Linear regression, predictor contains same components as depedent variable

#1
I came across this model: y(t) = f (y(t-1), X)
where
y ia a centered 3-month average, for example, for aug'12, y=average of Jul, Aug and Sep average.
X is other predictor.

This model is fitted on monthly data. As you can see, dependnet variable y(t) has two months overlap with predictor y(t-1): Aug and Sep values. The r-square is very high because of that.

Any problem with this model?

Thank you very much for your attention.