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I am trying to create a model that forecasts trading costs (using end of day data, so no intra day data). My trading cost (also called Implemented Shortfall (IS) is defined as such for a single stock,

IS = (vwap - open) / open

for the market as a whole,

IS = abs(IS_single_stock - IS_market_median)

Variables that I am looking at include a companies market cap, the daily spread, vwap, volume & a liquidity measure called liq_m.

Doing a simple linear regression of each variable against IS produced very low r-squares, below 0.1. Combining the variables did very little to to improve the results. The residual plots appear to have some pattern, the one below is similar for most of the variables, this is mcap vs IS residuals. residual plot, mcap vs IS

The normal probability plot also highlights that the residuals are not linear & have a left skew.

normal probability plot of residuals

In the literal I have read on implemented shortfall all the models are non linear models so this is not unexpected.

I am unsure though of how to proceed next i.e. how to select an appropriate non linear model for testing? The end goal is to have a model that allows me to forecast the cost of trading a certain company.

Below are two more plots. One is the daily plot of mcaps over time - a mean is used to calculate the mcap of the 100 companies used in the sample. Beneath that is the Implemented Shortfall again a mean is used in the plot.

mcap over time enter image description here

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  • $\begingroup$ Hi mHelpMe! So, do you need a list of non-linear model to forecast IS and, possibly, a paper that compare them, right? $\endgroup$ – Quantopik Apr 27 '15 at 14:47
  • $\begingroup$ that would be very helpful! As you can imagine its one thing knowing the model is non-linear quite another knowing the model specification $\endgroup$ – mHelpMe Apr 27 '15 at 14:50
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Trading costs are made of different components:

  • fees
  • bid-ask spread costs
  • market impact

The last two components (ba-spread plus market impact) have to be estimated using a regression. There is a consensus today about a square root law of the market impact, mainly the spread + market impact models usually have the following shape:

$$a \psi + \kappa \sigma\sqrt{Q\over ADV}$$

Where $\psi$ is the bid-ask spread, $\sigma$ the volatility of the considered instrument, and $ADV$ its average daily volume. $a$ and $\kappa$ are the parameters to estimate.

For details refer to Market Microstructure in Practice (p203-209 of 2nd edition) or most papers in the December 2015 issue of Market Microstructure and Liquidity.

To answer to the details of your question:

  • first you made a mistake in your definition of IS (Implement Shortfall), it should be $$IS = sign(order)\times {(decision - open) \over open}$$
  • then you can estimate $a$ and $\kappa$ minimizing $$\left(a \psi + \kappa \sigma\sqrt{Q\over ADV} - IS\right)^2$$ on your database of orders.
  • if you have any double on the square root, you can try a fit of $$\left(a \psi + \kappa \sigma\left({Q\over ADV}\right)^\gamma - IS\right)^2$$

A detail: $\psi, \sigma$ and $ADV$ are here for renormalization purpose from one instrument to another. You can use any statistics representative of these variables (long term averages, median, mean, etc).

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