Let's say I'm fitting data for example to ARMA(1,0,1)-model:

$x_t = \phi x_{t-1} + \epsilon_t + \theta \epsilon_{t-1}$.

Now I estimate the parameters $\phi$ and $\theta$ and solve some values for them, e.g.

$x_t = 0.7 x_{t-1} + \epsilon_t + 0.8 \epsilon_{t-1}$ (I just made the numbers up)

Let's say this the best model for my problem at hand and I begin forecasting.

So I make the forecast

$x_{t+1} = 0.7 x_{t} + \epsilon_{t+1} + 0.8 \epsilon_{t}$.

Now where I get confused is with the future value of the error term $\epsilon_{t+1}$. How can I solve this value? Should I predict the residuals also or should I use the expected value of $\epsilon_{t+1}$, which should be 0. Hope my question is clear =)

Thank you for any help =)

EDIT:

About the value of $\epsilon_{t+1}$. Should I estimate the value of $\epsilon_{t+1}$ by assuming (as in literature normally is assumed) that the noise process $\epsilon_t$ is normally distributed $\epsilon_t$ ~ $iidN(0,\sigma_{\epsilon}^2)$ and then use estimation techniques (Least squares, Maximum likelihood, Yule-Walker) to estimate the value for noise process variance $\widehat{\sigma}_{\epsilon}^2$ and then just evaluate value for $\epsilon_{t+1}$ ~ $iidN(0,\widehat{\sigma}_{\epsilon}^2)$ from the estimated Gaussian distribution?

Let's say I'm fitting data for example to ARMA(1,1)-model:

$x_t = \phi x_{t-1} + \epsilon_t + \theta \epsilon_{t-1}$.

Now I estimate the parameters $\phi$ and $\theta$ and solve some values for them, e.g.

$x_t = 0.7 x_{t-1} + \epsilon_t + 0.8 \epsilon_{t-1}$ (I just made the numbers up)

Let's say this the best model for my problem at hand and I begin forecasting.

So I make the forecast

$x_{t+1} = 0.7 x_{t} + \epsilon_{t+1} + 0.8 \epsilon_{t}$.

Now where I get confused is with the future value of the error term $\epsilon_{t+1}$. How can I solve this value? Should I predict the residuals also or should I use the expected value of $\epsilon_{t+1}$, which should be 0. Hope my question is clear =)

Thank you for any help =)

EDIT:

About the value of $\epsilon_{t+1}$. Should I estimate the value of $\epsilon_{t+1}$ by assuming (as in literature normally is assumed) that the noise process $\epsilon_t$ is normally distributed $\epsilon_t$ ~ $iidN(0,\sigma_{\epsilon}^2)$ and then use estimation techniques (Least squares, Maximum likelihood, Yule-Walker) to estimate the value for noise process variance $\widehat{\sigma}_{\epsilon}^2$ and then just evaluate value for $\epsilon_{t+1}$ ~ $iidN(0,\widehat{\sigma}_{\epsilon}^2)$ from the estimated Gaussian distribution?

Let's say I'm fitting data for example to ARMA(1,0,1)-model:

$x_t = \phi x_{t-1} + \epsilon_t + \theta \epsilon_{t-1}$.

Now I estimate the parameters $\phi$ and $\theta$ and solve some values for them, e.g.

$x_t = 0.7 x_{t-1} + \epsilon_t + 0.8 \epsilon_{t-1}$ (I just made the numbers up)

Let's say this the best model for my problem at hand and I begin forecasting.

So I make the forecast

$x_{t+1} = 0.7 x_{t} + \epsilon_{t+1} + 0.8 \epsilon_{t}$.

Now where I get confused is with the future value of the error term $\epsilon_{t+1}$. How can I solve this value? Should I predict the residuals also or should I use the expected value of $\epsilon_{t+1}$, which should be 0. Hope my question is clear =)

Thank you for any help =)

Let's say I'm fitting data for example to ARMA(1,0,1)-model:

$x_t = \phi x_{t-1} + \epsilon_t + \theta \epsilon_{t-1}$.

Now I estimate the parameters $\phi$ and $\theta$ and solve some values for them, e.g.

$x_t = 0.7 x_{t-1} + \epsilon_t + 0.8 \epsilon_{t-1}$ (I just made the numbers up)

Let's say this the best model for my problem at hand and I begin forecasting.

So I make the forecast

$x_{t+1} = 0.7 x_{t} + \epsilon_{t+1} + 0.8 \epsilon_{t}$.

Now where I get confused is with the future value of the error term $\epsilon_{t+1}$. How can I solve this value? Should I predict the residuals also or should I use the expected value of $\epsilon_{t+1}$, which should be 0. Hope my question is clear =)

Thank you for any help =)

EDIT:

About the value of $\epsilon_{t+1}$. Should I estimate the value of $\epsilon_{t+1}$ by assuming (as in literature normally is assumed) that the noise process $\epsilon_t$ is normally distributed $\epsilon_t$ ~ $iidN(0,\sigma_{\epsilon}^2)$ and then use estimation techniques (Least squares, Maximum likelihood, Yule-Walker) to estimate the value for noise process variance $\widehat{\sigma}_{\epsilon}^2$ and then just evaluate value for $\epsilon_{t+1}$ ~ $iidN(0,\widehat{\sigma}_{\epsilon}^2)$ from the estimated Gaussian distribution?