8 Correcting for Serial Correaltion with Strictly Exogenous Regressors
Note that the corrections mentioned below should NOT be used when the explanatory variables include lagged dependent variables.
The idea is to apply the generalized least-squares (GLS) method to estimate the parameters in a linear regression model in which the errors are serially correlated. Specifically, the errors are assumed to follow a first-order autoregressive process.
8.1 GLS
Assumes
\[ y_t = \beta_1x_{t1} + \beta_2x_{t2} + \cdots + \beta_Kx_{tK} + u_t , \tag{8.1}\]
where the errors follow the AR(1) model
\[ u_t = \rho u_{t-1} + e_t . \tag{8.2}\]
Because the problem in (8.1) is serial correlation in \(u_t,\) it makes sense to transform the equation to eliminate the serial correlation.
For \(t\ge 2\)
\[ y_{t-1} = \beta_1x_{t-1,1} + \beta_2x_{t-1,2} + \cdots + \beta_Kx_{t-1,K} + u_{t-1} , \tag{8.3}\]
Multiply (8.3) by \(\rho\) and subtract it from (8.1), we get
\[ y_t - \rho y_{t-1} = \beta_1(x_{t1} - \rho\, x_{t-1,1}) + \beta_2(x_{t2} - \rho\, x_{t-1,2}) + \cdots + \beta_K(x_{tK} - \rho\, x_{t-1,K}) + e_t , \] where \(e_t = u_t - \rho u_{t-1}.\) Hence, we have removed the serial correaltion in the error.
We rewrite the transformed equation as
\[ \tilde{y_t} = \beta_1\tilde{x}_{t1} + \beta_2\tilde{x}_{t2} + \cdots + \beta_K\tilde{x}_{tK} + e_t,\; t\ge 2, \tag{8.4}\] where \[ \begin{split} \tilde{y}_t &= y_t - \rho y_{t-1} \\ \tilde{x}_{tj} &= x_{tj} - \rho x_{t-1,j},\; j=1,\ldots,K. \\ \end{split} \] Applying OLS to (8.4), the estimator is BLUE. Since the errors in the transformed equation are serially uncorrelated, we can apply the usual \(t\) statistics and \(F\) statistics for hypothesis testing.
8.2 FGLS
In practice, since we don’t know \(\rho,\) we need to get a consistent estimator – this is done by feasible GLS (FGLS).
Steps:
Run OLS on the original equation
\[ \text{Regress } y_t \text{ on } x_{t1},x_{t2},\ldots,x_{tK} \text{ and obtain } \hat{u}_t , \] for \(t=1,2,\ldots,n.\)
Run the regression of
\[ \hat{u}_t \text{ on } \hat{u}_{t-1} \text{ and obtain } \hat{\rho} \] for \(t=2,\ldots,n\)
Apply the transformation as in (8.4), where
\[ \begin{split} \tilde{y}_t &= y_t - \hat\rho y_{t-1}, \; t\ge 2 \\ \tilde{x}_{tj} &= x_{tj} - \hat\rho x_{t-1,j},\; j=1,\ldots,K. \\ \end{split} \]
Apply OLS to the transformed equation
\[ \text{Regress } \tilde{y}_t \text{ on } \tilde{x}_{t1}, \tilde{x}_{t2}, \ldots, \tilde{x}_{tK} , \] and obtain new estimates of \(\beta_1,\) \(\ldots,\) \(\beta_K.\) The usual standard errors, \(t\) statistics, and \(F\) statistics are asymptotically valid.
This procedure is called the Cochrane-Orcutt (CO) Estimation, which has one less observation after the transformation.
Prais-Winsten (PW) Estimation uses a specific transformation at \(t=1:\)
\[ \sqrt{1-\rho^2} y_1 = \beta_1(\sqrt{1-\rho^2} x_{11}) + \beta_2(\sqrt{1-\rho^2} x_{12}) + \cdots + \beta_K(\sqrt{1-\rho^2} x_{1K}) + \sqrt{1-\rho^2}u_1 . \] Therefore the PW estimation has one more observation than the CO estimation.
In practice, both the Cochrane-Orcutt and Prais-Winsten methods are used in an iterative scheme.
After we obtained the new estimates of \(\beta_1,\) \(\ldots,\) \(\beta_K,\) after step 4, we can produce the fitted values
\[ \hat{y}_t = \hat\beta_1x_{t1} + \hat\beta_2x_{t2} + \cdots + \hat\beta_Kx_{tK} , \]
and then \(\rho\) is reestimated using a new set of residuals based on the following regression:
\[ y_t - \hat y_t = \rho(y_{t-1}-\hat y_{t-1}) + v_t . \]
Then transform the data using the newly obtained estimate of \(\rho\) and estimate the transformed equation by OLS.
We can repeat the whole process many times, until the estimate of \(\rho\) changes by very little from the previous iteration.
For estimation in the presence of both autocorrelation and lagged \(y\), refer to Chap 12.9.4, Econometric Analysis, 5e by Greene, pp. 277.
References:
- Chap 12.8, 12.9, Econometric Analysis, 5e by Greene, pp. 271–276
- Chap 12.3, Introductory Econometrics: A Modern Approach, 7e by Jeffrey M. Wooldridge