Estimation of regression with autocorrelated errors

Question

In a book it is written that,

In regression work we typically assume that the observational errors are pairwise uncorrelated. But in most time series data , the successive residuals have tendency to correlated with themselves.If we find autocorrelated errors, we need to modify the regression procedure to remove the effect of autocorrelated errors. Typically this is done by an appropriate transformation of the variables used in the regression estimation procedure.

Consider a multiple regression model with autocorrelated errors $$Y_t=\beta_0+\beta_1X_{1t}+\beta_2X_{2t}\ldots+\beta_pX_{pt}+\epsilon_t\ldots(1)$$

and the same regression model at time ${t-1}$ is $$Y_{t-1}=\beta_0+\beta_1X_{1,{t-1}}+\beta_2X_{2,{t-1}}\ldots+\beta_pX_{p,{t-1}}+\epsilon_{t-1}\ldots(2)$$

Multiplying both sides of this equation by correlation coefficient $\rho$,the correlation between adjacent errors gives $$\rho Y_{t-1}=\rho \beta_0+\rho \beta_1X_{1,{t-1}}+\rho \beta_2X_{2,{t-1}}\ldots+\rho \beta_pX_{p,{t-1}}+\rho \epsilon_{t-1}$$

Then we subtract this equation from the first equation to obtain $Y_t-\rho Y_{t-1}=\beta_0(1-\rho)+\beta_1(X_{1t}-\rho X_{1,{t-1}})+\beta_2(X_{2t}-\rho X_{2,{t-1}})\ldots+\beta_p(X_{pt}-\rho X_{p,{t-1}})+u_t\ldots(3)$

where $u_t=\epsilon_t-\rho \epsilon_{t-1}$ has uniform variance and is not autocorrelated.

$\bullet$ My first question is : Why do they multiply $\rho $ in model $(2)$? Why not in model $(1)$?

$\bullet$ My second question is :Are the data of independent variable $X$ in model$(1)$ same as the data of independent variable $X$ in model $(2)$ ? If this data depends on time, then they seem to be not equal data. If so then my parameters $\beta_0,\beta_1,\ldots\beta_p$ are not same for the model $(1)$ and $(2)$. Then how can i write model $(3)$? More specifically, how can i take $\beta_0,\beta_1,\ldots\beta_p$ as common factor ?

$\bullet$ My last question is : how does $u_t=\epsilon_t-\rho \epsilon_{t-1}$ have uniform variance and why is not autocorrelated?

Answer 1

Answers:

1) This is done so that you will get representation in eq. (3).

2) Beta-parameters are time invariant and only values of X variables will be different since they are at different lags. Then you can arrange eq. (1) and (2) in form of (3).

3) Usually $\epsilon_{t}=\rho \epsilon_{t-1}+u_{t}$ where $u_{t}\sim iid(0,\sigma_u)$ which means that $u_{t}$ is not autocorrelated and has uniform variance.

Transformation in eq. (3) is known as Cochrane-Orcutt transformation and with handling of initial values it is known as Prais-Winsten transformation.