Use of cross correlation between hydrological time series to improve estimates of lag one autoregressive parameters

The problem considered requires estimation of parameters characterizing a serially correlated hydrologic time series {yt}, where data are also available from a longer time series {xt}, itself serially correlated and cross correlated with {yt}. On the assumption that both series are lag one autoregressions (when {yt} is characterized by three parameters μy, β, and σϵ2), large-sample variances for the autoregressive parameter β are derived from the short series {yt} alone or from both series {xt} and {yt}, and the relative information is investigated numerically. It is concluded that, when the two serial correlations are approximately equal and the length of {xt} is twice that of {yt}, the gain in precision of the estimate β is about 4% when the cross correlation is 0.2, about 15% when the cross correlation is 0.4, about 31% when the cross correlation is 0.6, and about 49% when the cross correlation is 0.8. The equations giving maximum likelihood (ML) estimates are examined, and a relatively simple numerical technique is developed for one particular case of some practical importance. Small-sample properties of both ML estimates and an estimate derived from work by Matalas are investigated, and tentative conclusions are that (1) the modified Matalas estimates have smaller mean square error than ML estimates, and (2) ML estimates of μy and β derived from both series have smaller mean square error than ML estimates of μy and β derived from the single series only, the converse being true for σϵ2.