# Prais–Winsten estimation

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, 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.

## Theory

Consider the model
    $y_{t}=\alpha +X_{t}\beta +\varepsilon _{t},\,$
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 $\varepsilon _{t}$ is the error term. The error term can be serially correlated over time: $\varepsilon _{t}=\rho \varepsilon _{t-1}+e_{t},\ |\rho |<1$ and $e_{t}$ is white noise. In addition to the Cochrane–Orcutt transformation, which is
    $y_{t}-\rho y_{t-1}=\alpha (1-\rho )+(X_{t}-\rho X_{t-1})\beta +e_{t},\,$
for t = 2,3,...,T, the Prais-Winsten procedure makes a reasonable transformation for t = 1 in the following form:
    ${\sqrt {1-\rho ^{2}}}y_{1}=\alpha {\sqrt {1-\rho ^{2}}}+\left({\sqrt {1-\rho ^{2}}}X_{1}\right)\beta +{\sqrt {1-\rho ^{2}}}\varepsilon _{1}.\,$
Then the usual least squares estimation is done.

## Estimation procedure

First notice that
$\mathrm {var} (\varepsilon _{t})=\mathrm {var} (\rho \varepsilon _{t-1}+e_{it})=\rho ^{2}\mathrm {var} (\varepsilon _{t-1})+\mathrm {var} (e_{it})$
Noting that for a stationary process, variance is constant over time,
$(1-\rho ^{2})\mathrm {var} (\varepsilon _{t})=\mathrm {var} (e_{it})$
and thus,
$\mathrm {var} (\varepsilon _{t})={\frac {\mathrm {var} (e_{it})}{(1-\rho ^{2})}}$
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:
    $\mathrm {cov} (\varepsilon _{t},\varepsilon _{t+h})=\rho ^{h}\mathrm {var} (\varepsilon _{t})={\frac {\rho ^{h}}{1-\rho ^{2}}},{\text{ for }}h=0,\pm 1,\pm 2,\dots \,.$
It is easy to see that the variance–covariance matrix, $\mathbf {\Omega }$, of the model is
    $\mathbf {\Omega } ={\begin{bmatrix}{\frac {1}{1-\rho ^{2}}}&{\frac {\rho }{1-\rho ^{2}}}&{\frac {\rho ^{2}}{1-\rho ^{2}}}&\cdots &{\frac {\rho ^{T-1}}{1-\rho ^{2}}}\\[8pt]{\frac {\rho }{1-\rho ^{2}}}&{\frac {1}{1-\rho ^{2}}}&{\frac {\rho }{1-\rho ^{2}}}&\cdots &{\frac {\rho ^{T-2}}{1-\rho ^{2}}}\\[8pt]{\frac {\rho ^{2}}{1-\rho ^{2}}}&{\frac {\rho }{1-\rho ^{2}}}&{\frac {1}{1-\rho ^{2}}}&\cdots &{\frac {\rho ^{T-3}}{1-\rho ^{2}}}\\[8pt]\vdots &\vdots &\vdots &\ddots &\vdots \\[8pt]{\frac {\rho ^{T-1}}{1-\rho ^{2}}}&{\frac {\rho ^{T-2}}{1-\rho ^{2}}}&{\frac {\rho ^{T-3}}{1-\rho ^{2}}}&\cdots &{\frac {1}{1-\rho ^{2}}}\end{bmatrix}}.$
Having $\rho$ (or an estimate of it), we see that,
    ${\hat {\Theta }}=(\mathbf {Z} ^{\mathsf {T}}\mathbf {\Omega } ^{-1}\mathbf {Z} )^{-1}(\mathbf {Z} ^{\mathsf {T}}\mathbf {\Omega } ^{-1}\mathbf {Y} ),\,$
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 (y$_t$, t = 1, 2, ..., T) and ${\hat {\Theta }}$ 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 $\mathbf {\Omega }$ can be decomposed as $\mathbf {\Omega } ^{-1}=\mathbf {G} ^{\mathsf {T}}\mathbf {G}$ with
    $\mathbf {G} ={\begin{bmatrix}{\sqrt {1-\rho ^{2}}}&0&0&\cdots &0\\-\rho &1&0&\cdots &0\\0&-\rho &1&\cdots &0\\\vdots &\vdots &\vdots &\ddots &\vdots \\0&0&0&\cdots &1\end{bmatrix}}.$
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 $\rho$ 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.