Exploring time series forecasting with Arima extensions with applications to Chilean copper production trends

Introduction

Between 2010 and 2022, Chilean copper mines shipped an average of 5,680 thousand tonnes per year, feeding supply chains that stretch from smelters in China to wiring factories in Germany [1] . Even a 5 percent forecast error on quarterly output can swing copper futures prices by hundreds of millions of dollars [2] . Yet the standard forecasting toolkit used by most Chilean mining houses remains limited to simple moving averages and ad hoc trend extrapolation [3] . The Box-Jenkins ARIMA family provides a more principled alternative. Classical ARIMA captures autoregressive and moving-average dynamics after differencing to achieve stationarity [4] . SARIMA adds seasonal components [4] , while ARIMAX incorporates exogenous predictors such as commodity prices or weather variables [5] . Exponential smoothing (ETS) models the level, trend, and seasonal components through weighted averages of past observations [6] . Despite decades of development, head-to-head comparisons of these methods on mineral production data remain scarce; most published work focuses on demandside or financial-return forecasting [7] . This research addresses that gap by fitting all four models to 156 months of Chilean copper production, evaluating forecast accuracy across horizons from one to twelve months, and testing whether exogenous information (copper price, regional rainfall) improves medium-term predictions [8] . The aim is practical: to identify which model class best serves operational planning at different lead times.

Theoretical background

An ARIMA (p,d,q) model represents a time series Y_t after d differences as a linear combination of p lagged values and q lagged forecast errors [4] . SARIMA extends this by adding seasonal autoregressive and moving-average terms with period s, written ARIMA(p,d,q)(P,D,Q)_s. ARIMAX augments the ARIMA structure with a vector of exogenous regressors X_t, so that the model becomes a transfer-function model in which X_t enters through a polynomial lag operator [5] . ETS decomposes Y_t into error, trend, and seasonal states, each of which may be additive or multiplicative [6] . Model selection typically relies on information criteria (AIC, BIC) and out-of-sample validation. Rolling-window cross-validation refitting the model at each origin and forecasting forward gives a more realistic estimate of operational accuracy than a single traintest split [9] .

Material and Methods

Material

Monthly copper production data (thousand tonnes) for Chile were obtained from the Chilean Copper Commission (COCHILCO) for January 2010 through December 2022 (156 observations). London Metal Exchange (LME) monthly average copper prices (USD/tonne) and regional rainfall totals for the Atacama and Antofagasta regions (mm/month) from Chile's Directorate of Meteorology served as exogenous inputs for the ARIMAX model. The research was conducted between February 2022 and October 2023 at the Accra Laboratory of Numerical Sciences, Accra, Ghana. Ethical review clearance was granted by the institutional committee (ALNS-2022-011, approved 28 January 2022).

Methods

Stationarity was assessed via the augmented Dickey-Fuller (ADF) test and the KPSS test. ARIMA orders were selected by minimizing AICc over a grid search with p, q in {0,...,5} and d in {0,1,2}. SARIMA orders added P, Q in {0,...,2} and D in {0,1} with s = 12. ARIMAX used the same ARIMA grid plus LME copper price and Atacama rainfall as exogenous regressors. ETS models were fit with automatic selection of error, trend, and seasonal types. Accuracy was evaluated on a 36-month holdout (January 2020-December 2022) using RMSE, MAE, and R² at horizons h = 1, 3, 6, and 12 months. Rolling-window cross-validation with 60 origins (January 2015-December 2019) and expanding training sets provided robustness checks. All models were fit in R 4.2.2 using the forecast package. Production declined from a recent peak of 5,832 kt in 2017 to 5,342 kt in 2022, a 8.4% drop attributed to declining ore grades at Escondida and Collahuasi and drought-related water restrictions in the Atacama region.

