I have a time series where $y_t > y_{t-1} 95\%$ of the time. It’s autocorrelation function is such that it’s decreasing (yt is most correlated with 1 lagged period. This is obvious from looking at it since it looks as if $y_t = y_{t-1}+$ (a small positive value) most of the time).
I’m looking however to model these shocks, $y_t-y_{t-1}$. This process has little correlation with lagged periods. It’s mean is slightly positive and most mass is from $[0,\mu]$. However I can see once it goes above mu, it tends to stay there for about 3 periods.
In addition to $y_t$, I also have $X_t$, a possible prediction vector. Is it best to just model $y_t- y_{t-1}$ as an ARMA with $\beta^T X_t$ thrown in there? Is there a wiser way I could think about this problem? How can I model when $y_{t}-y_{t-1}$ (the shock) will be small or large?
Follow up: If I had to model $y_t$ for $t \in [T+1,\dots,T+k]$ when I am only given information up to time $T$, how can I generalize? Since I'd also have to project $X_t$ as well here.