Correct T-Statistics for Overlapping Windows

# Correct T-Statistics for Overlapping Windows A time-series regression uses observations built from overlapping rolling sums of current and lagged iid increments. Consecutive observations therefore reuse many of the same increments. Assess whether ordinary OLS t-statistics are reliable, explain the usual direction of distortion, and give a standard-error correction that does not require refitting the regression coefficients. ### Constraints & Assumptions - Each rolling value spans `L > 1` consecutive increments and advances one period at a time. - The coefficient estimator satisfies the intended exogeneity condition. - The original design matrix and fitted residuals remain available. - The sample is long enough for asymptotic covariance estimates to be meaningful. ### Clarifying Questions to Ask - Is the overlapping sum in the dependent variable, a regressor, or both? - Are the underlying increments homoskedastic and truly iid? - Does the regression include additional serially correlated controls? - How large is `L` relative to the sample? ### What a Strong Answer Covers - Dependence created by shared increments despite iid primitives - The distinction between coefficient bias and standard-error bias - Why positive overlap commonly inflates absolute naive t-statistics - HAC or block-based inference using the existing fit - Bandwidth, finite-sample, and effective-sample-size considerations ### Follow-up Questions 1. Why is dividing the nominal sample size by `L` only an approximation? 2. How would you choose a HAC bandwidth? 3. What changes if the overlap induces negative rather than positive autocorrelation? ```hint Estimate long-run variance Keep the fitted coefficients, but replace the iid covariance estimate with one that includes lagged covariance of the regression score or residual process. ```