M-estimator

Cochrane–Orcutt estimation is a procedure in econometrics, which adjusts a linear model for serial correlation in the error term. It is named after statisticians Donald Cochrane and Guy Orcutt, who worked in the Department of Applied Economics, Cambridge (U.K.).

Theory

Consider the model

$y_t=\alpha +X_t\beta +{\epsilon }_t,$

where $y_t$ is the value of the dependent variable of interest at time t, $\beta$ is a column vector of coefficients to be estimated, $X_t$ is a row vector of explanatory variables at time t, and ${\epsilon }_t$ is the error term at time t.

If it is found via the Durbin–Watson statistic that 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, ${\epsilon }_t=\rho {\epsilon }_{t-1}+e_t,\left|\rho \right|<1$, with the errors {$e_t$} being white noise, then the Cochrane–Orcutt procedure can be used to transform the model by taking a quasi-difference:

$y_t-\rho y_{t-1}=\alpha (1-\rho )+\beta (X_t-\rho X_{t-1})+e_t.$

In this specification the error terms are white noise, so statistical inference is valid. Then the sum of squared residuals (the sum of squared estimates of $e_t^2$) is minimized with respect to $(\alpha ,\beta )$, conditional on $\rho$.

Estimating the autoregressive parameter

If $\rho$ is not known, then it is estimated by first regressing the untransformed model and obtaining the residuals {${\hat{\epsilon }}_t$}, and regressing ${\hat{\epsilon }}_t$ on ${\hat{\epsilon }}_{t-1}$, leading to an estimate of $\rho$ 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 residulals can be done once and the resulting value of $\rho$ 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 $\rho$ is observed.

See also

• Prais–Winsten transformation
• Newey–West estimator

Literature

• Cochrane and Orcutt. 1949. "Application of least squares regression to relationships containing autocorrelated error terms". Journal of the American Statistical Association 44, pp 32–61