Distributional Forecasting of the U.S. Stock Market with Generalised Additive Models for Location, Scale and Shape

Forecasting future expected returns out-of-sample is challenging due to some statistical characteristics, such as the stochastic and dynamic nature in the time series. Conventional machine learning techniques focus mainly on point forecasting, which cannot take distributional properties of the returns into account. The Generalised Additive Models for Location, Scale and Shape (GAMLSS) fills this gap by forecasting both the expected returns and the distribution of the returns, thereby making it easier to verify the statistically distributed properties and ensuring models’ validity over the out-of-sample periods. In this paper, we obtained a dataset from the Amit Goyal webpage consisting of 15 financial variables, each covering monthly observations from January 1970 to December 2022. The study has revealed the effectiveness of GAMLSS as a flexible distributional regression in forecasting the U.S. stock market out-of-sample using a rolling window with alternative recursive window methods, to ensure robustness in the analysis. The rolling approach efficiently dealt with any problem of structural dynamics or adverse economic conditions across business cycles. The GAMLSS models demonstrate statistical evidence of superiority over the conventional machine learning techniques across all out-of-sample forecasting windows. Notably, the inclusion of smoothing splines as potential smoothers in the GAMLSS helps to statistically improve the predictive task of the forecasting models. In addition, the GAMLSS models are more economically effective than the buy-and-hold trading strategy, which relies solely on the risk-free treasury bills. Thus, the out-of-sample rolling window forecasts produced by the GAMLSS models tend to be more promising in guaranteeing the fate of investors and portfolio managers while undertaking risky investments with target expectations in a real-time market setting.