does it make sense to regress the changes in a bonds YTM against changes in the YTM of a bond index to get som measure of a bonds beta?
The beta concept is rather easy for stocks, but more complex for bonds. We know that stocks tend to grow long term and rise in price, which compensates investors for their risk. So generally investors want a positive beta to the stock market, to capture this long-run price appreciation tendency. For bonds, there is no long-term appreciation observable and the main source of risk is not a bond index, but interest rate risk, therefore no one uses betas to hedge bonds, but duration and convexitiy. (if its about hedging: Alternatively regress your (portfolio) bond return against the return of a bond-futures contract as they are more liquid to e.g. conduct a minimum variance hedge. Also it is important to note that if you do such a regression to take comparabale bonds in terms of credit rating, i.e. if you hedge a corporate bond portfolio with a gilt futures, you are exposed to an increase in the credit spread.)