The estimation of a lagged regression relation

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Abstract
The paper discusses a technique for estimating the matrices of coefficients, B(j), in a regression relation relating a vector time series, z(n), to lagged values, y(n—j), —p ≤ j ≤ q, of a second vector time series. The technique depends upon calculation of spectra and cross-spectra. Once these are computed the estimates B(j) are obtained successively without recalculation when an additional lag is introduced. When the residuals from the regression are generated by a linear process independent of y(n) it is shown that under some additional regularity conditions the estimates are asymptotically jointly normal, the variances and covariances of the elements of β(j) being independent of j and of p and q. The method of estimation is not efficient unless the spectra of the y process and the residual process are the same. Some idea of the magnitude of B(k) for lags for which computations have not been done can be obtained without doing these computations.

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