## 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