MSE and time delay relation
The zero time delay seems wrong there. Suppose you have an autoregressive model. A general formula could be:
$$y_t = \alpha + \sum_{i}\beta_i y_{t-i} + w_t$$
Now, if $i$ is zero then you will have
$$y_t = \alpha +\beta_0 y_t + w_t$$
which is really weird. It looks like you predict values at time $t$ from the values at time $t$. In plain English is like you predict the future values from the real values. In that case the error is zero, since it is not very hard to predict something from the real values. So, I think the offsets should start at $1$.