Considering the complexity pattern of the gold price, this paper adopts the decomposition-reconstruction-forecast-mergence scheme to perform the international gold price forecast. The original gold price data are decomposed into 12 intrinsic mode functions and a residual by the complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN) method, and then the 13 sequences are reconstructed into a high-frequency subsequence (IMFH), a low-frequency subsequence (IMFL), and the residual (Res). According to the different characteristics of the subsequences, the IMFL and Res are forecasted by the support vector regression (SVR) model. Besides, in order to further improve the prediction accuracy of IMFH, we have developed a novel hybrid method based on the support vector regression (SVR) model and the grey wolf optimizer (GWO) algorithm with SVR for predicting the IMFH of gold prices, i.e., the CEEMDAN-GWO-SVR model. This hybrid model combines the methodology of complex systems with machine learning techniques, making it more appropriate for analyzing relationships such as high-frequency dependences and solving complex nonlinear problems. Finally, the final result is obtained by combining the forecasting results of the three subsequences. The empirical results show that the proposed model demonstrates the highest prediction ability among all of the investigated models in a comparison of prediction errors with other individual models.

In this paper, we propose the model averaging estimation method for multiplicative error model and construct the corresponding weight choosing criterion based on the Kullback–Leibler divergence with a hyperparameter to avoid the problem of overfitting. The resulting model average estimator is proved to be asymptotically optimal. It is shown that the Kullback–Leibler model averaging (KLMA) estimator asymptotically minimizes the in-sample Kullback–Leibler divergence and improves the forecast accuracy of out-of-sample even under different loss functions. In simulations, we show that the KLMA estimator compares favorably with smooth-AIC estimator (SAIC), smooth-BIC estimator (SBIC), and Mallows model averaging estimator (MMA), especially when some nonlinear noise is added to the data generation process. The empirical applications in the daily range of S&P500 and price duration of IBM show that the out-of-sample forecasting capacity of the KLMA estimator is better than that of other methods.