In the last post, I showed you cross-correlation time-series factor momentum. This strategy times factors by utilizing auto-cross-correlations in factor data (see Gupta & Kelly (2019) for a comprehensive study). This week, I will extract factor predictability from auto-cross-correlations with deep learning. I use the data of Jensen, Kelly & Pedersen (2021), which is kindly provided by Professor Bryan Kelly on his homepage. Consider we are attempting to predict the value factor using past data of highly negative/positive cross-correlated factors. With supervised machine learning, this breaks down to:

The factor `i`

‘s (in our case `i`

is the value factor) return `r`

is predicted using last month’s returns of the driving factors j_{i,t+1}_{D}. A factor `j`

is considered as a driving factor if it belongs to the highest or lowest decile of cross-correlation coefficients `R`

. I use a simple deep neural network with 5 layers in a pyramid shape (16/8/4/2/1). The input layer uses a tanh activation function, the output layer a linear activation function. All other layers use ReLu activation. No further regularization is applied. I use factor data from January 1980 to December 1999 to train the model and data from January 2000 to December 2020 to test the data in a simple backtest. In the figure below we find the squared error of the neural network predictions compared to predictions using mean train-set returns. To rule out randomness in the initialization we repeat the prediction 50 times and keep the median as the final prediction._{i,d}

When we benchmark the predicted returns with the mean train-set returns we achieve a `R`

of ^{2}_{OOS}`0.103`

, a considerable performance. There are different applications for utilizing the predictions. Below I plot the performance of an investment strategy that uses the predicted returns, instead of prior month returns to form time-series factor momentum for the value factor. As you can see the performance is moderate in this setup.