Yeah just make it up. Use the distributions in scipy.stats to write down the data generating process you want to model, generate some random data, then see if you can recover the original values.
I guess it depends what is inside the alpha. I assumed it was just a constant term.
It will model a very specific type of dependency between the time steps, a deterministic linear trend. That is, it will account for the fact that scores go up over time.
I would say your model is quite complex. I would begin with a dirt simple slope-intercept model:
R_{i,t} \sim Poisson (\exp(\alpha + \beta \cdot t))
Pool all the data and just estimate two parameters. Make sure things work the way you expect, and that you aren’t getting -infinity in your predictions. After that works, add in your feature matrix, but only estimate one set of parameters for all the games and all the times, so you go up to k+1 parameters. Then if things are still looking good try to get fancier.
I mean that if I’m watching the match and 1 goal has been scored, if I get up to get another beer, by the time I come back, my best prediction of the number of goals (given that I didn’t see the last couple minutes) is 1. Maybe 2, but definitely not 3, 4, or 5, and certainly not 0!. The score at the previous time step informs my guess at the current time step. I should definitely not make a guess independent of the previous score; it that case I might guess 0 when I get back. That would be a pretty terrible guess.