Cochrane–Orcutt estimation

Consider the model

${\displaystyle y_{t}=\alpha +X_{t}\beta +\varepsilon _{t},\,}$

where ${\displaystyle y_{t}}$ is the value of the dependent variable of interest at time t, ${\displaystyle \beta }$ is a column vector of coefficients to be estimated, ${\displaystyle X_{t}}$ is a row vector of explanatory variables at time t, and ${\displaystyle \varepsilon _{t}}$ is the error term at time t.

If it is found, for instance 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,[2] ${\displaystyle \varepsilon _{t}=\rho \varepsilon _{t-1}+e_{t},\ |\rho |<1}$, with the errors {${\displaystyle e_{t}}$} being white noise, then the Cochrane–Orcutt procedure can be used to transform the model by taking a quasi-difference:

${\displaystyle y_{t}-\rho y_{t-1}=\alpha (1-\rho )+(X_{t}-\rho X_{t-1})\beta +e_{t}.\,}$

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 ${\displaystyle e_{t}^{2}}$) is minimized with respect to ${\displaystyle (\alpha ,\beta )}$, conditional on ${\displaystyle \rho }$.

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

It has to be noted, though, that 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.[9]

• Hildreth–Lu estimation
• Newey–West estimator
• Prais–Winsten estimation
• Feasible generalized least squares

• Davidson, Russell; MacKinnon, James G. (1993). Estimation and Inference in Econometrics. Oxford University Press. pp. 327–373. ISBN 0-19-506011-3.
• Fomby, Thomas B.; Hill, R. Carter; Johnson, Stanley R. (1984). "Autocorrelation". Advanced Econometric Methods. New York: Springer. pp. 205–236. ISBN 0-387-96868-7.
• Hamilton, James D. (1994). Time Series Analysis. Princeton: Princeton University Press. pp. 220–225. ISBN 0-691-04289-6.
• Johnston, John (1972). Econometric Methods (Second ed.). New York: McGraw-Hill. pp. 259–265.
• Kmenta, Jan (1986). Elements of Econometrics (Second ed.). New York: Macmillan. pp. 302–317. ISBN 0-02-365070-2.

• Econometrics lecture (topic: Cochrane–Orcutt procedure) on YouTube by Mark Thoma.