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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?

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Take an ARIMA(1,0,1) for simplicity: \begin{equation} y_t = \phi_0 + \phi_1 y_{t-1} + \theta_1 \epsilon_{t-1} + \epsilon_t. \end{equation} 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}

In practice, you will substitute the MLE estimates for the parameter values. Given the known distribution, you can build prediction interval. In other words, you will neglect the uncertainty due to the fact that parameters are estimated and not known. In other words, the prediction interval isn't a function of sample size, although in practice the fact that you rely on an asymptotic argument to replace parameter values with their MLE estimates does mean that it would bounce around if you added or subtracted observations from your sample.

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  • $\begingroup$ 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. $\endgroup$ – mark leeds Mar 26 at 6:04
  • $\begingroup$ 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. $\endgroup$ – Stéphane Mar 26 at 16:38
  • $\begingroup$ I totally agree that your example was good for illustration and that estimating these things can be quite messy for sure. $\endgroup$ – mark leeds Mar 26 at 18:38

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