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A OP's second expression corresponds to $$\Delta p = P(Y=1|D=1,X=\bar{X}) - P(Y=1|D=0,X=\bar{X}),$$ which is $$\Delta p = \Lambda(\beta_0 + \beta_1 + \beta_2 \bar{X}) - \Lambda(\beta_0 + \beta_2 \bar{X}),$$ where $\Lambda(z) = e^z/(1+e^z)$. Rationale for dividing by p OP's first expression corresponds to $(\Delta p)/p$, where $p$ is the proportion of 1'...

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The word "noise" implies that there is error in the measurement of household debt, but there is no good reason to believe this is so. In fact, I argue that a finer time period would give you more detail, and a more accurate measurement of the correlation between two variables. You don't want to find a moving average because that throws away valuable ...

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There are many methods to forecast exchange rates- look at the series of influential papers by Meese and Rogoff for a primer (https://scholar.harvard.edu/files/rogoff/files/51_jie1983.pdf). To answer your particular question- what you describe seems to be driven by multicollinearity. For instance, returns on stocks will be correlated with the inflation ...

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