Assume That You Had Found Correlation of the Residuals Across

Essay

Assume that you had found correlation of the residuals across observations.This may
happen because the regressor is ordered by size.Your regression model could therefore
be specified as follows: $$\begin{array} { c } Y _ { i } = \beta _ { 0 } + \beta _ { 1 } X _ { i } + u _ { i } \\u _ { i } = \rho u _ { i - 1 } + v _ { i } ; | \rho | < 1 .\end{array}$$
Furthermore, assume that you had obtained consistent estimates for $\beta _ { 0 } , \beta _ { 1 } , \rho$ . If asked to make a prediction for $Y$ , given a value of $X \left( = X _ { j } \right)$ and $\hat { u } _ { j - 1 }$ , how would you proceed? Would you use the information on the lagged residual at all? Why or why not?