Cochrane-Orcutt estimate

The Cochrane-Orcutt estimation (CO) is an iterative estimation method that is mainly used in econometrics and with which one can estimate error terms of the first-order autocorrelation and strictly exogenous variables in a multiple linear regression model . It was named after statisticians Donald Cochrane and Guy Orcutt .

Basics

The following is based on the general linear regression model:

${\displaystyle y_{t}=x_{t1}\beta _{1}+x_{t2}\beta _{2}+\dotsb +x_{tK}\beta _{K}+\varepsilon _{t}=\mathbf {x} _{t}^{T}\mathbf {\beta } +\varepsilon _{t},\quad t=1,2,\dots ,T}$

With

${\displaystyle \mathbf {x} _{t}={\begin{pmatrix}\ x_{t1}\\\ x_{t2}\\\vdots \\\ x_{tk}\\\vdots \\\ x_{tK}\end{pmatrix}}_{(K\times 1)}}$ and ${\displaystyle \;\;\;\mathbf {\beta } ={\begin{pmatrix}\beta _{1}\\\beta _{2}\\\vdots \\\beta _{k}\\\vdots \\\beta _{K}\end{pmatrix}}_{(K\times 1)}}$

, with , d. H. the error terms are serially correlated over time . ${\displaystyle \forall t\colon \operatorname {E} (\varepsilon _{t},)=0}$ ${\displaystyle \forall t\neq s\colon \operatorname {Cov} (\varepsilon _{t},\varepsilon _{s})=\operatorname {E} (\varepsilon _{t}\varepsilon _{s})\neq 0}$

Representations:

${\displaystyle y_{t}}$are observed random variables .

${\displaystyle x_{tk}}$ are observable, not random, known variables.

${\displaystyle \beta _{1},\beta _{2},\beta _{3},\dotsc ,\beta _{k}}$are unknown scalar parameters.

${\displaystyle \varepsilon _{t}}$ are unobservable random variables.

${\displaystyle \mathbf {x} _{t}^{T}}$is the transposed vector of the regressors

If by the Durbin-Watson - test statistic is determined that the error terms over time auto-correlated , then the "normal" is statistical inference useless because the standard error a bias has.

To avoid this problem, the residuals must be transformed. If there is a stationary autoregressive process of the first order, the error terms are modified as follows:

${\displaystyle \varepsilon _{t}=\rho \varepsilon _{t-1}+\nu _{t},\ |\rho |<1}$,

where the error term represents white noise . It is assumed that the error term is in turn dependent on another error term . The following assumptions are made about the newly added error term: ${\displaystyle \varepsilon _{t}}$ ${\displaystyle \varepsilon _{t}}$ ${\displaystyle \nu _{t}}$

• The expected value of the additive error terms is zero: ${\displaystyle \forall t\colon \operatorname {E} (\nu _{t})=0}$
• The error terms are uncorrelated : ${\displaystyle \forall t\neq s\colon \operatorname {Cov} (\nu _{t},\nu _{s})=\operatorname {E} (\nu _{t}\nu _{s})=0}$
• and have a constant variance (homoscedasticity) , ${\displaystyle \forall t\colon \operatorname {Var} (\nu _{t})=\operatorname {E} [(\nu _{t}-\operatorname {E} (\nu _{t}))^{2}]=\sigma _{\nu }^{2}=\mathrm {const.} }$

d. H. it applies . ${\displaystyle \forall t:\nu _{t}\sim (0,\sigma ^{2}\mathbf {I} _{T}))}$

These assumptions form the general linear statistical model with first order autoregressive perturbation. Together with , it is now the goal to estimate the unknown parameter . To estimate, one is again interested in the statistical properties and the mean and variance of . To derive these terms, one assumes that the process was initiated in the past and has been running for a long time. It is also assumed that the condition is met. If this condition is met, the first order autoregressive process is stationary. Stationary means that the mean, the variance and the covariance of themselves do not change over time, i.e. are constant. ${\displaystyle {\boldsymbol {\beta }}}$ ${\displaystyle \rho }$ ${\displaystyle {\boldsymbol {\beta }}}$ ${\displaystyle \varepsilon _{t}}$ ${\displaystyle -1<\rho <1}$ ${\displaystyle \varepsilon _{t}}$

