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In summary, our model design is relatively reliable in the context of the known empirical patterns such as the existence of jumps, the volatility smile and the negative correlation between asset returns and volatilities.
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In contrast, if the average return on a CB portfolio is zero we should reject the FFC-model because there is no linear relation between exposures and asset returns.
Despite the sizeable literature on the link between order imbalance and asset returns, previous empirical evidence on seasonal imbalance patterns is scarce: Chang and Shie (2011) document that buying pressure in the Taiwanese futures market clusters in the last 30 minutes of a trading day.
However, in order to explain pro-cyclical leverage it assumes that bad news increases volatility, that is, it assumes an inverse relationship between first and second moments of asset returns.
"The role of systematic risk in shaping individual and aggregate behavior provides a more direct biological channel through which the relationship between systematic financial risk and expected asset returns can arise.
Asset returns and their volatilities.
Most existing methods that aim at compromising between portfolio performance (e.g., expected return) and its risk (e.g., volatility or shortfall probability) need some statistical model of the asset returns.
For asset returns you do not do it.
The weak guarantee applies as long as the asset returns are realized within the prescribed uncertainty set, while the strong guarantee applies for all possible asset returns.
Unconditional vs. conditional expected asset returns.
The historical behavior of asset returns.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com