# Prais–Winsten estimation

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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. 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

$y_t = \alpha + X_t \beta+\varepsilon_t,\,$

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 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

$\sqrt{1-\rho^2}y_1 = \alpha\sqrt{1-\rho^2}+\left(\sqrt{1-\rho^2}X_1\right)\beta + \sqrt{1-\rho^2}\varepsilon_1. \,$

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:

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

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

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

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

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

where $\mathbf{Z}$ is a matrix of observations on the independent variable (Xt, t = 1, 2, ..., T) including a vector of ones, $\mathbf{Y}$ is a vector stacking the observations on the dependent variable (Xt, 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 general least square estimation procedure sketched above is helpful. The inverse of $\mathbf{\Omega}$ can be decomposed as $\mathbf{\Omega}^{-1}=\mathbf{G}'\mathbf{G}$ 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 may be used to make the estimation feasible. See Cochrane–Orcutt estimation.