ARIMA Models

ECON 3209 · Week 20, Lecture 1 · Kerala Agricultural University

Department of Development Economics, KAU

Autumn 2026

Learning Outcomes

By the end of this lecture, you should be able to:

  1. define AR, MA, and ARIMA models and interpret the orders p, d, and q
  2. write the compact ARIMA equation using backshift notation
  3. follow the Box-Jenkins workflow for identification, estimation, and checking
  4. estimate simple ARIMA models in Python and compare orders with AIC
  5. produce and interpret forecasts for a Kerala-style time series

From AR and MA to ARIMA

  • AR(p): current value depends on past values.
  • MA(q): current value depends on current and past shocks.
  • ARIMA(p,d,q): combines autoregressive terms, differencing, and moving-average terms.

\[\phi(B)(1-B)^d Y_t = \theta(B)\varepsilon_t\]

Meaning of p, d, q

  • \(p\): number of autoregressive lags.
  • \(d\): number of differences needed for stationarity.
  • \(q\): number of moving-average lags.
  • Example: ARIMA(1,1,1) means first-difference the series once, then model the differenced series with one AR term and one MA term.

Box-Jenkins Methodology

  1. Plot the data and test stationarity.
  2. Difference if needed.
  3. Use ACF/PACF and theory to suggest \(p\) and \(q\).
  4. Estimate candidate models.
  5. Compare AIC/BIC and residual diagnostics.
  6. Forecast and evaluate.

Prepare a Forecasting Series

Fit an ARIMA Model

Searching over Small Orders

Kerala Interpretation

For Kerala paddy production, the differencing order often handles trend, while AR and MA terms capture short-run persistence in shocks due to weather, prices, and policy changes.

Forecasting with ARIMA

How to Choose the Final Model

  • Respect stationarity evidence and use the right differencing order.
  • Prefer parsimonious models that pass residual checks.
  • Compare AIC/BIC and out-of-sample forecast performance.
  • Avoid choosing orders mechanically from one graph alone.

What ARIMA Can and Cannot Do

  • ARIMA is strong for short-run univariate forecasting.
  • It does not explain structural causal mechanisms by itself.
  • If major policy or climate regime changes occur, forecast performance may weaken.
  • Still, it is a foundational tool in applied time-series work.

Exercise

Estimate ARIMA(1,1,1) and ARIMA(2,1,0) on the paddy series. Which model has the lower AIC, and what five-year forecast does it produce?

Summary

  • ✅ ARIMA combines autoregression, differencing, and moving-average dynamics in one framework.
  • ✅ The Box-Jenkins process links identification, estimation, checking, and forecasting.
  • ✅ AIC/BIC and diagnostics help narrow down plausible orders.
  • ✅ ARIMA is a central tool for practical short-run forecasting.

Next Lecture

  • We evaluate forecasts using MAE, RMSE, MAPE, and rolling windows.
  • You will compare competing forecasts out-of-sample.
  • Model estimation matters, but forecast evaluation decides usefulness.