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| '''Cochrane–Orcutt estimation''' is a procedure in [[econometrics]], which adjusts a [[linear model]] for [[serial correlation]] in the [[error term]]. It is named after [[statistician]]s [[Donald Cochrane (economist)|Donald Cochrane]] and [[Guy Orcutt]], who worked in the Department of Applied Economics, [[University of Cambridge|Cambridge (U.K.)]].
| | Andera is what you can contact her but she by no means truly liked that title. For a while I've been in Alaska but I will have to transfer in a year or two. Invoicing is my profession. She is really fond of caving but she doesn't have the time recently.<br><br>my web site :: [http://www.skullrocker.com/blogs/post/10991 online psychics] |
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| ==Theory==
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| Consider the model
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| :<math>y_t = \alpha + X_t \beta+\varepsilon_t,\,</math>
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| where <math>y_{t}</math> is the value of the [[dependent variable]] of interest at time ''t'', <math>\beta</math> is a column [[Vector (geometry)|vector]] of coefficients to be estimated, <math>X_{t}</math> is a row vector of [[explanatory variable]]s at time ''t'', and <math>\varepsilon_t</math> is the [[error term]] at time ''t''.
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| If it is found via the [[Durbin–Watson statistic]] that the error term is [[serial correlation|serially correlated]] over time, then standard [[statistical inference]] as normally applied to [[ordinary least squares|regressions]] is invalid because [[standard error]]s are estimated with [[bias (statistics)|bias]]. To avoid this problem, the residuals must be modeled. If the process generating the residuals is found to be a [[stationary process|stationary]] first-order [[autoregressive model|autoregressive structure]], <math>\varepsilon_t =\rho \varepsilon_{t-1}+e_t,\ |\rho| <1 </math>, with the errors {<math>e_t</math>} being [[white noise]], then the Cochrane–Orcutt procedure can be used to transform the model by taking a quasi-difference:
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| :<math>y_t - \rho y_{t-1} = \alpha(1-\rho)+\beta(X_t - \rho X_{t-1}) + e_t. \,</math>
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| 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 <math>e_t^2</math>) is minimized with respect to <math>(\alpha,\beta)</math>, conditional on <math>\rho</math>.
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| ==Estimating the autoregressive parameter==
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| If <math>\rho</math> is not known, then it is estimated by first regressing the untransformed model and obtaining the residuals {<math>\hat{\varepsilon}_t</math>}, and regressing <math>\hat{\varepsilon}_t</math> on <math>\hat{\varepsilon}_{t-1}</math>, leading to an estimate of <math>\rho</math> 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 <math>\rho</math> 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 <math>\rho</math> is observed.
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| ==See also==
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| * [[Prais–Winsten transformation]]
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| * [[Newey–West estimator]]
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| ==Literature==
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| * 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
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| {{DEFAULTSORT:Cochrane-Orcutt Estimation}}
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| [[Category:Econometrics]]
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| [[Category:Time series analysis]]
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| [[Category:Regression with time series structure]]
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Andera is what you can contact her but she by no means truly liked that title. For a while I've been in Alaska but I will have to transfer in a year or two. Invoicing is my profession. She is really fond of caving but she doesn't have the time recently.
my web site :: online psychics