Suppose that the Federal Reserve had raised interest rate by 0.25% last week 17Sep2015. What is an estimated rise in the interest rise of the 10-year Treasury? Which futures contract should one use to make this estimation? How does one calculate the estimation?
Not sure this is a quantitative finance question, since it's more or less a judgment call.
There is no futures contract that you can use to make this estimation; instead, it requires an understanding of the Fed, what's going on in the economy, what's priced in by the market, what's the positioning profile of different players, etc.
Assuming the Fed hiked, it's not even clear that 10-year Treasury yield would necessarily rise. It may very well be the case that the market becomes so concerned about future growth potential that yield ended up declining.
If you do want to depend on quantitative methods, the best you can do is to look at how much yields moved during historical hikes. Even this is not reliable. The economic cycles differ; how much of the hike was priced prior to the decision being announced also make a big difference.
One thing you could do is look at the fed funds futures curve and look at what maturity has a 25bps hike priced in as a certainty. Then you could look at the 10y future that corresponds to that maturity date and present value it. The difference between that calculated value and the prompt future could be a decent theoretical estimate. I agree with haginile though, this would mainly be a theoretical exercise as there are too many other variables that would change in the event of a surprise hike.
There are very plausible investment situations where large local or global money managers are obliged to, say buy ten year bond notes with sufficient demand to substantially lift the price when concurrently the economic discussion at an FOMC meeting results in a monetary policy hike.
The question raises a tricky comparison of overnight indexing rates (Fed Funds) with a ten year investment instrument. Clearly the participants' objective in each market are different.
You may prefer to look at ten year IRS, potentially ten year OIS-indexed IRS where the underlying floating rate is sufficiently close to (or proxied by) Fed Funds rates.
Try a statistical regression of ten year note yields against fed fund rates. Add an instrument or two, if linear regression, and I'd suggest 2y10y curve as a proxy for the economic regime and USD FI fund inflows or M3 money supply to account for investment flow.