Confusion about error term in ARMA-model when predicting future values

Question

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?

Answer 1

So I make the forecast

$x_{t+1}=0.7x_t+ϵ_{t+1}+0.8ϵ_t$.

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

Yes, you are right, but you mean an ARMA(1,1) or ARIMA(1,0,1). So you want to calculate the one-period forecasts of an ARMA(1,1), which is the same as an ARIMA(1,0,1).

the one-period forecast is given by: $\hat{x}_{t+1|t}=E(\phi_1x_t + \delta + \epsilon_{t+1} + \theta \epsilon_t|x_t,...,x_1)$ In your case it seems, that you have no constant term?

So this gives in your case

$\hat{x}_{t+1|t}=E(0.7 x_t + \epsilon_{t+1} + 0.8 \epsilon_t|x_t,...,x_1)$

Since $E(\epsilon_{t+1})=0$ this gives

$\hat{x}_{t+1|t}=0.7x_t+0.8\epsilon_t$

About the value of ϵt+1. Should I estimate the value of ϵt+1 by assuming (as in literature normally is assumed) that the noise process ϵt is normally distributed ϵt ~ iidN(0,σ2ϵ) and then use estimation techniques (Least squares, Maximum likelihood, Yule-Walker) to estimate the value for noise process variance σˆ2ϵ and then just evaluate value for ϵt+1 ~ iidN(0,σˆ2ϵ) from the estimated Gaussian distribution?

I am not sure: You mean you want to calculate the "variance" of the forecast to get prediction intervals? This is done by calculating the mean squared error. To do the following you have to know the principals of time series analysis, I just give a short summary and application in your case:

The forecast error is given by

$MSE(\hat{x}_{t+I|t})=E((x_{T+I}-\hat{x}_{T+I|T})^2)=(1+\Psi_1^2+...+\Psi_{I-1}^2)\sigma_\epsilon^2$

The $\Psi$ are obtained as follows:

$a(L)x_t=b(L)\epsilon_t$

In your case of an specific ARMA(1,1): $(1-0.7L)x_t=(1+0.0.8L)\epsilon_t$ This can be written as (look at a time series book):

$(1-0.7L)x_t * (1+ \Psi_1L+ \Psi_2L^2+...)=1+0.8L$

you can solve this and obtain the $\Psi$ values. This is not necessary in this case, since you only need the MSE of the one step ahead prediction. So $MSE(\hat{x}_{t+I|t})=E((x_{T+I}-\hat{x}_{T+I|T})^2)=(1+\Psi_1^2+...+\Psi_{I-1}^2)\sigma_\epsilon^2$ reduces in case of $I=1$ to $MSE(\hat{x}_{t+1|t})=\sigma_\epsilon^2$

the $\sigma_\epsilon^2$ can be obtained from your output in R/STATA or whatever you use.