Cochrane–Orcutt estimation is a procedure in econometrics, which adjusts a linear model for serial correlation in the error term.
Developed in the 1940s, it is named after statisticians Donald Cochrane and Guy Orcutt.
is a row vector of explanatory variables at time t, and
is the error term at time t. If it is found, for instance via the Durbin–Watson statistic, that if the error term is serially correlated over time, then standard statistical inference as normally applied to regressions is invalid because standard errors are estimated with bias.
To avoid this problem, the residuals must be modeled.
If the process generating the residuals is found to be a stationary first-order autoregressive structure,[2]
} being white noise, then the Cochrane–Orcutt procedure can be used to transform the model by taking a quasi-difference: In this specification the error terms are white noise, so statistical inference is valid.
The transformation suggested by Cochrane and Orcutt disregards the first observation of a time series, causing a loss of efficiency that can be substantial in small samples.
[3] A superior transformation, which retains the first observation with a weight of
was first suggested by Prais and Winsten,[4] and later independently by Kadilaya.
is not known, then it is estimated by first regressing the untransformed model and obtaining the residuals {
and making the transformed regression sketched above feasible.
(Note that one data point, the first, is lost in this regression.)
This procedure of autoregressing estimated residuals can be done once and the resulting value of
can be used in the transformed y regression, or the residuals of the residuals autoregression can themselves be autoregressed in consecutive steps until no substantial change in the estimated value of
The iterative Cochrane–Orcutt procedure might converge to a local but not global minimum of the residual sum of squares.
[6][7][8] This problem disappears when using the Prais–Winsten transformation instead, which keeps the initial observation.