Correlation and coefficient with overlapping data

I have a time series data, with 5-period rates sampled at 1 period intervals. Essentially $$r_i = x_{i+5} - x_i ; ~~ i=1,...,N-5$$ This creates an overlapping data problem for regression. As far as I understand OLS estimates are unbiased but inefficient in this case. And we have methods like GLS or Newey-West to correct this.

Are there similar properties of the correlation coefficient as well?
Are there good ways to estimate the coefficient other than simply bootstrapping?

I face this in a primarily learning scenario, and don't want to go for Newey-West like estimators.

• Welcome to CV! I corrected the formatting of your question. In your questions and answers you can use Tex formatting enclosed in dollar signs ($) on both sides of equation. This makes a formula more readable. – Tim Dec 22 '14 at 14:26 • Do you mean$x_{i+5}\$?, the current formula will only leave a constant of 5. – Penguin_Knight Dec 22 '14 at 14:27
• Strictly speaking, using Newey-West type of covariance matrix does not cure the problem of inefficiency of OLS estimators. Using an appropriate covariance matrix means we acknowledge that they are inefficient, that's it, but nothing is done to remove the inefficiency. (I hope I am not mistaken.) GLS might be a better solution because it alters the point estimates to gain efficiency. – Richard Hardy Dec 22 '14 at 20:37
• possible duplicate of Time series regression with overlapping data – RockScience Sep 4 '15 at 4:31
• possible duplicate of What is this method for seasonal adjustment calculation? – StasK Sep 5 '15 at 1:57