Business Analytics 101: Forecasting Time Series ARIMA Models Autoregressive Integrated Moving Average ACF PACF BoxJenkins

What This Is

ARIMA (Autoregressive Integrated Moving Average) models are a type of statistical model used for time series forecasting. They are essential in business analytics for predicting future values based on past observations. For example, a retail company might use ARIMA to forecast sales, helping them plan inventory and manage resources more effectively.

Key Formulas & Metrics

• ARIMA(p, d, q) = (1 - φ₁B - ... - φₚBⁿ)(1 - B)ᵈ(1 + θ₁B + ... + θₖBⁿ)Yᵗ – general ARIMA model formula, where p is the number of autoregressive terms, d is the degree of differencing, and q is the number of moving average terms.
• φ = ρ = Cov(Yᵗ, Yᵗ-1) / Var(Yᵗ) – autoregressive parameter, representing the correlation between a time series and its past value.
• θ = Cov(Yᵗ, εᵗ) / Var(εᵗ) – moving average parameter, representing the correlation between a time series and its past errors.
• ACF = ρ(k) = Cov(Yᵗ, Yᵗ-k) / Var(Yᵗ) – autocorrelation function, measuring the correlation between a time series and its lagged values.
• PACF = ρ(k) = Cov(Yᵗ, Yᵗ-k) / Var(Yᵗ) – partial autocorrelation function, measuring the correlation between a time series and its lagged values, while controlling for intermediate lags.
• MAE = (1/n) Σ|yᵢ − ŷᵢ| – mean absolute error, measuring the average absolute difference between forecasted and actual values.
• RMSE = √(1/n) Σ(yᵢ − ŷᵢ)² – root mean squared error, measuring the square root of the average squared difference between forecasted and actual values.
• AIC = n log(σ²) + 2p + 2q – Akaike information criterion, measuring the relative quality of a model.
• BIC = n log(σ²) + p log(n) + q log(n) – Bayesian information criterion, measuring the relative quality of a model.

Step-by-Step Procedure

1. Identify the time series: Determine if the data is stationary or non-stationary. If non-stationary, apply differencing (d) to make it stationary.
2. Plot the ACF and PACF: Use these plots to determine the order of the autoregressive (p) and moving average (q) components.
3. Estimate the model parameters: Use the identified orders (p, d, q) to estimate the model parameters (φ, θ).
4. Evaluate the model: Use metrics such as MAE, RMSE, and AIC to evaluate the model's performance.
5. Refine the model: Refine the model by adjusting the orders (p, d, q) or parameters (φ, θ) to improve its performance.

Common Mistakes

• Mistake: Confusing correlation with causation.
• Correction: Correlation does not imply causation. Use techniques such as Granger causality to establish causality.
• Mistake: Misinterpreting p-values.
• Correction: A small p-value does not necessarily mean that the null hypothesis is rejected. Consider the effect size and the context of the problem.
• Mistake: Using the wrong error metric for a business problem.
• Correction: Use metrics such as MAE or RMSE for forecasting problems, and metrics such as precision and recall for classification problems.

Software / Tool Tips

• Python with pandas and statsmodels: Use the ARIMA function from the statsmodels library to estimate ARIMA models.
• R: Use the arima function from the stats library to estimate ARIMA models.
• Excel: Use the FORECAST function to estimate simple moving average models.

Quick Practice Problem

A company wants to forecast sales for the next quarter. The historical sales data is: 100, 110, 105, 120, 115. What is the moving average forecast for period 4?

Answer: 115 Explanation: The moving average forecast for period 4 is the average of the last three values: (110 + 105 + 120) / 3 = 115.

Last-Minute Cram Sheet

• ARIMA(p, d, q) is a general formula for time series forecasting.
• ACF and PACF plots help determine the order of autoregressive and moving average components.
• MAE and RMSE are metrics for evaluating forecasting models.
• AIC and BIC are metrics for comparing the quality of different models.
• ⚠️ p-value is NOT the probability that H₀ is true – it's the probability of observing the data (or more extreme) if H₀ is true.
• ⚠️ Correlation does not imply causation.
• ⚠️ A small p-value does not necessarily mean that the null hypothesis is rejected.
• ⚠️ Use the right error metric for the business problem.