Imputing fitted values on panel data quite new to statistics and panel regressions, using it for a university project. I'm currently doing research with Mergers and Acquisitions data. For a defined time frame I have several M&A transactions, the respective deal volume, the industry and the country the target firm operates in as well as the year of the transaction. Now, I only have the deal volume for approx. 30% of the transactions. For all the others I'd like to impute the missing deal sizes by constructing fitted values. I need this, since I want to create panel data with country-year-industry pairs.
My exact question is; what is likely to get me the most appropriate fitted values and do I need to consider fixed effects for country, year, industry or is it easier to just construct/ generate fitted values for the entire data set? Or am I completely wrong and need to do a different kind of thing to impute the missing deal volumes?
My data looks similar to this:

 A: I see two approaches that you can take for your problem: (1) a model-based and a (2) purely predictive approach. Depending on which one you choose, keep in mind that imputing values with either approach introduces its underlying assumptions, which affects the validity of your consecutive analysis.
The model-based approach (1)
This approach requires a theoretical model that causally explains deal volume using variables as e.g. collected in your dataset. You might find such models in the financial or economic literature. Depending on how good the model is and how good you can implement it (do you have the required data at hand?), this approach has the benefit that you can argue that your imputed deal volume then reflects realistic values. However, as theories often are simplifications of the real world, the imputed values are only realistic to the degree to which the simplifying assumptions apply to the real world.
Specifically with regards to fixed effects: their use will depend on the theoretical model.
To sum up: a model-based approach can yield theoretically valid imputations, to the degree that the model is valid. The implementation will likely be non-trivial, depending on the available data.
The purely predictive approach (2)
Without a theoretical model, you can resort to finding the best predictions of deal volume for your sample by using any sort of predictive method. The choice is vast here, from panel-regression with different fixed effects up to machine-learning type methods. Imputing values in such a way introduces the assumption that the best-fitting predictions (i.e. with the lowest deviation from the true values in your given sample) are valid for you analysis (which they might not be). This approach can be more simple to implement, since you can resort to just using the data you have without worrying about other variables whcih might be required by a theoretical model.
Specifically with regards to fixed effects: here you would use fixed effects if they reduced the prediction error
To sum up: the purely predictive approach can be simpler to implement (given your data), but there is no real reason why the imputed values are valid.
