5.5 Feasiable Prais Winsten

Weighting Matrix

\[ \mathbf{w} = \left( \begin{array}{ccccc} \sqrt{1- \hat{\rho}^2} & 0 & 0 &... & 0 \\ -\hat{\rho} & 1 & 0 & ... & 0 \\ 0 & -\hat{\rho} & 1 & & . \\ . & . & . & . & 0 \\ 0 & . & 0 & -\hat{\rho} & 1 \end{array} \right) \]

  1. Estimate the following equation using OLS

\[ y_t = \mathbf{x}_t \beta + \epsilon_t \] and obtain the residuals \(e_t = y_t - \mathbf{x}_t \hat{\beta}\)

  1. Estimate the correlation coefficient for the AR(1) process by estimating the following by OLS (without no intercept)

\[ e_t = \rho e_{t-1} + u_t \]

  1. Transform the outcome and independent variables \(\mathbf{wy}\) and \(\mathbf{wX}\) respectively (weight matrix as stated).
  2. The FPW estimator is obtained as the least squared estimated for the following weighted equation

\[ \mathbf{wy = wX\beta + w\epsilon} \]

Properties of Feasiable Prais Winsten Estimator

  • The Infeasible PW estimator is under A1-A3 for the unweighted equation
  • The FPW estimator is biased
  • The FPW is consistent under A1 A2 A5 and

\[ E((\mathbf{x_t - \rho x_{t-1}})')(\epsilon_t - \rho \epsilon_{t-1})=0 \] + A3a is not sufficient for the above equation + A3 is sufficient for the above equation

  • The FPW estimator is asymptotically more efficient than OLS if the errors are truly generated as AR(1) process
    • If the errors are truly generated as AR(1) process then usual standard errors are valid
    • If we are concerned that there may be a more complex dependence structure of heteroskedasticity, then we use Newey West Standard Errors