Interpretation of β in case of Log-Lin model

[Prateek Bedi]

Hi,

In many papers, the dependent variable is transformed by taking natural log. For instance, consider the following model:

log(Y) = α + β X1 + ε

I understand that the interpretation of β is the percentage change in Y for a given unit change in X. However, some papers comment on the direction of relationship between X and Y (and not Log (Y)) on the basis of sign of β.

Through a simple simulation exercise, I have found out that it may be erroneous to comment on the direction of relationship between Y and X on the basis of sign of β in the above equation. Specifically, the signs of Cov (X,Y) and Cov (X, Log (Y)) may be different.

Please let me know if my finding is correct.

[Maarten Buis]

That does not sound right. Can you tell us more about the simulation? It is hard to spot the error in your code without seeing the code...

[Prateek Bedi]

Sure Maarten Buis. PFA an excel sheet in this regard. Request you to press Shift/Fn + F9 to simulate the data. You should get the concerned case with a different sign for relationship between Y and X as compared to the sign of relationship between Log(Y) and X.

Thanks. Waiting for your reply!

[Nick Cox]

Cross-posted at https://stats.stackexchange.com/ques...etween-x-and-y

Please note our policy on cross-posting, which is explicit in the FAQ Advice: You are asked to tell us about it.

[Prateek Bedi]

Cross-posted at https://stats.stackexchange.com/ques...etween-x-and-y

Please note our policy on cross-posting, which is explicit in the FAQ Advice: You are asked to tell us about it.

Apologies for the same, Nick Cox. I shall be careful in the future.

[Maarten Buis]

In your simulation the real relationship in the population is 0. That in combination with the small size of your dataset, means that small changes can shift the sign. In essence what you are seeing in your simulation is just random noise.

Here is a translation of your simulation to Stata (I don't think that this is the best simulation for this problem, but this is as close to the original simulation as possible):

clear all

program define sim, rclass
    drop _all
    set obs 17
    gen double y = runiform()
    gen double lny = ln(y)
    gen double x = rnormal(100,10)

    tempname b1 b2
    reg y x
    scalar `b1' = _b[x]
    reg lny x
    scalar `b2' = _b[x]
    return scalar normal = `b1'
    return scalar ln     = `b2'
end

simulate ln=r(ln) normal=r(normal), reps(1000) : sim
count if sign(ln) != sign(normal)
list if sign(ln) != sign(normal)

[Nick Cox]

I added another answer on Cross Validated, mostly to correct a misleading answer there (not Maarten's!);.

[Prateek Bedi]

I added another answer on Cross Validated, mostly to correct a misleading answer there (not Maarten's!);.

Looked at your answer, Nick. Agree with the points made by you. I have added a comment to your answer on Cross Validated.