Asset returns and inflation in different inflationary regimes


Currently I am writing my thesis and I am stuck on which statistical model would be good to use. I want look how well the returns of different asset classes comove with the inflation rate, at different levels of inflation. In a simplied form: Asset returns = a + Beta * inflation rate + e. If beta equals 1 the returns of the asset would perfectly comove with the inflation rate. I want to check if the asset classes show a different relation with inflation when the level of inflation differs, in particular three regimes; inflation below 1%, between 1% and 3% and above 3%. Also, it would be nice to be able to make distinctions between short and long run effects. I was thinking of running a threshold vector error correction model, but I am not sure if that would be advisable and how to implement it.

Best, Jelle