## The volume forecasting model

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,

*is the factor matrix,*

**F**(Txr)*is the matrix of factor loadings, and*

**Λ'**(Nxr)*is the error term.*

**e**(TxN)*stands for...*

**K**