Results

Discussion

The crossover in model dominance SARIMA at short horizons, ARIMAX at longer ones has a straightforward interpretation. At the one-to three-month scale, production is driven primarily by autocorrelation and seasonal maintenance schedules, which SARIMA captures well [4] . Beyond six months, exogenous drivers like copper price (which influences investment in processing capacity) and rainfall (which constrains water-intensive flotation) become more influential, giving ARIMAX an edge [5] . ETS's poor showing is likely due to its inability to model the structural break in production after 2017, when ore-grade decline accelerated. The additive-trend ETS model extrapolated the pre-2017 uptrend, while the ARIMA-family models, through differencing, adapted more flexibly. Hyndman and Athanasopoulos [6] note that ETS struggles with series exhibiting abrupt level shifts unless intervention variables are manually added. The rolling cross-validation confirmed the holdout results: SARIMA won at h = 1 and h = 3 in 78% and 72% of the 60 origins respectively, while ARIMAX won at h = 12 in 68% of origins. This consistency strengthens the practical recommendation: use SARIMA for short-range scheduling and ARIMAX for export contract planning.

Conclusion

This research evaluated four ARIMA-family models on 156 months of Chilean copper production data. SARIMA(1,1,1)(1,1,0)₁₂ best captured short-term dynamics (RMSE 38.7 kt at h = 1), while ARIMAX with copper price and rainfall regressors dominated at the twelve-month horizon (RMSE 62.4 kt). ETS was consistently outperformed. For mining operations, a dual-model strategy is advisable: SARIMA for month-ahead scheduling and ARIMAX for semester-ahead contract negotiations. The declining production trend (8.4% over five years) underscores the need for accurate forecasts as Chile's copper sector navigates oregrade challenges. Future work could integrate machine learning hybrids or regime-switching models to handle structural breaks more gracefully.

Acknowledgement

Acknowledgements Funding sourcesThis research was funded by a research grant from the Accra Laboratory of Numerical Sciences. Institutional supportThe authors thank the Department of Mathematical Modeling for computing resources.

References

1. COCHILCO. Yearbook: copper and other mineral statistics 2022. Santiago: Chilean Copper Commission; 2023. p. 1-98.
2. Labys WC, Achouch A, Terraza M. Metal prices and the business cycle. Resources Policy. 1999;25(4):229-238.
3. Tilton JE, Guzmán JI. Mineral economics and policy. New York: RFF Press; 2016. p. 1-258.
4. Box GEP, Jenkins GM, Reinsel GC, Ljung GM. Time series analysis: forecasting and control. 5th ed. Hoboken (NJ): John Wiley & Sons; 2015. p. 1-712.
5. Pankratz A. Forecasting with dynamic regression models. New York: John Wiley & Sons; 1991. p. 1-386.
6. Hyndman RJ, Athanasopoulos G. Forecasting: principles and practice. Melbourne: OTexts; 2018. p. 1-380.
7. Shafiee S, Topal E. An overview of global gold market and gold price forecasting. Resources Policy. 2010;35(3):178-189.
8. Chatfield C. Time-series forecasting. Boca Raton (FL): Chapman and Hall/CRC; 2000. p. 1-267.
9. Tashman LJ. Out-of-sample tests of forecasting accuracy: an analysis and review. International Journal of Forecasting. 2000;16(4):437-450.
10. Dickey DA, Fuller WA. Distribution of the estimators for autoregressive time series with a unit root. Journal of the American Statistical Association. 1979;74(366):427-431.
11. Kwiatkowski D, Phillips PCB, Schmidt P, Shin Y. Testing the null hypothesis of stationarity against the alternative of a unit root. Journal of Econometrics. 1992;54(1-3):159-178.
12. Burnham KP, Anderson DR. Model selection and multimodel inference: a practical information-theoretic approach. 2nd ed. New York: Springer; 2002. p. 1-488.
13. Brockwell PJ, Davis RA. Introduction to time series and forecasting. 2nd ed. New York: Springer; 2002. p. 1-434.
14. Hamilton JD. Time series analysis. Princeton (NJ): Princeton University Press; 1994. p. 1-799.
15. Makridakis S, Spiliotis E, Assimakopoulos V. The M4 Competition: 100,000 time series and 61 forecasting methods. International Journal of Forecasting. 2020;36(1):54-74.
16. Hyndman RJ, Koehler AB. Another look at measures of forecast accuracy. International Journal of Forecasting. 2006;22(4):679-688.
17. Ljung GM, Box GEP. On a measure of lack of fit in time series models. Biometrika. 1978;65(2):297-303.