# How to measure the sensitivity of a fund to a set of indices?

I'm trying to understand how recently created funds work and ultimately derive a sort of a probability distribution for their future returns, by approximating them with indices and then using the historical data of these indices to construct my distribution, instead of working on the limited data I have on these funds, notably by constructing conditional probability distributions during bear/bull markets. I am aware of the fact that since these funds have limited data (daily NAVs), my approach is biased, but ultimately my question is really about approximating funds with indices.

My first attempt was to use a very basic linear regression, with stepwise selection of explanatory variables. Sadly, even though most of my variables had very low p-values, my f-tests always rejected the null hypothesis, the overall error of the model, while normally distributed, was way too high, for reference my R-squared values were of the order of $$0.3$$ to $$0.5$$, when I used a dozen indices that are supposed to encompass the securities the funds are invested into.

My next intuition was to use a Kalman filter. My only experience with Kalman filters is with Scharz's futures pricing models, where you've got one, two or three joint diffusion processes (GBM, Ornstein-Uhlenbeck etc...) that are updated each day with observations.

I'm not exactly sure how to set up my Kalman filter but my idea was the following:

My hidden state variable $$\mathbf{x}_t$$ is the vector representing the sensitivity of the fund to each index at date $$t$$, $$\mathbf{H}_t$$ the line vector with the returns of my indices at time $$t$$, $$\mathbf{v}_t$$ the observation noise, which gives us our measurement equation

$$\mathbf{y}_t = \mathbf{H}_t\mathbf{x}_t + \mathbf{v}_t$$

So now my question is, how does one find a suitable state transition and covariance matrices? Should I use instead a nonlinear Kalman filter (EKF or UKF)? Are there other methods I should look into?