Fitted values of initial observations in auto.arima for non-stationary models

Question

If I understand correctly, the fitted values returned in the auto.arima of the forecast R package are the one-step ahead forecasts given by the model, once the parameters have been estimated (using all the training data).

Suppose that I have fitted a non-stationary ARIMA model, say an airline model $(0,1,1)(0,1,1)_{12}$. In this case, how are the first $13$ fitted values computed? Since we have no information about the initial state of the series, how can we compute a one-step ahead forecast for the first period? Note that the situation is quite different from the stationary case, where we can use the stationary mean and variance as the initial state.

I have seen that auto.arima relies on the arima function from the stats package, so I guess the question could also be rephrased as: how are the residuals for the first periods computed by the arima function in the non-stationary case?

Thank you very much in advance.

Answer 1

Looking in the source code of the arima function, it appears as though the initial conditions are treated as additional parameters to be estimated via Maximum Likelihood.

Consider the AR(1) model as an example: $$ \begin{align*} y_t &= \mu + \phi (y_{t-1} - \mu) + \varepsilon_t \\ \varepsilon_t & \sim \mathcal{N}(0,\sigma^2) \end{align*}$$

The estimated parameters would include $\mu,\phi,\sigma^2$, and also $y_0$.

Edit: It's possible that I'm misreading the source code. If so, someone please correct me!