Estimating : The CochraneOrcutt (CO) iterative procedure. As an illustration of this procedure, consider the two-variable model:

Estimating ρ: The Cochrane–Orcutt (C–O) iterative procedure. As an illustration of this procedure, consider the two-variable model:

Y_t = β₁ + β₂X_t + u_t …………………….. (1)

and the AR(1) scheme

u_t = ρu_t−1 + ε_t , −1 < ρ < 1 ……………. (2)

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

1. Estimate Eq. (1) by the usual OLS routine and obtain the residuals, û_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:

_ût = ρ̂û _t−1 + v_t ………………………….. (3)

which is the empirical counterpart of Eq. (2).

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

4. Since a priori it is not known if the ρˆ obtained from Eq. (3) is the best estimate of ρ, substitute the values of ˆ β∗1 and ˆ β∗2 obtained in step (3) in the original regression Eq. (1) and obtain the new residuals, say, û^∗_t as û^∗_t = Y_t – β̂^∗₁ – β̂^∗₂ X_t (4) which can be easily computed since Y_t , X_t , β̂^∗₁ , and β̂^∗₂ are all known.

5. Now estimate the following regression:

û^∗_t = ρ̂^∗û^∗_t−1 + w_t ………………………. (5)

which is similar to Eq. (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 recommendation 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 three iterations before we stopped.

a. Use the Cochrane–Orcutt iterative procedure to estimate ρ for the wages– productivity regression, Eq. (12.5.2). How many iterations were involved before you obtained the “final” estimate of ρ?

b. Using the final estimate of ρ obtained in (a), estimate the wages–productivity regression, dropping the first observation as well as retaining the first observation. What difference you see in the results?

c. Do you think that it is important to keep the first observation in transforming the data to solve the autocorrelation problem?