Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. It only takes a minute to sign up.
Sign up to join this community
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: $Y_{t+h} | \mathcal{F}_t \sim \mathcal{N}(Y_t, h\sigma^2)$
