# Prediction Intervals the Same in Multi-Step with R Forecast Package - Why?

Using the R forecast package for a multi-step time series forecasting exercise, I've noticed that sometimes I get the same prediction interval at each step (6 months, in my case) and sometimes, I get what would be more along the lines of my expectation -- a widening PI the further into the future I go - to account for the increasing uncertainty.

Why would a PI remain constant in a multi-step ts forecast?

Prediction intervals do not necessarily widen at longer horizons.

For example, take a model like:

$$Y_t = f(t) + \varepsilon_t, \quad \varepsilon_t \sim^{\text{iid}} \mathcal{N}(0,\sigma^2)$$

for an arbitrary (deterministic) function $$f$$.

Then $$Y_{t+h} | \mathcal{F}_t \sim \mathcal{N}(f(t+h), \sigma^2)$$ and prediction intervals have constant width. If you have a white noise model (i.e. $$f(t) = \mu$$), then the whole predictive distribution is the same for every horizon.

This could make sense for example when you have an accurate physical model for $$f(t)$$ and independent random measurement error at each time.

Prediction intervals widen when uncertainty "accumulates" over time, like for example in a random walk model:

$$Y_t = Y_{t-1} + \varepsilon_t$$

So that more and more shocks $$\varepsilon_{t+i}$$ are included at longer horizons:

$$Y_{t+h} = Y_t + \varepsilon_{t+1} + ... + \varepsilon_{t+h}$$

And intervals will widen: $$Y_{t+h} | \mathcal{F}_t \sim \mathcal{N}(Y_t, h\sigma^2)$$