Estimating ρ: The Cochrane-Orcutt (C-O) iterative procedure.‡

As an illustration of this procedure, consider the two-variable model:

Yt = β1 + β2 Xt + ut (1)

and the AR(1) scheme

ut = ρut-1 + εt , -1 ρ 1 (2)

Cochrane and Orcutt then recommend the following steps to estimate ρ.

1. Estimate (1) by the usual OLS routine and obtain the residuals, uˆ_t . Incidentally, note that you can have more than one X variable in the model.

2. Using the residuals obtained in step 1, run the following regression: uˆ_t = ρˆuˆ_t-1 + v_t (3)

which is the empirical counterpart of (2).*

3. Using ρˆ obtained in (3), estimate the generalized difference equation (12.9.6).

4. Since a priori it is not known if the ρˆ obtained from (3) is the best estimate of ρ, substitute the values of βˆ* and βˆ* obtained in step (3) in
the original regression (1) and obtain the new residuals, say, uˆ_t * as uˆ_t * = Y_t - βˆ₁* - βˆ₂* X_t (4)

which can be easily computed since Yt , Xt , βˆ₁* , and βˆ₂∗ are all known.

5. Now estimate the following regression:

uˆ_t* = ρˆ_t-1* uˆ *+ w_t (5)

which is similar to (3) and thus provides the second round estimate of ρ

Since we do not know whether this second-round estimate of ρ is the best estimate of the true ρ, we go into the third-round estimate, and so on. That is why the C-O procedure is called an iterative procedure. But how long should we go on this (merry) go-round? The general recom- mendation is to stop carrying out iterations when the successive estimates of ρ differ by a small amount, say, by less than 0.01 or 0.005. In our wages-productivity example, it took about seven iterations before we stopped.

a. Using software of your choice, verify that the estimated ρ value of 0.8919 for Eq. (12.9.16) and 0.9610 for Eq. (12.9.17) are approxi- mately correct.

b. Does the rho value obtained by the C-O procedure guarantee the global minimum or just the local minimum?
c. Optional: Apply the C-O method to the log-linear wages-productivity regression given in (12.5.2), retaining the ?rst observation as well as

dropping it. Compare your results with those of regression (12.5.1).