Oxford MPhil+DPhil Economics - quant research?

That sounds like pure research. I think you will want to be a bit more applied instead of math for math’s sake. Do you think there’s a common heteroskedaticity issue across much of quant research? What if (as is reality) linear regression is just one tool in a much larger toolkit some of which is newer and sexier and probably more of a focus these days? AI?

Not to say that deep math knowledge isn’t important it’s just that you should probably try to stay closer to applications to discuss in an interview

Examples - novel data sets and novel way of crunching those datasets to get a forecast/signal. Slippage models for transaction costs or risk models to inform portfolio construction, optimizing for return solely vs Sharpe vs return subject to risk constraint.

Modeling the half life of forecasts when to retire, when to procedurally tweak and test again. AI - LLM for sentiment, or extraction of unstructured data, or reinforcement learning approaches because the market s just one big game. Novel derivatives or novel approaches to price derivatives or other markets where the underlying is clearly no longer log normal or continuous and discrete in weird ways. Bootstrapping inputs, credit curves, etc from noisy data and estimated the error bars.

It may well be that whatever QR shop you go to has heavy reliance on linear regression where adjustments for hereroskedaticity timely is important and this improves a key tool they use but you need to be not narrowly focused but broadly open to using the various tools to accomplish the end goal which is to make more money more consistently. Just some random examples. IANAQ so not looking to get into a math debate but hopefully you see what I mean by applications