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I need to analyse the effect of change in units in independent variable. I understand that a change in independent variable units would lead to a change in the slope but would leave intercept unchanged. However, how do I analyse the same for a double log model? Does anything change with this particular specification? Also, the answer key I have suggests that the intercept would change and not the slope and I can really not seem to understand how. It would be great help if someone could help me with this. Thanks!

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The formula for the slope is $\hat{\beta}=\frac{Cov(x,y)}{Var(x)}$, and the formula for the intercept is $\hat{\alpha}=\bar{y}-\hat{\beta}\bar{x}$. So, the original estimates will be $$\hat{\beta}=\frac{Cov(log(P),log(X))}{Var(log(P))}$$ and $$\hat{\alpha}=\bar{log(X)}-\hat{\beta}\bar{log(P)}$$

To get the new estimates, we replace $P$ with $kP$, where $k=1000$. So, we get $$\hat{\beta}_{new}=\frac{Cov(log(kP),log(X))}{Var(log(kP))}=\frac{Cov(log(k)+log(P),log(X))}{Var(log(k)+log(P))}=\frac{Cov(log(P),log(X))}{Var(log(P))}=\hat{\beta}$$ and $$\hat{\alpha}_{new}=\bar{log(X)}-\hat{\beta}\bar{log(kP)}=\bar{log(X)}-\hat{\beta}\bar{(log(k)+log(P))}= - \hat{\beta}log(k)+\hat{\alpha}$$ So $\hat{\beta}$ remains the same but $\hat{\alpha}$ changes.

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  • $\begingroup$ Thank You for the answer! I have a little doubt, though. I am not sure why you've changed the dependent variable as well, since the question only suggests change in prices. Is there a reason behind this? Although I don't think this changes the answer. $\endgroup$ – S.Rana Mar 28 at 18:03
  • $\begingroup$ I'm sorry, you're right. I'll make that edit. Glad this was helpful. $\endgroup$ – Noah Mar 28 at 18:34

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