ARIMA Models

• Explain p, d, q — autoregression, differencing, moving-average-error terms
• Choose orders from ACF/PACF and fit diagnostics
• Forecast with residuals-checked intervals

• It's the statistical gold standard: it produces point forecasts AND proper prediction intervals.
• Its assumptions are explicit — you can defend an ARIMA to a statistician or an auditor.
• Properly specified, it often beats fancy models on smooth, stationary-enough data.
• Knowledge of ARIMA transfers to ETS, SARIMA and the family tree.

Autoregression says the next note is partly a weighted sum of previous notes; the moving-average term models the 'echoes' of recent mistakes. Differencing is like turning the recording so you play the changes, not the absolute level.

• d = number of differences to stationarity: y_t' = y_t − y_{t-1} (repeat until ADF tests stationary).
• p = AR order: y_t' depends on y_{t-1}', ..., y_{t-p}' with coefficients φ.
• q = MA order: y_t' depends on recent white-noise errors ε_{t-1}..ε_{t-q} with coefficients θ.
• Choose orders: decay with lags in ACF and p significant spikes in PACF → AR(p); q similarly for MA.
• After fitting (MLE, Box-Jenkins), verify residuals are white noise (no ACF) and compute forecasts ± intervals.

• A utility's mid-term (3-month) load forecast uses ARIMA(2,1,2) because the flat weekly structure favors parsimony over ML.
• A crypto volume forecaster picks order via AIC scan nightly, refitting fast on rolling windows.
• A marketing team uses ARIMA's prediction intervals to size campaign response budgets instead of point estimates.

• Seek the simplest (smallest AIC/BIC) order whose residuals pass white-noise tests.
• Always difference with d from the ADF result, not 'let's try d=1'.
• Prefer SARIMA when any regular seasonality lurks — plain ARIMA will leak it into residuals.
• Verify out-of-sample backtest, not just in-sample fit, before trusting the order pick.

• Fitting ARIMA to a trending series without differencing and reading spurious significance.
• Selecting order by 'biggest p/q for best AIC' — model complexity races residual noise.
• Ignoring residual autocorrelation after fit; 'the AIC was low' alone doesn't validate.
• Forecasting beyond the regime the series came from (COVID-style breaks) with an order fitted on the old world.

• Residuals with a big ACF spike at lag 12? You missed seasonality — add SARIMA terms.
• Diagnostic p-values 'too beautiful' (0.99) on tiny samples — check the test isn't underpowered.
• Convergence warnings → scale the series (log or multiply) before fitting.
• Compare a grid of (p, q) in (0..4)² with AIC — cheap and catches obvious misses.