Depends which data you have available. In general, difference in pricing between private and publicly traded bonds is likely to be not only due to difference in liquidity, but also due to the difference in credit risk. You could set up some pricing model, and see which factors are relevant.

I agree with Peter.

One potential solution, that is costly in terms of obtaining and arranging the data, would be to compare same credit risk, same maturity publicy traded and private debt.

Even in that case, the private debt is not traded, so you will not have a market price to compare, but you could restrict your analysis to new issuances. So, lets say that a company is issuing private debt, and at the same time, that company has some bonds traded in the market. At the moment of issuing the new private debt, if the credit risk is the same (no covenants added, or other guarantees, or even a different pay off structure) you could assume that the difference between the yield offered by the new (private) debt and the yield quoted for the bonds is a liquidity premium.

But if you want to create a top down approach, taking data for public databases on traded and private debt, you will find a lot of issues. Companies will probably be different, there will be different average credit risk ratings, even different average maturities or durations, different covenants (it is tipical that private debt includes covenants or further guarantees in one way or another, but that is not necessary present in a public issue). So taking a top down approach requires a lot of work going into the details of the actual differences due to actually different debt structuration (apart of the obvious liquidity premium). That top down approach will take time and require some, maybe a lot of, assumptions.

Best

Carlos

I am actually thinking of writting economeric model on bond pricing using multi factor model or the arbitrage pricing theory of Ross. If you got time you can contact me and we can probably co author a paper on this topic.