ARIMA Exercises in R: 25 Real-World Practice Problems

Explanation: autoplot() from forecast dispatches a tidy ggplot2 time plot for ts objects, so you avoid hand-coding ggplot(). Eyeballing the series first is mandatory: the growing seasonal amplitude here is the visual signal that a log transform will help before any ARIMA fitting. Skipping the plot is the most common rookie mistake.

Explanation: Applying log() to a ts object preserves the tsp attribute, so frequency and start/end dates carry over. Log is the right transform when seasonal variance scales with the mean (multiplicative seasonality). Box-Cox via BoxCox.lambda() is the principled alternative when the right exponent is unclear, but for AirPassengers, log is the textbook choice.

Explanation: The p-value 0.35 is far above 0.05, so we cannot reject the unit-root null: Nile is non-stationary in its raw form and needs differencing before ARMA fitting. ADF assumes the alternative is stationarity around a constant, so a series with a known break (1899) violates that. KPSS via kpss.test() flips the null and is a useful sanity check whenever ADF returns ambiguous results.

Explanation: ndiffs() runs the chosen unit-root test repeatedly, differencing once and re-testing, until stationarity is achieved (or a max of 2). For co2, one regular difference flattens the long trend; the residual seasonal pattern is then handled with nsdiffs(). Picking d by eye is unreliable : ndiffs() is the deterministic, reproducible choice and is exactly what auto.arima() calls internally.

Explanation: Differencing removes the deterministic upward trend but leaves seasonal structure intact, which is why the ACF still shows a huge spike at lag 12. The job of the analyst is to read both plots together: cutting tails on the PACF suggest AR orders, cutting tails on the ACF suggest MA orders. A persistent lag-12 spike is the giveaway that a seasonal MA term is also needed.

Explanation: For an AR(p) process, the partial autocorrelation function cuts off after lag p while the ACF tails off exponentially. The PACF here has two significant bars then collapses to noise, matching the AR(2) DGP. This pattern-matching rule (PACF cutoff for AR, ACF cutoff for MA) is the Box-Jenkins core diagnostic; auto.arima() automates it but you should be able to do it by hand for small models.

Explanation: Where AR(p) leaves a clean cutoff in the PACF, MA(q) does the same for the ACF. The theoretical ACF of MA(1) is non-zero only at lag 1, equal to theta/(1+theta^2) = 0.7/1.49 = 0.47, which is what you see. If you see ACF cutting off at lag 2 instead, suspect MA(2). Mixed ARMA(p,q) processes show neither clean cutoff and need information criteria to disentangle.

Explanation: Series A shows the ACF decaying geometrically (0.8, 0.64, 0.51, ...) while the PACF cuts off cleanly at lag 1 : classic AR(1). Series B is the mirror: ACF spike only at lag 1, PACF tails off geometrically (with alternating sign) : classic MA(1). The duality is pi_k non-zero for finite k in MA, and rho_k non-zero for finite k in AR. When in doubt, plot both and let the cutoff side win.

Explanation: Arima() from forecast wraps stats::arima() but adds a constant for differenced series and a clean summary() printout. The AR1 coefficient is small and only marginally significant (z = 2.5); the MA1 coefficient is large and very significant. In practice, this points toward a simpler ARIMA(0,1,1) parameterisation, which is the IMA(1,1) random-walk-with-noise model that Nile is famous for.

Explanation: Default stepwise = TRUE greedily walks the model space, which is fast but can miss optima. Forcing stepwise = FALSE enumerates more candidate orders, and approximation = FALSE does exact MLE rather than the conditional sum of squares used during the search. Slower, but the AICc you report is exact, which matters for journal-quality results. For 90 percent of production use, the default stepwise is fine.

Explanation: Hand-rolling the grid is what auto.arima() does internally and is a useful exercise for understanding the trade-off between log-likelihood and parameter count. AICc penalises both, with a small-sample correction over AIC. Wrap the fit in tryCatch() because some (p,q) combinations fail to converge or violate stationarity. The winner here is typically ARIMA(3,1,1) or ARIMA(1,1,1) depending on optimiser settings.

Explanation: A drift term in an ARIMA model with d = 1 adds a deterministic linear trend on top of the random walk component. The estimated drift 0.0100 per month corresponds to roughly 12 percent compound annual growth in passengers, which matches the visual slope of the log series. Without include.drift = TRUE, the differenced series is forced to have mean zero, biasing forecasts downward for a clearly trending series.

Explanation: nsdiffs() runs a seasonal unit-root test (default OCSB) at the series frequency. A return of 1 means take one seasonal difference: diff(x, lag = 12) for monthly data. Combined with ndiffs() from Exercise 1.4, you now have both d = 1 and D = 1, the canonical airline-model differencing pattern. Going to D = 2 is rarely justified and usually overdifferences.

