I have a regression with a linear and quadratic term. The linear coefficient is not significant but the quadratic is significant. Does it make sense to have no linear effect but a quadratic effect in a regression model?
Really, it depends on the nature of the data-generating process. All it means is that the response is correlated with the square of the variable more than with the untransformed variable. The two may well be partially collinear over the range of your data. If this were the case, and if the variable truly had a quadratic influence on your response, then the coefficients on both would become significant as the dataset grew (assuming that the true model also had a linear component).
You may or may not want to use semiparametrics for routine data analysis, but they are certainly useful for getting some intuition about your dataset. Here is something you might look at using
mgcv in R:
install.packages('mgcv') library(mgcv) data = read.csv("yourdata.csv") mod = gam(y~s(x)+others, data=data) plot(mod, select=1)
y is your response variable,
x is the thing you're not sure about the functional form of, and
others are (potentially many) other variables in your regression. The last line plots the estimated not-necessarily-linear function. Either you can just use this estimate (though you'd probably want to read up on the method first), or it can guide you on which polynomial terms to include in a linear model.