# Prais–Winsten estimation

(Redirected from Prais-Winsten transformation)

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]

## Theory

Consider the model

${\displaystyle y_{t}=\alpha +X_{t}\beta +\varepsilon _{t},\,}$

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

${\displaystyle y_{t}-\rho y_{t-1}=\alpha (1-\rho )+\beta (X_{t}-\rho X_{t-1})+e_{t}.\,}$

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

${\displaystyle {\sqrt {1-\rho ^{2}}}y_{1}=\alpha {\sqrt {1-\rho ^{2}}}+\left({\sqrt {1-\rho ^{2}}}X_{1}\right)\beta +{\sqrt {1-\rho ^{2}}}\varepsilon _{1}.\,}$

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:

${\displaystyle \mathrm {cov} (\varepsilon _{t},\varepsilon _{t+h})={\frac {\rho ^{h}}{1-\rho ^{2}}},{\text{ for }}h=0,\pm 1,\pm 2,\dots \,.}$

Now is easy to see that the variance–covariance matrix, ${\displaystyle \mathbf {\Omega } }$, of the model is

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

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

${\displaystyle {\hat {\Theta }}=(\mathbf {Z} ^{\mathsf {T}}\mathbf {\Omega } ^{-1}\mathbf {Z} )^{-1}(\mathbf {Z} ^{\mathsf {T}}\mathbf {\Omega } ^{-1}\mathbf {Y} ),\,}$

where ${\displaystyle \mathbf {Z} }$ is a matrix of observations on the independent variable (Xt, t = 1, 2, ..., T) including a vector of ones, ${\displaystyle \mathbf {Y} }$ is a vector stacking the observations on the dependent variable (Xt, t = 1, 2, ..., T) and ${\displaystyle {\hat {\Theta }}}$ 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 ${\displaystyle \mathbf {\Omega } }$ can be decomposed as ${\displaystyle \mathbf {\Omega } ^{-1}=\mathbf {G} ^{\mathsf {T}}\mathbf {G} }$ with[3]

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

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 ${\displaystyle \rho }$ is not known, a recursive procedure may be used to make the estimation feasible. See Cochrane–Orcutt estimation.

## References

1. ^ Prais, S. J.; Winsten, C. B. (1954). "Trend Estimators and Serial Correlation" (PDF). Cowles Commission Discussion Paper No. 383. Chicago.
2. ^ Johnston, John (1972). Econometric Methods (2nd ed.). New York: McGraw-Hill. pp. 259–265.
3. ^ Kadiyala, Koteswara Rao (1968). "A Transformation Used to Circumvent the Problem of Autocorrelation". Econometrica. 36 (1): 93–96. JSTOR 1909605.
• 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.