation of these two effects from a STATA perspective, that would be very useful.