# Relationship between Data Size and Arima Prediction Interval Width?

When we use Arima model to acquire Interval Predictions, will the width of prediction intervals decrease if we use more data (longer history) to fit the model?

Take an ARIMA(1,0,1) for simplicity: $$$$y_t = \phi_0 + \phi_1 y_{t-1} + \theta_1 \epsilon_{t-1} + \epsilon_t.$$$$ Typically, this is estimated by maximimum likelihood which requires us to make an assumption about the distribution of $$\epsilon_t$$. Most of the time, people pick a Gaussian distribution and impose homoskedasticity, i.e., they say $$\epsilon_t \sim N(0,\sigma)$$.
For simplicity, we'll do the one-step ahead prediction interval, conditional on $$(y_T, \epsilon_T)$$: \begin{align} y_{T+1} | (y_T, \epsilon_T) &\sim N(\mu_T, \Sigma_T) \\ \mu_T &= E_T(y_{T+1}) = \phi_0 + \phi_1 y_T +\theta_1 \epsilon_T \\ \Sigma_T &= var_T(y_{T+1}) = \sigma^2. \end{align}
• Nice answer. Also, note that, assuming the model was correct ( the underlying DGP was ARIMA(1, 0, 1)), then the variance estimate of the error term should converge to its true value as the estimation length gets longer and longer. Using Stephane's example, the error variance estimate should converge to $\sigma^2$. Of course, you never know what the true underlying DGP is, so it's not a terribly useful statement. – mark leeds Mar 26 at 6:04
• While it is true that there is a model selection problem involved, you are obligated to make some assumptions to resolve the problem of characterizing forecast uncertainty. The approach I proposed is the simplest way to answer the question, as well as the most common way to do it in practice. It clearly isn't the most robust way to do it. Moreover, if $|\phi_1|$ is close to 1, we'd need to have a discussion about the pesky prior implicit in the frequentist approach with regards to initial conditions. – Stéphane Mar 26 at 16:38