# Market impact power law fitting confusion

In many market impact papers such as "Anomalous price impact and the critical nature of liquidity in financial markets" by Tóth et al (2018), there is a standard power-law relation in the form of

$$MI = Y\sigma(\frac{Q}{V})^{\delta}$$

where $$\sigma$$ is volatility, $$Q$$ is volume traded, $$V$$ is market volume and $$MI$$ is the price impact. $$Y$$ and $$\delta$$ are constants to be fit, where $$\delta$$ is usually from 0.4~0.7.

Taking logs of both sides: $$\log(MI) = \log(Y)+\log(\sigma)+\delta\log(\frac{Q}{V})$$, where we can first ignore $$Y$$ to fit for $$\delta$$, and once we have a value for $$\delta$$, we can then fit for $$Y$$.

One definition of price impact $$MI$$ is simply $$\frac{S_{avg}-S_{0}}{S_{0}}*sgn(Q)$$, where $$S_{avg}$$ is the average execution price of all the individual child orders, $$S_{0}$$ is the price before the first execution and $$sgn(Q)$$ is 1 for buy orders, -1 for sell orders.

However, this seems to restrict buy order samples to $$S_{avg} > S_{0}$$, and sell order samples to $$S_{0} > S_{avg}$$, as we require $$MI>0$$ for $$\log(MI)$$ to work. For instance, if a buy order was being sliced out during a period T (e.g. 1 hour) where the market was consistently moving down, then $$S_{avg}$$ will definitely be less than $$S_{0}$$, and $$MI <0$$. Does that mean we can't use this sample in the log fit? This seems quite impractical as there isn't anything 'wrong' about the order, and such orders are quite commonly observed.

This will result in a drop of roughly ~50% of the samples I have, which isn't ideal as I don't have that many samples to begin with. Not sure if I'm going about this correctly, any advice would be appreciated.