explanatory variable and squared explanatory variable

[Francisco Nobre]

Hello,

In my model specification I have as independent variables aveage (average age) and aveage²
aveage has a negative sign and aveage2 has a positive sign.
How can I interpret these results?

Thank you

[Carlo Lazzaro]

Francisco:
first of all, you should make you code more compact in that respect:

c.age##c.age

- as per FAQ, please post what you typed and what Stata gave you back, so that interested listers can reply more positively. Thanks.

[Richard Williams]

Pages 2-4 of

https://www3.nd.edu/~rwilliam/stats2/l61.pdf

include a discussion of squared terms in a model.

[Nick Cox]

See this thread: https://www.statalist.org/forums/for...quadratic-term

Plot the two terms together to see what is happening. See where the turning point lies in relation to your data and wonder whether it makes sense.

[Marcos Almeida]

Just as a side note after Carlo's helpful advice.

To start, I recommend to center the variable age before creating the squared term.

There is a formula to get the point which divides the interaction curve in 2, I mean, when the effect modification "changes" according to the values: -b/2a

When the quadratic interaction term is negative, the curve shows concavity downwards.

The commands -margins - and - marginsplot - are excellent for the interpretation of interactions.

Edited: crossed with Richard's and Nick's replies.

[Francisco Nobre]

Thank you all for the help!

[River Huang]

Including the squared term of `aveage' allows your dependent variable (say, y) and `aveage' to be (possibly) inverted-U or U-shaped related. A formal test of this can be implemented by (ssc install) utest. Please help utest for more detailed information, and read the paper: J. T. Lind and H. Mehlum: With or without U? The appropriate test for a U shaped relationship. Oxford Bulletin of Economics and Statistics 72(1): 109-18 (2010).

[mohina saxena]

Hello All,

I need to ask something which could be very trivial but its very important for me to clarify this. I would be grateful if someone can please clarify this. In my model when I include the square term of a variable then both the levels and the quadratic term are significant but have the "same" positive sign. I further test for its inclusion using

test x c.x#c.x

and find it significant as well.

Since,the sign of the coefficient of x^2 is positive then it implies that effect of x will increase on y with further increase in x. Thus, the effect of x on y increases at an increasing rate. As such there is no-non linearity, so should I add the quadratic term in my model?

In fact I must mention if I remove the quadratic term, then R-square falls from 0.60 to 0.37 (I am doing Pooled OLS) and some of the other explanatory variables of my model looses their significance ( I am not targeting significant results, but this is the "confounding" effect that is happening). Apart from this, earlier I was also having an interaction of variable x with some other explanatory term which was insignificant with inclusion of x^2 but after dropping x^2 the interaction becomes significant(I can get that dropping square term is causing such significance of interaction)

My x variable is firm size and outcome is firm performance so theoretically speaking it is important to control for its quadratic effect but I am confused should i include it even when sign is not changed although both level and square are significant?

Please help.

[Dario Maimone Ansaldo Patti]

Aiken and West (1991) "Multiple regression: Testing and interpreting interactions" could be a useful reading to understand your points.

[Richard Williams]

There is no reason both x and x^2 can't have positive coefficients, There is still some point where increases in x switch from causing decreases in y to causing increases (although it may or may not be within the observed range of the data.) See pages 2-4 of

https://www3.nd.edu/~rwilliam/stats2/l61.pdf

"As such there is no-non linearity." Not so. Linearity means straight line. You don't have a straight line with a squared term. I think you are confusing monotonic inreases with linear increases.

[mohina saxena]

Many thanks Sir for the clarification and directing me to a much needed reference.
I got where I was going wrong in my understanding. Agreed, there is a non-linearity in the model (not only because of squared term) but especially when both the terms are significant. So, I am proceeding with inclusion of both the terms in level as well as squared term in my regression.

regards,
Mohina