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I am estimating the following model:

$\ln(y) = \alpha + \beta_1x + \beta_2x^2 $

$\hat\beta_2$ is insignificant while $\hat\beta_1$ is significantly different from zero. However they are jointly significant.

Is it still correct to keep $x^2$ and interpret the semi-elasticity?

Thank you!

Edit: $y$ is a continuous measure of income. $x$ is an ordinal variable measuring numeracy. There is economic evidence of decreasing returns to numeracy hence why I am estimating this model. Unfortunately, the ordinal nature of the variable makes it difficult to plot a relationship.

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    $\begingroup$ Without knowing anything more it is hard to tell - statistical significance alone does not say much. It would be better if you could describe your data and model fit in greater detail. $\endgroup$ – Tim Apr 10 '15 at 11:50
  • $\begingroup$ y is a continuous measure of income. x is an ordinal variable measuring numeracy. There is economic evidence of decreasing returns to numeracy hence why I am estimating this model. Unfortunately, the ordinal nature of the variable makes it difficult to plot a relationship. Thanks! $\endgroup$ – Ben Apr 10 '15 at 11:55
  • $\begingroup$ It's certainly correct to keep the quadratic term - p-values, standard errors, &c. will be valid, whereas following a procedure of refitting a regression after dropping "insignificant" terms biases them. See here. It depends on the goals of your modelling whether dropping it might help you: estimating a coefficient is more informative than merely labelling it "insignificant" & ignoring it, but it may be that you'd improve predictive power by using the simpler model. The PRESS ... $\endgroup$ – Scortchi Apr 10 '15 at 12:11
  • $\begingroup$ ... & AIC statistics can be useful for estimating how models compare in this respect; & for selecting a model to use, if you keep the number of comparisons small. (BTW please add extra info. by editing your question rather than in comments.) And, as you've just said the predictor's ordinal, have you considered representing it using orthogonal polynomial contrasts (you can still drop high-order terms if you want). $\endgroup$ – Scortchi Apr 10 '15 at 12:12
  • $\begingroup$ Thank you, I have made the edits. I will research orthogonal polynomial contrasts, admittedly I have never heard of them. Yes, it seems that keeping it will be more useful for explanatory purposes. While it does improve predictive power to remove it the difference is quite small so it seems to better to keep it. Thanks for you help! $\endgroup$ – Ben Apr 10 '15 at 12:55

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