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I have the same results for two regression

1) y= number of board directors in t+1 x= a dummy variable with 1 if there is at least an institutional investor among the shareholders in t.

2) y= a dummy variable with 1 if there is at least an institutional investor among the shareholders in t+1 X= number of board directors in t

The coefficient of the two regression are positive and significant.

I want to know which conclusion can I write regarding the causality of the relation.

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Correlation does not imply causation is a statistics mantra. Regression does not tell you anything about causation. For a simple counter-example, imagine you used ice-creams sales to predict sunny weather and the model would work like a charm. Does this mean that eating ice-creams causes sunny weather? This is just a simple example and you could argue that the casual is the opposite, but for more examples you can check tylervigen.com, or do some random searches on Google correlate.

Check also the following threads to learn more:

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