Wavelet-L2E Stochastic Volatility Models: an Application to the Water-Energy Nexus

Forecasting commodity markets are difficult due to the time-varying nature and complexity of the financial return series representing these markets. Over the past few decades, statistical tools have come about to remedy these challenges. These methods focus on identifying the time-varying behavior and incorporating the characteristics within proposed models. This paper augments a series of well-known stochastic volatility models by first applying the WaveL2E thresholding method. The WaveL2E denoises non-stationary time series by dynamic multivariate complex wavelet thresholding combined with multivariate minimum distance mixture density estimation. Volatility clustering within the signal is estimated through the models’ ability to identify abrupt changes in the mean behavior as well as accurately threshold groups of outliers. Through optimization and recovery of the mixture density parameters as time series, the WaveL2E simultaneously identifies the signal and the time-dependent variance components. After an overview of the accuracy of the proposed models, we develop dynamic forecasts of the recovered signal. Our forecasted results show that the new WaveL2E stochastic volatility models are promising forecasting tools for highly persistent returns. Finally, we demonstrate the utility of our methods by studying the exchange-traded funds (ETF) for water and energy, identifying a quarterly lead-lag relationship.