Explanation: diff(x, lag = 12) produces x_t - x_{t-12}, which kills any cycle of period 12. The remaining series shows a year-over-year change, which for CO2 hovers between roughly 1 and 3 ppm and drifts upward as emissions accelerate. To make it fully stationary, you typically apply a further diff() (regular difference of lag 1), giving the combined (1-B)(1-B^12) filter used in SARIMA models.

Explanation: The airline model factorises the dependence as a non-seasonal IMA(1,1) on top of a seasonal IMA(1,1). Both MA coefficients are large and significant: the non-seasonal MA captures short-run noise smoothing while the seasonal MA captures slow drift in the seasonal pattern itself. Until you can beat AICc of this 2-parameter model on a monthly business series, you do not need anything more complicated.

Explanation: With seasonal = TRUE (default), auto.arima() infers the period from frequency(co2) = 12 and searches over both seasonal and non-seasonal orders simultaneously. The chosen (1,1,1)(1,1,2)[12] model has 5 estimated parameters and an AICc near 185 : much better than the airline model on this dataset because the seasonal cycle in CO2 is more nuanced than in passenger traffic. Always trust the algorithm to find structure you would miss by eye.

Explanation: A p-value above 0.05 means you cannot reject the null of no autocorrelation in residuals at lags 1 through 24, so the residuals behave like white noise : exactly what an adequate model should produce. The fitdf adjustment is critical: forgetting it inflates df and biases the test toward not rejecting. Rule of thumb: set lag to about 2 * frequency for monthly data, never lower than 10.

Explanation: checkresiduals() is the one-stop diagnostic you should run after every ARIMA fit. The function picks a sensible lag based on the data frequency, applies the right df correction automatically, and plots ACF + histogram so you can visually verify no autocorrelation and approximate normality of residuals. If the p-value is below 0.05 or the histogram is skewed, revisit your model : usually a missing seasonal term or a structural break.

Explanation: When the model ignores the trend, every residual carries the deterministic upward drift, so successive residuals are highly correlated and the ACF stays above the confidence band for dozens of lags. This is the visual signature of underdifferencing. The fix in this case is to add d = 1 and a seasonal difference, which is exactly what the airline model does. Recognising this pattern instantly is what separates competent forecasters from novices.

Explanation: Standard ARIMA prediction intervals assume Gaussian residuals; if the Q-Q plot bends sharply in the tails, the 95 percent intervals reported by forecast() will be too narrow. For the airline model on AirPassengers, the residuals are very close to normal with mild heavy tails : acceptable for most business use. If normality is grossly violated, switch to bootstrap intervals via forecast(..., bootstrap = TRUE).

Explanation: forecast() returns point forecasts, 80 percent intervals, and 95 percent intervals by default. Because the model was fit on log() of the series, forecasts are in log space; exponentiate (exp(ex_6_1$mean)) to recover the original passenger units. To preserve correct prediction intervals after back-transformation, pass lambda = 0 and biasadj = TRUE directly when fitting Arima() instead of pre-logging : this is the cleaner workflow.

Explanation: Prediction intervals widen with horizon because the forecast variance accumulates the unit-step variance over time for ARIMA models. The 80 / 95 ratio of 1.28 / 1.96 standard deviations is fixed under Gaussian assumptions, so checking it is a quick sanity test that nothing exotic was applied. In tabular form like this you can paste it straight into a report or compare against actuals when they arrive.

Explanation: Holdout RMSE is the single most honest forecast metric: it measures error on data the model never saw during fitting. Computing it on the log scale gives a roughly proportional error (a value of 0.04 means about 4 percent average deviation in passengers). Always exponentiate back when reporting to stakeholders so the number is in the same units as the original series; never present log-scale errors to non-technical readers.

Explanation: Comparing ARIMA against ETS on a holdout is the gold-standard benchmark for univariate forecasts. For series like Nile with no obvious seasonality and a single structural break, the two families produce nearly indistinguishable forecasts because both reduce to simple exponential smoothing in disguise. When in doubt, ensemble them (average the two forecasts) : this typically beats either model on its own.

Explanation: Rolling-origin evaluation refits the model at every step, which mimics how forecasts are actually deployed in production: every new observation arrives, the model retrains, and the next forecast is issued. Compare against tsCV() in forecast for a vectorised version, but the explicit loop here is more transparent and easier to extend with multi-step horizons or window-limited training. The squared-error vector lets you compute any aggregate metric (MAE, RMSE, MASE).