I want to interpret the models (2) Pool variable. It equals 1 if the house has a pool and 0 if not. The relation between the dependent variable and the Pool variable is a log-linear that means ∆y/y = b1*∆X whereby X is a dummy variable D so the change ∆D can only be 1 or -1 if I am not wrong. So if a house has a pool the price will rise by 7.1% and if house has already a pool and it gets destroyed or something like this the price will decrease by 7.1%. Is this interpretation correct?
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$\begingroup$ Welcome to Cross Validated! Do you take $\log(y)$ and then run a linear regression using those values, or do you run a generalized linear model with a $\log$ link function? $\endgroup$– DaveFeb 11 at 16:02
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$\begingroup$ Ty Dave. I didnt ran this regression it is from the Stock and Watson "Introduction to Econometrics "book. I think you take ln(y) and then run a regression with the listed variables. So the model is just ln(y) = b0 + b1*ln(size) + b2*Pool + ... + the other dummies. $\endgroup$– BlankerHansFeb 11 at 16:05
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1$\begingroup$ Your language is potentially confusing, because it strongly suggests the model is causal. It is not. In fact, if a pool were destroyed, that would likely hugely change the value of the house, depending on the circumstances. The term only estimates the average difference in log prices between houses with and without pools, accounting for ("controlling for") all other factors. $\endgroup$– whuber ♦Feb 11 at 16:54
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The interpretation of the coefficient of pool in the second model is: Compared to houses that have no pool, the geometric mean of the price for houses with pools is $100(\exp(0.071) - 1)\% = 7.36\,\%$ higher, all else being held constant. In other words, $\exp(\beta)$ is the ratio of the geometric means of pool/no pool (or generally, current level/reference level of the categorical variable). For more information, see this page.
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$\begingroup$ I see that this is mathematically more precise. In lecture we used the approximation that ln((y+∆y)/y) is roughly (y+∆y) for small changes. We derived it like this: ln(y+∆y) = b0 + b1*ln(size) + b2*(Pool+∆Pool) + ... + bi * (other dummy variables) | - ln(y) => ln((y+∆y)/y) = b2*∆Pool then using the approximation => ∆y/y = b2*∆Pool I think that is how I get to 7.1% increase if the house has a pool? Isnt it? :D $\endgroup$ Feb 11 at 16:36
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1$\begingroup$ @BlankerHans Yes, that's the approximation that Stock and Watson explain and use in their book. $\endgroup$ Feb 11 at 16:48
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$\begingroup$ So regarding this approximation my initial interpretation is (approximatly) correct? $\endgroup$ Feb 11 at 17:00
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$\begingroup$ @BlankerHans As far as I follow Stock and Watson, yes, that seems to be the case, but see whuber's comment to your question. $\endgroup$ Feb 11 at 17:10