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In the book "An Introduction to Statistical Learning with Applications in Python, Trevor Hastie et al., Springer", there's the following paragraph:

The left-hand panel of Figure 1.2 displays two boxplots of the previous day’s percentage changes in the stock index: one for the 648 days for which the market increased on the subsequent day, and one for the 602 days for which the market decreased.

Percentage change in S&P Index

Figure 1.2 (the left-hand panel)

It's not obvious to me why then the authors concluded:

The two plots look almost identical, suggesting that there is no simple strategy for using yesterday’s movement in the S&P to predict today’s returns.

Suppose in the boxplot for the "Up" bucket on the right, the median drops to $-4$ for example, then my understand is that in this case there's some strategy to predict today's return based on yesterday's movement. If so then what is that and why it's not applicable for the case in the image?

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Dave is correct (+1) but didn't explicitly say what the strategy should be. If there was a median of 4% drop in the S and P after an up day, then the strategy would be to sell immediately after an up day.

But ... this strategy would be quickly discovered. People would start selling more in the late afternoons of good days and then they wouldn't be so good.

The authors of the book are correct that the plots indicate there is no simple strategy based on yesterday's change. But lots of other work on the stock market indicates that this statement is much too narrow. The book is really making a point about statistics, not markets.

Also, your hypothetical that the S & P drops by 4% after an up day is rather far-fetched.

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As it stands, the distributions are just about identical. Given an “up” day yesterday, the distribution of returns today is about the same as the distribution of returns for the day after a “down” day.

If the “up” plot were shifted downward to have the median around $-4$ while the “down” plot remained the same, that would not be the case. The distributions would be dramatically different. In that case, the values you would expect to see after a “down” day might be like $1, -1.4, 0.7, 0.8, -0.2$, while the values you would expect to see after an up day would be like $-3, -6, -3.5, -4.5$. Instead of a mix of increases and decreases of comparable magnitude after a “down” day, and “up” day would strongly suggest a forthcoming “down” day.

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