Prais–Winsten estimation

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| 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_{t})=\rho^2 \mathrm{var}(\varepsilon_{t-1}) +\mathrm{var}(e_{t})

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

(1-\rho^2 )\mathrm{var}(\varepsilon_t)= \mathrm{var}(e_{t})

and thus,

\mathrm{var}(\varepsilon_t)=\frac{\mathrm{var}(e_{t})}{(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 below:

: \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.

References

References

1. (1954). "Trend Estimators and Serial Correlation". Cowles Commission Discussion Paper No. 383.
2. Johnston, John. (1972). "Econometric Methods". McGraw-Hill.
3. Kadiyala, Koteswara Rao. (1968). "A Transformation Used to Circumvent the Problem of Autocorrelation". [[Econometrica]].
4. (1978). "A Maximum Likelihood Procedure for Regression with Autocorrelated Errors". [[Econometrica]].
5. Amemiya, Takeshi. (1985). "Advanced Econometrics". Harvard University Press.