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. 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.

Contents

Theory

Consider the model

y_t = \alpha %2B X_t \beta%2B\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}%2Be_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)%2B\beta(X_t - \rho X_{t-1}) %2B 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}%2B\left(\sqrt{1-\rho^2}X_1\right)\beta %2B \sqrt{1-\rho^2}e_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%2Bh})=\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 maybe used to make the estimation feasible. See Cochrane–Orcutt estimation.

References