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I came across a paper that uses a panel data of US states (from 2000-2010) with the following model:

$y_{it}=b_{1}x_{it} +b_{2}x_{it}*D_{it}+other vars + \alpha_{t} + \gamma_{i}$

where, $x$ and $y$ are continuous vars and $D$ is a dummy variable that takes a value of 1 if there is a decrease in $x$ from one year to another year and zero otherwise (i.e., for an increase or no change in $x$). The paper then interprets coefficient on $x_1$ as the effect of the increase in $x$ on $y$ and $b_{1}+b_{2}$ as the effect of a decrease in $x$.I was wondering whether this is a valid interpretation.

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You could get closer to a valid interpretation if the two effects were adequately distinguished. Eg if b1 was multiplied by (1-D), so that it was only active for increases. As is, b1 is in effect for both increases and decreases, but b2 is only in effect for decreases. This would result in some sort of mixture, wherein b1 represents both the effect of an increase and a decrease in x on y, but more so the increase, and b2 represents some portion of the effect of a decrease. Summing them together won't give you only the effect of a decrease, so I would say no, it's not a valid interpretation.

Even with a simple respecification as described, the claim that x is an 'effect' on y requires many assumptions...

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