Modigliani risk-adjusted performance

In econometrics, Prais–Winsten estimation is a procedure meant to take care of the serial correlation of type AR(1) in a linear model. It is a modification of Cochrane–Orcutt estimation in the sense that it does not lose the first observation and leads to more efficiency as a result.

Theory

Consider the model

$y_t=\alpha +X_t\beta +{\epsilon }_t,$

where $y_t$ is the time series of interest at time t, $\beta$ is a vector of coefficients, $X_t$ is a matrix of explanatory variables, and ${\epsilon }_t$ is the error term. The error term can be serially correlated over time: ${\epsilon }_t=\rho {\epsilon }_{t-1}+e_t,\left|\rho \right|<1$ and $e_t$ is a white noise. In addition to the Cochrane–Orcutt procedure transformation, which is

$y_t-\rho y_{t-1}=\alpha (1-\rho )+\beta (X_t-\rho X_{t-1})+e_t.$

for t=2,3,...,T, Prais-Winsten procedure makes a reasonable transformation for t=1 in the following form

$\sqrt{1-{\rho }^2}{y}_1=\alpha \sqrt{1-{\rho }^2}+\left(\sqrt{1-{\rho }^2}{X}_1\right)\beta +\sqrt{1-{\rho }^2}{\epsilon }_1.$

Then the usual least squares estimation is done.

Estimation procedure

To do the estimation in a compact way it is directive to look at the auto-covariance function of the error term considered in the model above:

$\mathrm{c}\mathrm{o}\mathrm{v}({\epsilon }_t,{\epsilon }_{t+h})=\frac{{\rho }^h}{1-{\rho }^2},\text{ for }h=0,\pm 1,\pm 2,\dots .$

Now is easy to see that the variance-covariance, $\mathrm{\Omega }$, of the model is

$\mathrm{\Omega }=\left[\begin{array}{ccccc}\frac{1}{1-{\rho }^2}& \frac{\rho }{1-{\rho }^2}& \frac{{\rho }^2}{1-{\rho }^2}& \cdots & \frac{{\rho }^{T-1}}{1-{\rho }^2}\\ \frac{\rho }{1-{\rho }^2}& \frac{1}{1-{\rho }^2}& \frac{\rho }{1-{\rho }^2}& \cdots & \frac{{\rho }^{T-2}}{1-{\rho }^2}\\ \frac{{\rho }^2}{1-{\rho }^2}& \frac{\rho }{1-{\rho }^2}& \frac{1}{1-{\rho }^2}& \cdots & \frac{{\rho }^{T-2}}{1-{\rho }^2}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ \frac{{\rho }^{T-1}}{1-{\rho }^2}& \frac{{\rho }^{T-2}}{1-{\rho }^2}& \frac{{\rho }^{T-3}}{1-{\rho }^2}& \cdots & \frac{1}{1-{\rho }^2}\end{array}\right].$

Now having $\rho$ (or an estimate of it), we see that,

$\hat{\mathrm{\Theta }}=({\mathbf{Z}}^{\prime }{\mathrm{\Omega }}^{-1}\mathbf{Z}{)}^{-1}({\mathbf{Z}}^{\prime }{\mathrm{\Omega }}^{-1}\mathbf{Y}),$

where $\mathbf{Z}$ is a matrix of observations on the independent variable (X_t, t = 1, 2, ..., T) including a vector of ones, $\mathbf{Y}$ is a vector stacking the observations on the dependent variable (X_t, t = 1, 2, ..., T) and $\hat{\mathrm{\Theta }}$ includes the model parameters.

Note

To see why the initial observation assumption stated by Prais-Winsten (1954) is reasonable, considering the mechanics of general least square estimation procedure sketched above is helpful. The inverse of $\mathrm{\Omega }$ can be decomposed as ${\mathrm{\Omega }}^{-1}={\mathbf{G}}^{\prime }\mathbf{G}$ with

$\mathbf{G}=\left[\begin{array}{ccccc}\sqrt{1-{\rho }^2}& 0& 0& \cdots & 0\\ -\rho & 1& 0& \cdots & 0\\ 0& -\rho & 1& \cdots & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& \cdots & 1\end{array}\right].$

A pre-multiplication of model in a matrix notation with this matrix gives the transformed model of Prais-Winsten.

Restrictions

The error term is still restricted to be of an AR(1) type. If $\rho$ is not known, a recursive procedure may be used to make the estimation feasible. See Cochrane–Orcutt estimation.

References

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• Wooldridge, J. (2008) Introductory Econometrics: A Modern Approach, 4th Edition, South-Western Pub. ISBN 0-324-66054-5 (p. 435)