Predicting Value in AR(1) Model

Consider an autoregressive model of order 1, denoted as AR(1), represented by the equation:

$$Y_t = \phi Y_{t-1} + \epsilon_t$$

where:

• $$Y_t$$ = the value of the time series at time t
• $$\phi$$ = the autoregressive parameter
• $$\epsilon_t$$ = white noise error term at time t, which is independently and identically distributed with a mean of zero

Assuming the parameter $$\phi$$ is estimated to be 0.8 from a given time series data, an analyst wants to predict the value of the time series for time t=2 (i.e., $$Y_2$$) given that the observed value at time t=1 (i.e., $$Y_1$$) is 10. What will be the predicted value of $$Y_2$$ using the AR(1) model?