I am facing issue around using a log form of a variable in regression as independent variable , that is, using log(z) and [log(z)]^2 as independent variables in regression specification along with other independent variables. people using it to explore non-linear kind of relationship. In my view, the [log(z)]^2 should ideally be redundant because of following
y=Bo + B1*log(z) + B2*[log(z)]^2+ ∑B_k*X_i + e ------ (1)
∂y/∂z= B1*(1/z)+2* B2*(1/z)
∂y/∂z= (B1+2*B2)(1/z)
Since we would end up at same specification of marginal impact of z even if specified above regression without the term [log(z)]^2 i.e.
y=Bo + B1*log(z) + ∑BkXi+ e --------- (2)
∂y/∂z= B1*(1/z)
So thinking about log(z) as if it is z is essentially creating lot of problem in the process. The problem, however, is that the expression (1) gives slightly higher R-squared and there is lot of adjustment in co-efficient value of constant term to accommodate the square term. My understanding is that the quadratic term of log(z) confuses the analysis more than it solves it. I would love to see responses my view of using quadratic term. Or Please let me know if I am doing something very silly and can not see through that.
Intelligent idea, but it does not work. The problem is, that ∂y/∂z is only applicable for very small differences (exactly: for one point z). In general, y=ax+bx²+... is chosen, because one expects an optimum of y. If you calculate (a+2b)x, you get a monotonuous (increasing or decreasing) function, i.e. there is no optimum. But even if both coefficients have the same sign, there are likely considerable differences (although both functions are (very near to) logarithmic.
Anton Rainer that is my point one should not use [log(z)]^2 either by itself or along with log(x) because mathematically because marginal impact is same for all these specifications.
But when we do it shows minor increment in R-squared and people think it is an improvement. but it is not.
Regards
Zak
Now I got your point.
You are right, that y=a*ln(z)+b*(ln(z))² is a senseless specification. At least, I cannot imagine any economic (but also physical, biological etc.) relation where it could make sense. Whereas for y=ax+bx², one can find an optimum (unless a and b have the same sign, otherwise I would hesitate to choose this specification), there is no such optimum in the above equation (∂y/∂z=0 only if a+2b=0). Including (ln(z))² in an econometric estimation will lead to a slightly higher R², but certainly to spurious results.
By the way, I am, in general, against estimating in absolute values or their logs. Estimations in differences or percentage changes are much "sharper".
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