This section explains the intra-day volume forecasting model proposed by Bialkowski, J., Darolles, S., and Le Fol, G. (2008).

They use CAC40 data to test their model, including the turnover of every stock in the index as of September 2004. Trades are aggregated into 20-minute time slots, resulting in 25 observations each day.

Turnover is decomposed into two additive components. The first one is the seasonal component (the U shape) that represents the expected level of turnover on an average day for each stock. Given that every day is a little different from the average, there is a second one, the dynamic component, which shows the expected deviation from the average on a specific day.

The decomposition is carried out using the factor model of Bai, J. (2003). The initial problem is as follows:

Here, the X (TxN)-sized matrix contains the initial data, F (Txr) is the factor matrix, Λ' (Nxr) is the matrix of factor loadings, and e (TxN) is the error term...