Cochrane–Orcutt Estimation
   HOME

TheInfoList



OR:

Cochrane–Orcutt estimation is a procedure in
econometrics Econometrics is the application of statistical methods to economic data in order to give empirical content to economic relationships.M. Hashem Pesaran (1987). "Econometrics," '' The New Palgrave: A Dictionary of Economics'', v. 2, p. 8 p. 8†...
, which adjusts a
linear model In statistics, the term linear model is used in different ways according to the context. The most common occurrence is in connection with regression models and the term is often taken as synonymous with linear regression model. However, the term ...
for
serial correlation Autocorrelation, sometimes known as serial correlation in the discrete time case, is the correlation of a signal with a delayed copy of itself as a function of delay. Informally, it is the similarity between observations of a random variable as ...
in the
error term In mathematics and statistics, an error term is an additive type of error. Common examples include: * errors and residuals in statistics, e.g. in linear regression In statistics, linear regression is a linear approach for modelling the relati ...
. Developed in the 1940s, it is named after
statistician A statistician is a person who works with theoretical or applied statistics. The profession exists in both the private and public sectors. It is common to combine statistical knowledge with expertise in other subjects, and statisticians may wor ...
s Donald Cochrane and Guy Orcutt.


Theory

Consider the model :y_t = \alpha + X_t \beta+\varepsilon_t,\, where y_ is the value of the
dependent variable Dependent and independent variables are variables in mathematical modeling, statistical modeling and experimental sciences. Dependent variables receive this name because, in an experiment, their values are studied under the supposition or dema ...
of interest at time ''t'', \beta is a column
vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
of coefficients to be estimated, X_ is a row vector of
explanatory variable Dependent and independent variables are variables in mathematical modeling, statistical modeling and experimental sciences. Dependent variables receive this name because, in an experiment, their values are studied under the supposition or demand ...
s at time ''t'', and \varepsilon_t is the
error term In mathematics and statistics, an error term is an additive type of error. Common examples include: * errors and residuals in statistics, e.g. in linear regression In statistics, linear regression is a linear approach for modelling the relati ...
at time ''t''. If it is found, for instance via the
Durbin–Watson statistic In statistics, the Durbin–Watson statistic is a test statistic used to detect the presence of autocorrelation at lag 1 in the residuals (prediction errors) from a regression analysis. It is named after James Durbin and Geoffrey Watson. The ...
, that the error term is serially correlated over time, then standard statistical inference as normally applied to regressions is invalid because
standard error The standard error (SE) of a statistic (usually an estimate of a parameter) is the standard deviation of its sampling distribution or an estimate of that standard deviation. If the statistic is the sample mean, it is called the standard error ...
s are estimated with
bias Bias is a disproportionate weight ''in favor of'' or ''against'' an idea or thing, usually in a way that is closed-minded, prejudicial, or unfair. Biases can be innate or learned. People may develop biases for or against an individual, a group ...
. 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, \varepsilon_t =\rho \varepsilon_+e_t,\ , \rho, <1 , with the errors being
white noise In signal processing, white noise is a random signal having equal intensity at different frequencies, giving it a constant power spectral density. The term is used, with this or similar meanings, in many scientific and technical disciplines, ...
, then the Cochrane–Orcutt procedure can be used to transform the model by taking a quasi-difference: :y_t - \rho y_ = \alpha(1-\rho)+(X_t - \rho X_)\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 e_t^2) is minimized with respect to (\alpha,\beta), conditional on \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. A superior transformation, which retains the first observation with a weight of \sqrt was first suggested by Prais and Winsten, and later independently by Kadilaya.


Estimating the autoregressive parameter

If \rho is not known, then it is estimated by first regressing the untransformed model and obtaining the residuals , and regressing \hat_t on \hat_, leading to an estimate of \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 \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 \rho is observed. It has to be noted, though, that the iterative Cochrane–Orcutt procedure might converge to a local but not
global minimum In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given r ...
of the residual sum of squares. This problem disappears when using the Prais–Winsten transformation instead, which keeps the initial observation.


See also

* Hildreth–Lu estimation * Newey–West estimator * Prais–Winsten estimation *
Feasible generalized least squares In statistics, generalized least squares (GLS) is a technique for estimating the unknown parameters in a linear regression model when there is a certain degree of correlation between the residuals in a regression model. In these cases, ordina ...


References


Further reading

* * * * *


External links

* by
Mark Thoma Mark Allen Thoma (born December 15, 1956) is a macroeconomist and econometrician and a professor of economics at the Department of Economics of the University of Oregon. Thoma is best known as a regular columnist for ''The Fiscal Times'' through ...
. {{DEFAULTSORT:Cochrane-Orcutt Estimation Autocorrelation Curve fitting Regression with time series structure