Prediction Intervals (PI)

A prediction interval is a range for a future observation. In the definition given by Hyndman and Athanasopoulos, it “gives an interval within which we expect \(y_t\) to lie with a specified probability.” Their assessment of why it matters is blunter than most: “point forecasts can be of almost no value without the accompanying prediction intervals.”

The contrast with a confidence interval is the usual starting confusion, and it is a difference of target rather than of method.

• Confidence interval: “I am 95% confident the mean weight is between 68 and 72 kg.” About a population quantity — see confidence intervals.
• Prediction interval: “I am 95% confident the next person’s weight will be between 55 and 85 kg.” About one future outcome, and therefore wider, because it carries the spread of individuals as well as the uncertainty in the estimate.

A symmetric interval under distributional assumptions

For an approximately normal forecast distribution, a symmetric interval uses a central forecast, a coverage multiplier, and an estimated forecast standard deviation. This is one construction, not the definition of every prediction interval; asymmetric or discrete distributions can instead use lower and upper quantiles.

\[\hat{y}_{T+h|T} \pm c\,\hat\sigma_h\]

Here \(\hat\sigma_h\) is the estimated standard deviation of the \(h\)-step-ahead forecast distribution and \(c\) “depends on the coverage probability.” Assuming normality the multipliers are the familiar ones: 1.28 for 80%, 1.64 for 90%, 1.96 for 95%, 2.58 for 99%.

For ordinary linear regression at a point \(x_0\) the same structure appears with an extra term:

\[\hat{y}_0 \pm t_{1-\alpha/2,\,n-p}\cdot\hat\sigma\sqrt{1 + h_0}\]

where the t critical value is the positive \(1-\alpha/2\) quantile with \(n-p\) degrees of freedom under the normal, independent, constant-variance linear-regression error model, and \(h_0 = x_0^\top(X^\top X)^{-1}x_0\) is the leverage of the new point. The \(1\) is what makes it a prediction interval rather than a confidence interval: it is the variance of the individual future observation, which a confidence interval for the mean does not include.

Keep the two quantities apart when reading any of this. One standard deviation is not a 95% interval, and a “±” figure with no stated coverage level says nothing about coverage. The multiplier is doing visible work here, and it is often dropped in summaries.

How far ahead: usually wider, not always

The textbook statement is careful, and the qualifier is the interesting part: “a common feature of prediction intervals is that they usually increase in length as the forecast horizon increases… That is, \(\sigma_h\) usually increases with \(h\) (although there are some non-linear forecasting methods which do not have this property).”

So the width comes from how errors propagate in the particular model, not from the calendar. The four benchmark methods make the differences concrete — with \(\hat\sigma\) the residual standard deviation, \(m\) the seasonal period, and \(k\) the integer part of \((h-1)/m\):

Method	\(h\)-step forecast standard deviation	Behavior with horizon
Mean	\(\hat\sigma\sqrt{1 + 1/T}\)	Flat — no \(h\) in it at all
Naïve	\(\hat\sigma\sqrt{h}\)	Grows without bound
Seasonal naïve	\(\hat\sigma\sqrt{k+1}\)	Flat within a season, steps up at each full period
Drift	\(\hat\sigma\sqrt{h(1 + h/(T-1))}\)	Grows faster than the naïve method

The first row is the clean counterexample to “further ahead means wider.” Forecasting with the mean of independent observations at \(T = 100\) gives a forecast standard deviation of \(\sqrt{1 + 1/100} \approx 1.005\hat\sigma\) at every horizon — a 95% interval of about \(\pm 1.97\hat\sigma\) at one step and at twelve steps alike. The naïve method at twelve steps, by contrast, is \(\sqrt{12} \approx 3.46\) times its one-step width. Note also that “when \(h = 1\) and \(T\) is large, these all give the same approximate value \(\hat\sigma\)” — the methods only separate as you look further out.

