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In econometrics, Prais–Winsten estimation is a procedure meant to take care of the serial correlation of type AR(1) in a linear model. Conceived by Sigbert Prais and Christopher Winsten in 1954,[1] it is a modification of Cochrane–Orcutt estimation in the sense that it does not lose the first observation, which leads to more efficiency as a result and makes it a special case of feasible generalized least squares.[2]

## Contents

* 1 Theory
* 2 Estimation procedure
* 3 Note
* 4 Restrictions
* 5 References
* 6 Further reading

## Theory

Consider the model

[math]\displaystyle{ y_t = \alpha + X_t \beta+\varepsilon_t,\, }[/math]

where [math]\displaystyle{ y_{t} }[/math] is the time series of interest at time t, [math]\displaystyle{ \beta }[/math] is a vector of coefficients, [math]\displaystyle{ X_{t} }[/math] is a matrix of explanatory variables, and [math]\displaystyle{ \varepsilon_t }[/math] is the error term. The error term can be serially correlated over time: [math]\displaystyle{ \varepsilon_t =\rho \varepsilon_{t-1}+e_t,\ |\rho| \lt 1 }[/math] and [math]\displaystyle{ e_t }[/math] is white noise. In addition to the Cochrane–Orcutt transformation, which is

[math]\displaystyle{ y_t - \rho y_{t-1} = \alpha(1-\rho)+(X_t - \rho X_{t-1})\beta + e_t, \, }[/math]

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

[math]\displaystyle{ \sqrt{1-\rho^2}y_1 = \alpha\sqrt{1-\rho^2}+\left(\sqrt{1-\rho^2}X_1\right)\beta + \sqrt{1-\rho^2}\varepsilon_1. \, }[/math]

Then the usual least squares estimation is done.

## Estimation procedure

First notice that

[math]\displaystyle{ \mathrm{var}(\varepsilon_t)=\mathrm{var}(\rho\varepsilon_{t-1}+e_{it})=\rho^2 \mathrm{var}(\varepsilon_{t-1}) +\mathrm{var}(e_{it}) }[/math]

Noting that for a stationary process, variance is constant over time,

[math]\displaystyle{ (1-\rho^2 )\mathrm{var}(\varepsilon_t)= \mathrm{var}(e_{it}) }[/math]

and thus,

[math]\displaystyle{ \mathrm{var}(\varepsilon_t)=\frac{\mathrm{var}(e_{it})}{(1-\rho^2 )} }[/math]

Without loss of generality suppose the variance of the white noise is 1. To do the estimation in a compact way one must look at the autocovariance function of the error term considered in the model blow:

[math]\displaystyle{ \mathrm{cov}(\varepsilon_t,\varepsilon_{t+h})=\rho^h \mathrm{var}(\varepsilon_t)=\frac{\rho^h}{1-\rho^2}, \text{ for } h=0,\pm 1, \pm 2, \dots \, . }[/math]

It is easy to see that the variance–covariance matrix, [math]\displaystyle{ \mathbf{\Omega} }[/math], of the model is

[math]\displaystyle{ \mathbf{\Omega} = \begin{bmatrix} \frac{1}{1-\rho^2} & \frac{\rho}{1-\rho^2} & \frac{\rho^2}{1-\rho^2} & \cdots & \frac{\rho^{T-1}}{1-\rho^2} \\\\[8pt] \frac{\rho}{1-\rho^2} & \frac{1}{1-\rho^2} & \frac{\rho}{1-\rho^2} & \cdots & \frac{\rho^{T-2}}{1-\rho^2} \\\\[8pt] \frac{\rho^2}{1-\rho^2} & \frac{\rho}{1-\rho^2} & \frac{1}{1-\rho^2} & \cdots & \frac{\rho^{T-3}}{1-\rho^2} \\\\[8pt] \vdots & \vdots & \vdots & \ddots & \vdots \\\\[8pt] \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{bmatrix}. }[/math]

Having [math]\displaystyle{ \rho }[/math] (or an estimate of it), we see that,

[math]\displaystyle{ \hat{\Theta}=(\mathbf{Z}^{\mathsf{T}}\mathbf{\Omega}^{-1}\mathbf{Z})^{-1}(\mathbf{Z}^{\mathsf{T}}\mathbf{\Omega}^{-1}\mathbf{Y}), \, }[/math]

where [math]\displaystyle{ \mathbf{Z} }[/math] is a matrix of observations on the independent variable (Xt, t = 1, 2, ..., T) including a vector of ones, [math]\displaystyle{ \mathbf{Y} }[/math] is a vector stacking the observations on the dependent variable (yt, t = 1, 2, ..., T) and [math]\displaystyle{ \hat{\Theta} }[/math] includes the model parameters.

## Note

To see why the initial observation assumption stated by Prais–Winsten (1954) is reasonable, considering the mechanics of generalized least square estimation procedure sketched above is helpful. The inverse of [math]\displaystyle{ \mathbf{\Omega} }[/math] can be decomposed as [math]\displaystyle{ \mathbf{\Omega}^{-1}=\mathbf{G}^{\mathsf{T}}\mathbf{G} }[/math] with[3]

[math]\displaystyle{ \mathbf{G} = \begin{bmatrix} \sqrt{1-\rho^2} & 0 & 0 & \cdots & 0 \\\ -\rho & 1 & 0 & \cdots & 0 \\\ 0 & -\rho & 1 & \cdots & 0 \\\ \vdots & \vdots & \vdots & \ddots & \vdots \\\ 0 & 0 & 0 & \cdots & 1 \end{bmatrix}. }[/math]

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 [math]\displaystyle{ \rho }[/math] is not known, a recursive procedure (Cochrane–Orcutt estimation) or grid-search (Hildreth–Lu estimation) may be used to make the estimation feasible. Alternatively, a full information maximum likelihood procedure that estimates all parameters simultaneously has been suggested by Beach and MacKinnon.[4][5]

## References

1. ↑ Prais, S. J.; Winsten, C. B. (1954). "Trend Estimators and Serial Correlation". Cowles Commission Discussion Paper No. 383 (Chicago). https://cowles.yale.edu/sites/default/files/files/pub/cdp/s-0383.pdf.
2. ↑ Johnston, John (1972). Econometric Methods (2nd ed.). New York: McGraw-Hill. pp. 259–265. ISBN 9780070326798. https://books.google.com/books?id=aBOaAAAAIAAJ&pg=259.
3. ↑ Kadiyala, Koteswara Rao (1968). "A Transformation Used to Circumvent the Problem of Autocorrelation". Econometrica 36 (1): 93–96. doi:10.2307/1909605.
4. ↑ Beach, Charles M.; MacKinnon, James G. (1978). "A Maximum Likelihood Procedure for Regression with Autocorrelated Errors". Econometrica 46 (1): 51–58. doi:10.2307/1913644.
5. ↑ Amemiya, Takeshi (1985). Advanced Econometrics. Cambridge: Harvard University Press. pp. 190–191. ISBN 0-674-00560-0. https://books.google.com/books?id=0bzGQE14CwEC&pg=PA190.

## Further reading

* Judge, George G.; Griffiths, William E.; Hill, R. Carter; Lee, Tsoung-Chao (1980). The Theory and Practice of Econometrics. New York: Wiley. pp. 180–183. ISBN 0-471-05938-2.
* Kmenta, Jan (1986). Elements of Econometrics (Second ed.). New York: Macmillan. pp. 302–320. ISBN 0-02-365070-2. https://archive.org/details/elementsofeconom0003kmen/page/302.

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