Y being correlated to X is not the same thing as autocorrelation. The former shows that X predicts Y, the later that the residuals are correlated with each other I believe, because of the nature of time series data commonly.
Unless you are using X and Y differently than commonly done. I am not certain of the logic of predicting a present Y from a future X. One of the requirements logically (not statistically) for X to predict Y is that X must occur before Y.
I am guessing you are using notation different than I have encountered in the past.