Dear Yue Teng:

There is nothing wrong. These results usually arrive when the underlying -true- relationship is concave (e.g. linear term positive and cuadratic term negative). A linear approximation must have a smaller coefficient (than that obtained in a second order polinomial for the linear term) to minimize RSS. The higher the concavity index, the higher the difference between linear term coefficients.

Hope this helps

You are basically weighting the observations of log(gdp).  This implies that you believe that periods where GDP increases are more important than periods where GDP decreases.  Additionally, the two variables are highly correlated, so you have a multi-colinearity problem.  I would drop one of the two variables from the regression if I were you.

When creating a polynomial you can achieve less correlated terms ( and get a more interpretable intercept) by taking away the mean of the log(gdp) before creating the quadratic ( that is log(GDP) - mean(log(GDP)). But I think Demian's answer is the correct one. Plot the data and overlay the linear and quadratic fitted function to see what is going on, 

You could also try a generalized additive model which has a much more flexible specification than the parametric form of a polynomial see

https://www.researchgate.net/publication/225303313_Moving_out_of_the_Linear_Rut_The_Possibilities_of_Generalized_Additive_Models

and 

https://www.researchgate.net/publication/225303308_Generalized_Additive_Models_Graphical_Diagnostics_and_Logistic_Regression

Article “Moving Out of the Linear Rut: The Possibilities of Generali...

Article Generalized Additive Models, Graphical Diagnostics, and Logi...

Following Dr. Jones advices, you could also try with different higher order polynomials and transformations (first differences, 1/log(gdp), etc.), and choosing among them by means of some authomatic model selection technique (stepwise regression, ´PGETS, RETINA, etc.). For such an objective, Stata users have our GSREG open source code available (https://ideas.repec.org/c/boc/bocode/s457737.html and http://www.stata-journal.com/sjpdf.html?articlenum=st0383).

Best regards

Thank you all so much for the excellent answers! also sorry for not responsing earlier. In the regression, the purpose to add the squared GDP is to capture the inverse-U Kuznets effect. Adding the squared term also makes the constant term very large. I will try Prof. Jones' method to take away the mean of log GDP before creating square.

But I am not certain about how to interpret such a large coefficient of the linear Log GDP. My regression is a level-log model. So for example, if the coefficient of (linear) natural Log GDP is 45, can I say that an 10% increase in GDP is associated with an increase in dependent variable by 45*Ln(1.1)=4.29? Here I take natural Log.

And does it make sense to separately interpret the coefficient of the linear term, after adding its squared term?

Thank you so much again!

The thing to do is to plot the raw data against log GDP and then plot the linear and quadratic predictions.

The numerical size of the effect will be in part determined by the scale of measurement of your predictor and outcomes and you have not told us about that.

If you theory suggests a u-shape then fitting a polynomial seems like a good idea.

Hi Yue, I have the same question but in a log-log framework. Have you worked out how to interpret the coefficient on GDP?

Many thanks,

Mingyue

in my case also, by adding the square and cubic forms of log GDP, the coefficient is increasing, my question in this context is that when I use fixed effect panel, coefficients of GDP and its square and cubic form are not increasing so as compared to pooled mean group method (PMG/AMG), is there any connection with lags? Yue Teng Kelvyn Jones Demian Tupac Panigo

I do not think lagged values will come into it unless you have included variables in that form.

I prefer RE to FE because you can then model both between country and within country between occasion variation and include country variables to try and account for the former:

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I am also struggling with similar issue, that is, using log(z) and [log(z)]^2 as independent variables in regression specification along with other independent variables. 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 using quadratic term. Or Please let me know if doing something very silly and can not see through that.

Adding a squared log GDP term to a regression model can potentially affect the coefficient of the log GDP variable. This effect is typically observed when the relationship between the dependent variable and the independent variable (log GDP in this case) is nonlinear.

When the relationship is linear, the coefficient of the log GDP variable represents the average change in the dependent variable associated with a one-unit change in the log GDP. However, when a squared log GDP term is added to the model, it introduces a curvature to the relationship between log GDP and the dependent variable.

Adding a squared log GDP term to a regression model can sometimes result in the coefficient of the log GDP variable appearing larger. This phenomenon is known as "attenuation bias" or the "compression effect."

The reason behind this effect lies in the correlation between the log GDP variable and its squared term. When the squared log GDP term is added to the model, it introduces a new variable that is highly correlated with the log GDP variable. This high correlation can lead to the coefficient of the log GDP variable being inflated.