Cochrane–Orcutt estimation

Consider the model

:$y\_t\; =\; \backslash alpha\; +\; X\_t\; \backslash beta+\backslash varepsilon\_t,\backslash ,$

where $y\_\{t\}$ is the value of the dependent variable of interest at time t, $\backslash beta$ is a column vector of coefficients to be estimated, $X\_\{t\}$ is a row vector of explanatory variables at time t, and $\backslash varepsilon\_t$ 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,{{cite book |last=Wooldridge |first=Jeffrey M. |authorlink=Jeffrey Wooldridge |year=2013 |title=Introductory Econometrics: A Modern Approach |location=Mason, OH |publisher=South-Western |edition=Fifth international |pages=409–415 |isbn=978-1-111-53439-4 }} , 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:

:$y\_t\; -\; \backslash rho\; y\_\{t-1\}\; =\; \backslash alpha(1-\backslash rho)+(X\_t\; -\; \backslash rho\; X\_\{t-1\})\backslash beta\; +\; e\_t.\; \backslash ,$

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 $(\backslash alpha,\backslash beta)$, conditional on $\backslash rho$.

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.{{cite journal |first1=Potluri |last1=Rao |first2=Zvi |last2=Griliches |authorlink2=Zvi Griliches |title=Small-Sample Properties of Several Two-Stage Regression Methods in the Context of Auto-Correlated Errors |journal=Journal of the American Statistical Association |volume=64 |issue=325 |year=1969 |pages=253–272 |doi=10.1080/01621459.1969.10500968 |jstor=2283733 }} A superior transformation, which retains the first observation with a weight of $\backslash sqrt\{(1-\backslash rho^\{2\})\}$ was first suggested by Prais and Winsten,{{Cite journal | last1=Prais | first1=S. J. | last2=Winsten| first2=C. B. | title=Trend Estimators and Serial Correlation | year=1954 |url=https://cowles.yale.edu/sites/default/files/files/pub/cdp/s-0383.pdf |location=Chicago |journal=Cowles Commission Discussion Paper No. 383 }} and later independently by Kadilaya.{{cite journal |first=Koteswara Rao |last=Kadiyala |title=A Transformation Used to Circumvent the Problem of Autocorrelation |journal=Econometrica |volume=36 |issue=1 |year=1968 |pages=93–96 |doi=10.2307/1909605 |jstor=1909605 }}

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

The iterative Cochrane–Orcutt procedure might converge to a local but not global minimum of the residual sum of squares.{{Cite journal | doi = 10.1016/0165-1765(80)90055-5| title = The Cochrane-Orcutt procedure numerical examples of multiple admissible minima| journal = Economics Letters| volume = 6 |issue=1| pages = 43–48| year = 1980| last1 = Dufour | first1 = J. M.| last2 = Gaudry | first2 = M. J. I.| last3 = Liem | first3 = T. C.}}{{cite journal |first1=Leslie T. |last1=Oxley |first2=Colin J. |last2=Roberts |title=Pitfalls in the Application of the Cochrane‐Orcutt Technique |journal=Oxford Bulletin of Economics and Statistics |volume=44 |issue=3 |year=1982 |pages=227–240 |doi=10.1111/j.1468-0084.1982.mp44003003.x }}{{Cite journal | doi = 10.1007/BF01973194| title = A warning on the use of the Cochrane-Orcutt procedure based on a money demand equation| journal = Empirical Economics| volume = 8| issue = 2| pages = 111–117| year = 1983| last1 = Dufour | first1 = J. M.| last2 = Gaudry | first2 = M. J. I.| last3 = Hafer | first3 = R. W.| s2cid = 152953205}} This problem disappears when using the Prais–Winsten transformation instead, which keeps the initial observation.{{cite journal |first1=Howard |last1=Doran |first2=Jan |last2=Kmenta |title=Multiple Minima in the Estimation of Models With Autoregressive Disturbances |journal=Review of Economics and Statistics |volume=74 |issue=2 |year=1992 |pages=354–357 |doi=10.2307/2109671 |jstor=2109671 |hdl=2027.42/91908 |hdl-access=free }}

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

{{Reflist|30em}}

• {{cite book |last1=Davidson |first1=Russell |last2=MacKinnon |first2=James G. |author2link=James G. MacKinnon |title=Estimation and Inference in Econometrics |location= |publisher=Oxford University Press |edition= |year=1993 |isbn=0-19-506011-3 |pages=327–373 }}
• {{cite book |last1=Fomby |first1=Thomas B. |last2=Hill |first2=R. Carter |last3=Johnson |first3=Stanley R. |chapter=Autocorrelation |pages=205–236 |title=Advanced Econometric Methods |location=New York |publisher=Springer |year=1984 |isbn=0-387-96868-7 }}
• {{cite book |last=Hamilton |first=James D. |authorlink=James D. Hamilton |title=Time Series Analysis |location=Princeton |publisher=Princeton University Press |edition= |year=1994 |isbn=0-691-04289-6 |pages=220–225 }}
• {{cite book |last=Johnston |first=John |authorlink=John Johnston (econometrician) |title=Econometric Methods |location=New York |publisher=McGraw-Hill |edition=Second |year=1972 |pages=259–265 }}
• {{cite book |last=Kmenta |first=Jan |authorlink=Jan Kmenta |chapter= |pages=[https://archive.org/details/elementsofeconom0003kmen/page/302 302–317] |title=Elements of Econometrics |location=New York |publisher=Macmillan |year=1986 |edition=Second |isbn=0-02-365070-2 |url-access=registration |url=https://archive.org/details/elementsofeconom0003kmen/page/302 }}

• {{YouTube|id=hX9dBODrX4Y&index=10&list=PLD15D38DC7AA3B737#t=21m19s|title=Econometrics lecture (topic: Cochrane–Orcutt procedure)}} by Mark Thoma.

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Category:Autocorrelation

Category:Curve fitting

Category:Regression with time series structure