PhD problem: a new student has the same idea as me I am in second year of my PhD. I have a plan for my second paper, I did a full scale literature review etc. But then I put it on hold to finish my first paper.

A new student started this term and is coincidentally pursuing the idea for my second paper!

I highly doubt we can both have the same paper. 

I know that co-authoring is a possibility. But, is there grounds for me to argue that she should find a new topic, instead of me. As it will mean that I wasted a large chunk of my first year on this topic. 

We plan to use data for the same country, which means that we will ultimately end up with the same statistical model, because the data available is not the best, so there is not much leeway on what model I use. 

From what the student has told me, her statistical plan (is the same as mine), which is also the popular method in the literature field.
You should probably talk to your advisor and possibly arrange a meeting with the other student, her advisor, your advisor and yourself. You need to sit together and discuss how your work and her work are going to differ, because nobody has anything to gain from two competing PhD students undermining each other within the same research group. Once you have fixed boundaries, you will be free to go your own way and collaborate on the common parts, which will benefit both of you.