The same distinction appears in ARIMA models, where it is governed by differencing: “for stationary models (i.e., with \(d = 0\)) they will converge, so that prediction intervals for long horizons are all essentially the same. For \(d \ge 1\), the prediction intervals will continue to grow into the future.”

Two practical consequences. State the horizon with the interval — 500 units next week, 95% PI [420, 580] is a claim about one week, and a single interval reported for a whole forecast path hides whatever the real pattern is. And derive the width from the model rather than assuming a shape, because “it must be wider further out” is true of some models and false of others.

What the interval does not cover

This is where prediction intervals are most often oversold, and the authors of the standard reference say it about their own method: “as with most prediction interval calculations, ARIMA-based intervals tend to be too narrow.”

Their explanation lists exactly what is left out. “This occurs because only the variation in the errors has been accounted for. There is also variation in the parameter estimates, and in the model order, that has not been included in the calculation. In addition, the calculation assumes that the historical patterns that have been modeled will continue into the forecast period.“

Read the last clause carefully, because it is the one that matters for business forecasting. A competitor entering, a pricing change, a supply disruption, a pandemic — these are the “future shocks” an interval is casually assumed to include, and they are changes to the pattern rather than draws from the error distribution. An interval estimated from a stable past does not quantify a change the past did not contain: if the relationship shifts, the interval is wrong rather than merely narrow.

The qualifier is worth stating, though, because uncertainty that has been modeled explicitly is inside the interval. A model with price as an input carries that input’s uncertainty if you propagate it; a scenario forecast reports a separate interval per assumed future; an intervention or regime-switching model represents the breaks it was built to represent. So the honest claim is conditional on the model and the scenario — not that structural change is universally excluded, and not that it is covered.

The residual assumptions, and checking them

Multi-step calculations “assume that the residuals are uncorrelated,” and the interval machinery generally assumes they are “uncorrelated and normally distributed. If either of these assumptions does not hold, then the prediction intervals may be incorrect.” The recommendation that follows is a procedure rather than a caveat: “always plot the ACF and histogram of the residuals to check the assumptions before producing prediction intervals.”

Autocorrelation is the common violation in real series — see temporal autocorrelation — and ignoring it can distort coverage in either direction. Where normality is the problem rather than correlation, there is a direct remedy: “when a normal distribution for the residuals is an unreasonable assumption, one alternative is to use bootstrapping, which only assumes that the residuals are uncorrelated with constant variance.” Uncorrelated residuals need not be independent; ordinary residual resampling needs stronger conditions such as suitably iid innovations, while dependence may call for block or model-based resampling. See bootstrap.

And then measure the coverage instead of trusting the label. Hold out periods, count how often the actual value fell inside the interval, and compare with the level claimed. A “95%” interval that contains the outcome 70% of the time is a finding you can act on; the number on the label is not evidence about the number in practice. Scoring the whole distribution rather than one interval is the more complete version of the same check — see probabilistic forecasts and forecast error.

Intervals from quantiles instead

The other route does not go through a standard deviation at all: predict the quantiles you want as bounds. For a continuous calibrated distribution, the 0.05 and 0.95 quantiles give a 90% interval; discrete distributions can have greater coverage at those bounds. As a numerical example, bounds of 420 and 580 give [420, 580]. This assumes nothing about the shape of the distribution, which is the appeal, and it requires a method that estimates quantiles directly. See quantile regression and quantile forecasts.

A worked reading, to close. A house-price model predicts \$300,000 with a 95% prediction interval of [\$250,000, \$360,000]. The correct interpretation is about a future observation: another house with these features would fall in that range 95% of the time if the model and its residual assumptions hold for that house. It is not a range for the average price of such houses — that is a narrower confidence interval — and it says nothing about a market that has moved since the model was fitted. Compare with a point forecast, which carries none of this and reads as certainty.

References

• Rob J Hyndman & George Athanasopoulos, Forecasting: Principles and Practice (3rd ed.), §5.5 “Distributional forecasts and prediction intervals”
• §9.8 “Forecasting”