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Say I have a model:

log(Y) ~ X1 + I(X1^2) 

1) Do I still need to take the log of Y if I am using the quadratic of X. I thought we need to take log to linearise the relationship between Y and X and therefore we don't need to include the squared term of X1. In a way, which of the following three models is the right model:

log(Y) ~ X1 + I(X1^2) 
Y ~ X1 + I(X1^2)  
log(Y) ~ X1 

2) Let's say my model is

Y ~ X1 + I(X1^2) 

Both the X1 and I(X1^2) comes out to be significant. How do I interpret my regression slopes? How do I make a statement saying.." one unit change in X changes Y by the value given by the regression slope?" How is the quadratic term interpreted?

Thanks

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For your first question, you should take log of Y if log of Y makes sense from a substantive point of view. One example where a log often makes sense is money because we often think of money in terms of proportions rather than absolute differences. The difference between a \$100,000 house and a \$150,000 house is large; the difference between a \$10,000,000 house and a \$10,050,000 house is rounding error. Decide whether to transform variables before you start modeling.

For your second question, you cannot make a statement of the type you give, because a 1 unit change in X will have a different effect on Y at different levels of X.

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