Cochrane–Orcutt estimation

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 D. Cochrane and G. H. Orcutt, who worked in the Department of Applied Economics, Cambridge (U.K.).

Theory

Consider the model

$y_t = \alpha + X_t \beta+\varepsilon_t,\,$

where y_t is the time series of interest at time t, β is a vector of coefficients, X_t is a matrix of explanatory variables, and ε_t is the error term. The error term can be serially correlated over time: $\varepsilon_t =\rho \varepsilon_{t-1}+e_t,\ |\rho| <1$. The Cochrane–Orcutt procedure transforms the model:

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

Then the sum of squared residuals $e_t^2$ is minimized with respect to (α,β), conditional on ρ.

Restrictions

If ρ is not known, then it is estimated by first getting the residuals of the model $\hat{\varepsilon}_t$ and regressing $\hat{\varepsilon}_t$ on $\hat{\varepsilon}_{t-1}$, leading to an estimate of ρ and making the transformed regression sketched above feasible. This procedure can be done until in consecutive steps of estimating the correlation coefficient no substantial change is observed. Note that this procedure is designed for an AR(1) error term structure and you would lose the first observation, which might be important for small samples.

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