I am using IE 9 for windows 7. I want to calculate my residency by using the calculator on their own web page, when I press the date button it doesn't working but in Google chrome it is working. What is the problem in IE 9.

If it works in chrome then use it there rather than wasting time with IE.
Chrome is a far superior web browser anyway.

I solved my problem by changing compatiability view setting in IE 9

IE unfortunately is a very sensitive browser. The best solution is to simply switch to a more reliable and stable browser such as Google Chrome, and that will be the end of your problems.