Data: pooled cross-sectional data. 6 countries. 3 years of cross sectional data (10 year gaps in between), with different samples of individuals in each year. The data are from three periods: 'before', 'during', and 'after' a shock.

Response: relative rainforest area
Main predictor: profits from logging industry

Goal: to estimate how the relationship between logging profits and rainforest area changes over time, while accounting for time-invariant country variables.

model (omitting subscripts for simplicity):

rainforest = b0 + b1 year2 + b2 year3 + b3 logging_profits + b4 (year2 * logging_profits) + b5 (year3 * logging_profits) + b6 country2... + b10 country6 + e

Question: including country dummies with panel data would yield estimates of b3, b4, b5 that hold constant any stable within-country factors. But my data are cross-sectional at 3 time points. Does the interpretation of the coefficients differ in this case? If so, how?