Do I have endogeneity problem if I use overlapping observations in AR(1) model? My dependent variable is $y_t=\frac{data_t}{data_{t-4}}-1$ with quarterly data and the model is standard AR(1): 
$$ y_t=\alpha_1 y_{t-1}+\beta x_t+\varepsilon_t. $$ 
I was told that due to overlapping observations I should have endogeneity problem and so I should use quarterly change instead of year-over-year change. I still couldn't see the correlation between the error term and the AR(1) term. Could someone help me on this?
 A: By using overlapping observations in an AR model, you end up having the same stuff on the left hand side as you have on the right hand side. If your data points span four quarters each, and the overlap is three quarters, then $y_t$ and $y_{t-1}$ have three quarters in common. That is, you could write $y_t=x_1+x_2+x_3+x_4$ and $y_{t-1}=x_0+x_1+x_2+x_3$. The overlapping component is $x_1+x_2+x_3$. 
You cannot think in terms " $\alpha_1$ tells what happens to $y_t$ when $y_{t-1}$ increases by 1" because generally you cannot move $y_t$ without moving $y_{t-1}$ (due to the overlapping component). 
From the perspective of whether the error term is orthogonal to the regressor, suppose there is a shock in a quarter that belongs both to the four-quarter period $t$ and the four-quarter period $t-1$. Then $y_{t-1}$ will be affected together with $y_t$. That is, the error term will not be orthogonal to the regressor.
Thus you do have an endogeneity problem. 
If you do not have a good reason for using year-over-year changes, why not use quarter-over-quarter changes? It should be as easy, and also free from the overlapping observations problem.
