Can time be squared to develop a curvilinear model of crop yield against time? I am developing a linear model of yield against time (33 years of yield data) where year is 1975,1976....2007. I want to know whether change in yield over time was linear or not. So I fitted a linear model of yield against year:
 mdl<-lm(yld ~ year,data=data)

In my second model, year raised to the power 2.
 mdl2<-lm(yld ~ year + I(year^2),data=data)
 anova(mdl,mdl2)

Model 2 gives a significant improvement over the linear model, so I accept the evidence of  curvature in the data. 
My question is: is this analysis correct? I mean can I actually square 'year' to develop my second model? here is the plot

Thanks a lot 
 A: No reason why you can't in principle. Stuff goes up, stuff goes down.
That said, did you plot your data? Just because it's statistically "significant" (that's really an evil term) doesn't mean it's substantively significant. And even if there's a curve, does it make sense to fit the curve? Does it make sense to fit a quadratic curve? You have one time series over 33 points. Plot the thing.
A: Clearly the relationship is curved.
You don't really have a theoretical justification for it being quadratic rather than something else.
Nevertheless, in the sense that you can regard any smooth curved relationship as locally approximated by a Taylor expansion, a quadratic "makes sense" as a form of model suggested by a second-order approximation to whatever the relationship is, one that's justifiable from the obvious stance that the form of model suggested by a first-order approximation is insufficient for your purposes. 
There are numerous caveats, including:


*

*you really can't extrapolate at all

*indeed you can't expect it to necessarily fit well over the range of data you have

*some people might start complaining about units
The last one is the easiest to deal with. The corresponding unit-adjustment is in the coefficient. 
That is, yes, you really can square 'year'.

It's better if you can come up with a theoretically justified, or at least biologically-plausible model for the yield vs time relationship, but that's not always possible.
There are other approaches to dealing with curved relationships for which you have no obvious model, such as splines or kernel smoothing.