${\displaystyle \varepsilon _{t}}$can be expressed as a weighted sum of a time series of uncorrelated and identically distributed error terms: for whose expected value applies: ${\displaystyle \varepsilon _{t}=\rho \varepsilon _{t-1}+\nu _{t}=\nu _{t}+\rho \nu _{t-1}+\rho ^{2}\nu _{t-2}+\ldots =\sum _{i=0}^{\infty }\rho ^{i}\nu _{t-i}}$

${\displaystyle \operatorname {E} (\varepsilon _{t})=\operatorname {E} \left(\sum _{i=0}^{\infty }\rho ^{i}\nu _{t-i}\right)=\sum _{i=0}^{\infty }\rho ^{i}\operatorname {E} [\nu _{t-i}]=0}$

The following applies to the variance: ${\displaystyle \operatorname {Var} (\varepsilon _{t})=\operatorname {E} [(\rho \varepsilon _{t-1}+\nu _{t})^{2}]={\frac {\sigma _{\nu }^{2}}{1-\rho ^{2}}}=\sigma _{\varepsilon }^{2}={\text{const.}}}$

and the covariance: ${\displaystyle \operatorname {Cov} (\varepsilon _{t},\varepsilon _{s})=\operatorname {E} (\varepsilon _{t}\varepsilon _{s})=\rho ^{s}{\frac {\sigma ^{2}}{1-\rho ^{2}}}=\rho ^{s}\operatorname {Var} (\varepsilon _{t})}$

The Cochrane-Orcutt procedure can be used to transform the model by a quasi-difference :

${\displaystyle y_{t}-\rho y_{t-1}=\beta _{1}(1-\rho )+\beta _{2}(x_{t}-\rho x_{t-1})+\nu _{t}}$.

In this specification, the error terms are white noise, so statistical inference is valid.

Thereafter, the sum of the squared residuals can be minimized with the least squares method in terms of , under the condition of . ${\displaystyle (\beta _{1},\beta _{2})}$ ${\displaystyle \rho }$

Estimation of the autoregressive parameter

If the parameter is not known, it is estimated by first regressing the untransformed model and then extracting the residuals . The regression from on gives an estimate for what makes the above-mentioned transformed regression feasible. Note that this regression will lose the first data value. ${\displaystyle \rho }$ ${\displaystyle {\hat {\varepsilon }}_{t}}$ ${\displaystyle {\hat {\varepsilon }}_{t}}$ ${\displaystyle {\hat {\varepsilon }}_{t-1}}$ ${\displaystyle \rho }$

This estimation process is carried out once. The value of thus obtained can be used in the transformed regression. However, the residuals of the autoregression of the residuals themselves can be subjected to autoregression in successive steps until no significant change in the estimated value of can be observed. ${\displaystyle \rho }$ ${\displaystyle y}$ ${\displaystyle \rho }$

It should be noted that the Cochrane-Orcutt iterative estimate can find a local minimum rather than the global minimum of the sum of squares of the residuals .

literature

• D. Cochrane, GH Orcutt: Application of least squares regression to relationships containing auto-correlated error terms. In: Journal of the American Statistical Association , Volume 44, Issue 245, 1949, pp. 32-61.
• John Black, Nigar Hashimzade, Gareth Myles (Eds.): A Dictionary of Economics. Oxford University Press, 2009, ISBN 978-0-19-923704-3 .

Individual evidence

1. ↑ Jeffrey M. Wooldridge: Introductory Econometrics A Modern Approach , p. 845
2. ↑ Jeffrey M. Wooldridge : Introductory Econometrics: A Modern Approach ( en ), Fifth international. Edition, South-Western, Mason, OH 2013, ISBN 978-1-111-53439-4 , pp. 409-415.
3. ↑ J.-M. Dufour, MJI Gaudry, TC Liem: The Cochrane-Orcutt procedure numerical examples of multiple admissible minima . In: Economics Letters . 6, No. 1, 1980, pp. 43-48. doi : 10.1016 / 0165-1765 (80) 90055-5 .
4. ↑ J.-M. Dufour, MJI Gaudry, RW Hafer: A warning on the use of the Cochrane-Orcutt procedure based on a money demand equation . In: Empirical Economics . 8, No. 2, 1983, pp. 111-117. doi : 10.1007 / BF01973194 .