Cochrane–Orcutt estimation

Cochrane–Orcutt estimation is a procedure in

Theory

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

If it is found, for instance via the

${\displaystyle \varepsilon _{t}=\rho \varepsilon _{t-1}+e_{t},\ |\rho |<1}$, with the errors {${\displaystyle e_{t}}$} being

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

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 ${\displaystyle \rho }$.

Inefficiency

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 ${\displaystyle {\sqrt {(1-\rho ^{2})}}}$ was first suggested by Prais and Winsten,^[4] and later independently by Kadilaya.^[5]

Estimating the autoregressive parameter

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

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

References