Time Series Forecasts and Intervals

A point forecast without an interval is half an answer. The exam wants both the predicted value and the band of plausible outcomes, and that band widens as you forecast further ahead.

The h-step-ahead forecast is the conditional expectation $\hat{Y}_{T+h} = E[Y_{T+h} \mid Y_T, Y_{T-1}, \dots]$. Build it by writing the model recursion at time $T+h$, then replacing every term with what you know.

For an AR(1) model $Y_t = c + \phi Y_{t-1} + \varepsilon_t$ with $\varepsilon_t$ white noise of variance $\sigma^2$:

$$\hat{Y}_{T+1} = c + \phi Y_T, \quad \hat{Y}_{T+h} = c + \phi \hat{Y}_{T+h-1}$$

For a random walk $Y_t = Y_{t-1} + \varepsilon_t$, every future forecast equals $Y_T$. The chain never decays because $\phi = 1$.

For a random walk with drift $Y_t = \delta + Y_{t-1} + \varepsilon_t$, the h-step forecast is $Y_T + h\delta$. The drift accumulates linearly.

For an MA(1) model $Y_t = \mu + \varepsilon_t + \theta \varepsilon_{t-1}$, only the one-step forecast carries a non-trivial shock term: $\hat{Y}_{T+1} = \mu + \theta \hat{\varepsilon}_T$. For $h \geq 2$ the forecast collapses to $\mu$ because no observed shock survives the lag structure.

Common mistakes

• Forgetting that future errors are zero in the point forecast. Plugging a non-zero value for $\varepsilon_{T+h}$ double-counts and biases $\hat{Y}$. Set future shocks to 0.
• Using $\sigma^2$ as the variance at every horizon. That works only at h = 1. For AR(1) at h = 2, the variance is $\sigma^2(1 + \phi^2)$, not $\sigma^2$.
• Forgetting that random-walk variance grows linearly. A 10-step horizon has variance $10\sigma^2$, not $\sigma^2$. Many candidates write the same interval at every horizon.

Bottom line

• Point forecast = conditional expectation given history. Use the fitted recursion: plug in known values, set future shocks to zero, iterate one step at a time.
• Forecast error variance grows with horizon h, governed by the model's shock-propagation structure.
• Random-walk forecast variance is $h\sigma^2$, growing linearly and unbounded.
• Stationary AR(1) variance converges to $\sigma^2/(1-\phi^2)$ as the horizon increases.

Exam shortcut

For any AR(1) with $|\phi| < 1$, the long-run forecast variance ceiling is $\sigma^2/(1-\phi^2)$: compute it once and you know the asymptote. For a random walk, multiply $\sigma^2$ by h and take the square root: SE $= \sigma\sqrt{h}$. If the question gives a forecast and a standard error and asks for the 95% interval, just compute $\hat{Y} \pm 1.96 \cdot SE$; no model recursion needed.

The full lesson (about 4,600 words, 31 min read) adds 7 worked examples, all 9 common mistakes, a self-check, free in the app.

Learning objectives

• 3c

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