Interpret time series output to make forecasts.

Software hands you a fitted ARIMA, a coefficient table, residual diagnostics, and a forecast. The exam asks you to read it, sanity check it, and produce a point forecast plus a 95% interval. Every number in the output has a job.

Why the output exists. A fitted time series model is a recipe: identify the structure (AR, MA, ARMA, ARIMA), estimate the coefficients, validate the assumptions, then project forward. The software prints every ingredient. Your job is to read the prints and answer two questions: is the model adequate, and what does it predict next?

Every estimation routine reports, for each parameter, the point estimate ${\hat{\varphi }}_i$ or ${\hat{\theta }}_j$, its standard error, a t-ratio, and a p-value. The standard error comes from the observed information matrix at the maximum likelihood estimate.

$$t_i=\frac{{\hat{\varphi }}_i}{\text{SE}({\hat{\varphi }}_i)}$$

KEY: A coefficient is statistically significant when $|t_i|>1.96$ (large samples) or roughly 2. Keep significant terms. Drop a non-significant lag only if dropping it does not worsen residual diagnostics.

Common mistakes

• Confusing intercept with mean. Software prints $c$, but the long-run mean is $\mu =c/(1-{\varphi }_1-\cdots -{\varphi }_p)$. Plugging $c$ where $\mu$ belongs in the mean-corrected forecast formula gives wrong answers. With $c=2.40$ and ${\varphi }_1=0.60$, reporting 2.40 instead of 6.00 as the long-run mean is a classic slip.
• Reading PACF cutoff for an MA process. MA(q) cuts off in the ACF at lag q and tails off in the PACF. Reversing this picks the wrong family. A clean ACF cutoff at lag 1 with a tailing PACF is an MA(1), not an AR(1).
• Treating Ljung-Box rejection as good news. A small p-value (below 0.05) rejects the white-noise null, meaning the model failed. Candidates sometimes write "p = 0.01, model fits well" because small p-values feel like a win.

Bottom line

• Model ID: white noise has flat ACF and PACF; AR(p) cuts off in PACF at lag p; MA(q) cuts off in ACF at lag q; ARMA tails off in both.
• Coefficient table: each row gives estimate, standard error, t-ratio, p-value. A term is significant when $|t|>1.96$; drop it if $|t|<2$ and diagnostics still pass.
• Long-run mean is $\mu =c/(1-{\varphi }_1-\cdots -{\varphi }_p)$, not the printed intercept $c$.
• Residual diagnostics: Ljung-Box p-value above 0.05 means no leftover autocorrelation; flat squared-residual ACF means constant variance holds, while spikes signal GARCH.

Exam shortcut

If the question gives a coefficient table and asks for the forecast, write the recursion ${\hat{Y}}_{n+1}=c+{\varphi }_1{Y}_n+{\theta }_1{\hat{\epsilon }}_n$ first, then iterate. Plug numbers only after the formula is on paper. If the question gives two models and asks which is better, compute AIC by $-2\ln L+2k$; if both $k$ match, pick the higher $\ln L$.

The full lesson (about 2,361 words, 16 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• D4

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