I learnt IS-LM model but I got curious whether it is applicable to the real world. I mean it's just an model, so I doubt that the realtionship between Y and r actually exists. Are there any real life examples that prove the model? I want to know if something like a recent world event or fiscal and monetary policies in sone countries actually affects output and interest rate.
There is a huge discussion about structural Macroeconomics models - especially on their theoretical and practical relevance - with IS-LM being on the forefront of the battle some 40+ years ago. Many people simply reject LM existence. Advanced level textbooks ignore this approach for two decades. However it ironically dominates introductory courses.
Here is a nice discussion from Bloomberg that claims that, perhaps, the most instructive paper on the matter was published in QJE back in 1992 - link. (Full text can be downloaded elsewhere.)
From personal experience I could add that IS-LM is a great tool to build initial schematic intuition about the issue in question even though its microfoundations are not clear at all (put it another way, we have no idea why does the model work, but it does).