So I have a small question about forecasting using an AR(1) model.
I have
$Y_t=4-0.6Y_{t-1}+e_t$ with {$e_t$} as W.N. with $\sigma^2_e=2$
I am asked to forecast $\hat{Y_t}(1)$ for which I am using the following equation...
$$\hat{Y_t}(1) = \mu + \phi*(Y_t - \mu)$$ where $$\phi=-0.6$$ $$\mu = \frac{4}{1-\phi} = \frac{4}{1-(-0.6)}= 2.5$$
my real question is if I am not given the last value in $Y_t$ how do I forecast $\hat{Y_t}(1)$?
What you might do is to develop a theoretical (guessed !) prior probability distribution (frequency distribution) for possible values of Y(t) and create/simulate via Monte Carlo a family of N possible values (Delphi Method). Then use your equation to predict the next value for each of the simulated(possible) values and then generate a histogram of those forecasts. Confidence limits for these forecasts can the based upon the actual distribution of outcomes.
We have recently incorporated that very useful feature into AUTOBOX ( a time series analysis package that I have helped to develop) and are using it to compute probability distributions ala http://probabilitymanagement.org/ for both univariate and causal models where exogenous variables are used. See here https://www.youtube.com/watch?v=-eBkpr3P27M
Why are you using two different equations (one including $\mu$ and another excluding it)? Be consistent. If you are subtracting $\mu$ from $y_{t-1}$ on the right hand side, don't forget to do this also on the left hand side ($y_t-\mu$); otherwise your forecasted value will be biased to the amount of $\mu$.
With regards to multistep forecasts, forecast iteratively and substitute observed values with forecasts where needed. E.g. if you don't know $y_t$, substitute it with its forecast.
