Time Series Modeling and Forecasting of Retail Sales Data: An Empirical Investigation Using ARIMA Methodology
Keywords:
ARIMA, Time Series, Retail Sales Forecasting, Box-Jenkins Methodology, Stationarity, ADF Test, ACF/PACF, Ljung-Box, BigMart, Demand Forecasting, Mean Retail Price IndexAbstract
This research paper presents a comprehensive empirical investigation into time series modeling and forecasting of retail sales data using the Autoregressive Integrated Moving Average (ARIMA) methodology, following the classical Box-Jenkins framework. The study employs transaction-level retail data from the BigMart outlet network spanning ten retail outlets across multiple location tiers, covering an observation period from 1985 to 2009. An annual retail price index is derived from 5,681 item-level Mean Retail Price (MRP) records, yielding a 25-year time series used as the primary analytical series.
The stationarity of the series was rigorously evaluated using the Augmented Dickey-Fuller (ADF) test, confirming that the original series is non-stationary (ADF = -2.6131, t-critical = -3.00 at 5%), while first differencing achieves stationarity (ADF = -3.0722). Autocorrelation Function (ACF) and Partial Autocorrelation Function (PACF) analysis identified significant autocorrelation at lag 1 for both functions, motivating an AR(1) process on differenced data. A systematic grid search across 54 ARIMA(p,d,q) candidate models identified ARIMA(2,0,0) as the minimum-AIC specification (AIC = 48.31), while ARIMA(1,1,0) was selected as the theoretically grounded forecasting model consistent with the stationarity analysis.
Out-of-sample forecasting on a 4-year holdout period (2006–2009) yielded a Mean Absolute Percentage Error (MAPE) of 1.10%, Root Mean Square Error (RMSE) of 1.8985, and Mean Absolute Error (MAE) of 1.5606 — outperforming Naive, Historical Mean, and Linear Trend benchmark models on directional accuracy. Ljung-Box residual tests confirm white noise residuals at both lag 5 (p = 0.5370) and lag 10 (p = 0.4910), validating model adequacy. A 5-year forward forecast (2010–2014) projects a gradual mean-reverting decline in the retail price index, converging toward the series long-run mean of 140.58 with widening 95% confidence intervals.
