Timing in State Space models

[niskrev]

I noticed a timing inconsistency in the Making a Custom Statespace Model.ipynb notebook.

To be consistent with the general form in (1)-(2), the ar(3) model should be:

$$x(t) = \rho_1 x_{t-1} + \rho_2 x_{t-2} + \rho_3 x_{t-3} + \varepsilon_{t-1}$$

instead of

$$x(t) = \rho_1 x_{t-1} + \rho_2 x_{t-2} + \rho_3 x_{t-3} + \varepsilon_{t}$$

Alternatively, in (1) instead of $\varepsilon_{t}$ it should be $\varepsilon_{t+1}$. I realize that this doesn't matter in many cases, but in some models there is a big difference between the new state being driven by the same period innovation or the one from the last period. I apologise if this is explained elsewhere in the documentation, I have only looked at this notebook.

[jessegrabowski]

You're absolutely right; this is a typo in the notebook. If you want to open a PR to fix it I'd appreciate it. Equation (1) ought to have $x_t$ on the left-hand side and $x_{t-1}$ on the right.

[niskrev]

Thank you @jessegrabowski. I created a pull request with these changes.

Also, I want to thank you for the great work you are doing!