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You should definitely adjust for inflation, as you should with all monetary variables. I wouldn't worry using the CPI, you could use an alternative indicator as a robustness check to see whether the choice of indicator affects your main conclusions.


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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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