Comparative Evaluation of ARIMA, LSTM and Hybrid ARIMA-LSTM Models for Forecasting the Apple Inc. Stock Price

Abstract

ABSTRACT: Accurate stock-price forecasting remains challenging due to the complex, nonlinear, and volatile nature of financial time series. This study developed and evaluated a hybrid ARIMA-LSTM model for forecasting Apple Inc. stock prices using 11,107 observations spanning December 1980 to January 2025. The dataset, obtained from Kaggle, used Close price as the target variable and Open, High, Low, Adjusted Close, and Volume as input features. Three forecasting models were evaluated: ARIMA(1,1,1), a stacked LSTM comprising two LSTM layers with 50 units each followed by two Dense layers, and a hybrid ARIMA-LSTM model that combines ARIMA-based linear forecasting with LSTM residual learning. The data were divided chronologically into 80% training and 20% testing sets, and model performance was assessed using MAE, MSE, RMSE, MAPE, and R².  ARIMA achieved the best overall performance, with an MAE of 1.3328, MSE of 4.3612, RMSE of 2.0884, MAPE of 1.25%, and R² of 0.9990. The LSTM recorded an MAE of 3.7500, MSE of 34.3950, RMSE of 5.8650, MAPE of 2.60%, and R² of 0.9920. The hybrid model achieved an MAE of 1.3610, MSE of 4.4570, RMSE of 2.1110, MAPE of 1.30%, and R² of 0.9990.