benchmark has modestly outperformed futures prices at horizons 3, 6 and 12 months. On the other hand, Bowman and Husain (2004) showed for several metals that models that incorporate futures prices in an error correction model performed better than models based entirely on historical data or judgment, in terms of directional accuracy and precision, particularly at longer forecast horizons. Complementing futures price data with other information—such as industrial production, exchange rate dynamics, commodity currencies, international metals stock index, structural breaks, and short-run common-cycle restrictions—has further improved forecast performance (Gong and Lin 2018; Issler, Rodrigues and Burjack 2014; Pincheira-Brown and Hardy 2019; Pincheira-Brown and Hardy 2021; Pincheira-Brown et al. 2021).

# **2.2.2 Univariate time series models**

Univariate time series models have performed better than no-change forecasts but underperformed other quantitative methods (Rubaszek, Karolak and Kwas 2020; Alipour, Khodaiari and Jafari 2019). For aluminum, copper, nickel, and zinc, univariate autoregressive models delivered significantly better forecasts than the no-change approach (Rubaszek, Karolak and Kwas 2020).

However, univariate time series models have generally underperformed other quantitative methods. For copper, the forecast performance of ARIMA and no-change forecasts was inferior to that of neural networks, dynamic averaging and selection models and stochastic differential equations (Alipour, Khodaiari and Jafari 2019; Buncic and Moretto 2015; Lasheras et al. 2015). For aluminum, VECMs have had better out-of-sample forecast accuracy than ARIMA and VAR models (Castro, Araujo and Montini 2013). For aluminum and nickel prices, a modified grey wave forecasting technique—a univariate technique that explicitly accounts for irregular fluctuations in time series—performed better than no-change or ARMA methods (Chen, He and Zhang 2016). For aluminum, copper, lead, and zinc, wavelet-autoregressive integrated moving average (ARIMA)-based models have outperformed traditional ARIMA models in terms of forecast accuracy (Kriechbaumer et al. 2014). For lead and zinc, He et al. (2015) found that a curveletbased multiscale forecasting approach was superior to traditional benchmark models such as ARMA and random walk. For lead, ARIMA models have generated slightly better forecasts than models based on lagged forward prices (Dooley and Lenihan 2005).

# **2.2.3. Multivariate time series models**

Multivariate time series models such as VARs have generally outperformed the no-change benchmark and, in many cases, univariate models. In a comprehensive analysis of price forecasts for copper, lead, nickel, tin and zinc, Issler, Rodrigues and Burjack (2014) using annual and monthly data from 1900 to 2010 found that model performance differs by data frequency and commodity. For annual data, univariate autoregressive models were the best for aluminum and copper prices _,_ while VARs produced the best results for lead and zinc, and VECMs had the best results for nickel and tin. But for monthly data, VECMs of all metal prices and U.S. industrial production showed superior forecasting performance. Castro, Araujo and Montini (2013) also 9