panel data and price effect


I am using a panel data (N firms are observed over T years).

My dependent is the gross-margin and I would like to estimate the impact of the volatility of prices on it, using a set of other explanatory variables and a vector of time dummies to capture trends and other unobserved macro effects.

My problem:
In each time period, as the volatility of prices is the same for all the N firms, there is no between firms variation. Therefore, there will be perfect collinearity, right?

Can we say that the variable "volatility" is simply going to add up to the time dummies?

In anyway, does it means that there are no ways of estimating the effect of the volatility of prices on firm with a random effect estimator or a fixed effect estimator?

Ideas of solutions:
- A friend of mine suggested that I use simply interaction variables: I could interact one of the explanatory variable with the variable volatility. The interpretation would be, for instance, "keeping the size of the firm constant, a 1% increase in volatility leads to a x% increase in gross-margin".

However, my opinion is that we have the same problem: no variation across the sample in each time period. Therefore, the interaction variables wouldn't capture anything and would only bring problem of perfect collinearity.

- Another solution would be to run N times series regressions, but my T is quite small (between 10 and 12 years).

I would be glad to hear what to you think of all that and if you know any techniques to estimate the impact of an undifferentiated variable such as price with a panel data.

Best regards,

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