A return-prediction coefficient varies through the day, but each intraday regression is too noisy on its own. How would you estimate the coefficients?

A researcher believes the previous 10-minute return predicts the next 10-minute return, with a coefficient that varies through the day: y(d,t) = beta_t * x(d,t) + noise, for day d = 0..D-1 and intraday bucket t = 0..T-1. He could run T separate OLS regressions, but the data set is small (D = 300 days, T = 100 buckets) and the signal is weak (at best about 5% correlation with the target). He hopes a method that shares information across neighbouring intraday times will help. You are given the 300 x 100 matrices x and y. How would you estimate the beta_t?