If I’m reading this right, you'll want to add one new row at the top of the table for 31-Dec-1962, then a variable for Beta1 following Var1, Beta2 following Var2, etc. The last row in each Beta would be NaN.
LAGMATRIX in Econometrics Toolbox will create lagged versions of all series at all lags in one call. Running the regressions with mldivide would then be a matter of indexing into the appropriate column to obtain each y_lag.
Writing successive betas to the table would be along the lines of Tbl.Betaj(i) = beta(i,j), where the beta(i,j) are the successive regression results.
