Since Po = 1 we get by successive substitution

Pi = P. = P = p. • •

Thus the lag correlations are all powers of p and decline geometrically.

These expressions can be used to derive the covariance matrix of the errors and using what is known as generalized least squares (GLS). We will not derive the expression for GLS here but will outline the minor changes that it implies. Consider the model

y, = a + x, + u, t=l,2,...,T (6.2)

M, = pM, + e, (6.3)

Except for the treatment of the first observation in the case of AR(1) errors as in (6.3), and the treatment of the first two observations in the case of AR(2) errors, and so on, the GLS procedure amounts to the use of transformed data, which are obtained as follows.*

Lagging (6.2) by one period and multiplying it by p, we get

py, , = ap + (3pj:, , + , , (6.4)

Subtracting (6.4) from (6.2) and using (6.3), we get

y, - /-1 = a(l - P) + - px, i) + e, (6.5)

Since e, are serially independent with a constant variance ct?, we can estimate the parameters in this equation by an OLS procedure. Equation (6.5) is often called the quasi-difference transformation of (6.4). What we do is transform the variables y, and x, to

y* = y, - py,-i / = 2, 3, . . . ,

x = X, - px,.,

and run a regression of y* on x*, with or without a constant term depending on whether the original equation has a constant term or not. In this method we use only (J-l) observations because we lose one observation in the process of taking differences. This procedure is not exactly the GLS procedure. The GLS procedure amounts to using all the observations with

y\ = VI - pyi (6.6)

x\ = Vl - px,

and regressing y* on jc* using the T observations.

In actual practice p is not known. There are two types of procedures for estimating p:

1. Iterative procedures.

2. Grid-search procedures,

The derivation involves the use of matrix algebra, which we have avoided. 4f the number of observations is large, the omission of these initial observations does not matter much.

Iterative Procedures

Among the iterative procedures, the eariiest was the Cochrane-Orcutt procedure. In the Cochrane-Orcutt procedure we estimate equation (6.2) by OLS, get the estimated residuals „ and estimate p by p = 2]

Durbin" suggested an alternative method of estimating p. In this procedure, we write equation (6.5) as

y, = a(l - p) + py,„, + Px, - Ppx, , + e, (6.7)

We regress y, on y, , x„ and x, , and take the estimated coefficient of y,, as an estimate of p.

Once an estimate of p is obtained, we construct the transformed variables y* and X* as defined in (6.6) and (6.6) and estimate a regression of y* on x*. The only thing to note is that the slope coefficient in this equation is p, but the intercept is a(l - p). Thus after estimating the regression of y* on x*, we have to adjust the constant term appropriately to get estimates of the parameters of the original equation (6.2). Further, the standard errors we compute from the regression of y* on x* are now "asymptotic" standard errors because of the fact that p has been estimated. If there are lagged values of as explanatory variables, these standard errors are not correct even asymptotically. The adjustment needed in this case is discussed in Section 6.7.

If there are many explanatory variables in the equation, Durbins method involves a regression in too many variables (twice the number of explanatory variables plus y,„). Hence it is often customary to prefer the Cochrane-Orcutt procedure until it converges. However, there are examples" to show that the minimization of 2] e; in (6.5) can produce multiple solutions for p. In this case the Cochrane-Orcutt procedure, which relies on a single solution for p, might give a local minimum, and even when iterated might converge to a local minimum. Hence it is better to use a grid-search procedure, which we will now describe.

Grid-Search Procedures

One of the first grid-search procedures is the Hildreth and Lu procedure suggested in 1960. The procedure is as follows. Calculate y* and x* in (6.6) for different values of p at intervals of 0.1 in the range - 1 < p < 1. Estimate the regression of y* on xj and calculate the residual sum of squares RSS in each

D. Coctirane and G. H. Orcutt, "Application of Least Squares Regressions to Relationships Containing Autocorrelated Error Terms," Journal of the American Statistical Association, 1949, pp. 32-61.

"J. Durbin, "Estimation of Parameters in Time Series Regression Models," Journal of the Royal Statistical Society, Series B. 1960, pp. 139-153.

"J. M. Dufour, M. J. 1. Gaudry, and T. C. Lieu, "The Cochrane-Orcutt Procedure: Numerical Examples of Multiple Admissible Minima," Economics Letters, 1980 (6), pp. 43-48. -Clifford Hildreth and John Y. Lu, Demand Relations with Autocorrelated Disturbances, AES Technical Bulletin 276, Michigan State University, East Lansing, Mich., November 1960.

case. Choose the value of p for which the RSS is minimum. Again repeat this procedure for smaller intervals of p around this value. For instance, if the value of p for which RSS is minimum is - 0.4, repeat this search procedure for values of p at intervals of 0.01 in the range -0.5 < p < -0.3.

This procedure is not the same as the maximum likelihood (ML) procedure. If the errors e, are normally distributed, we can write the log-likelihood function as (derivation is omitted)

where

T 1 Q

log L = const. - - log CT? -I- - log (1 - P) -

e = S [y, - py, , - a(l - p) - ?>ix, - px, ,)]2

(6.8)

Thus minimizing Q is not the same as maximizing log L. We can use the grid-search procedure to get the ML estimates. The only difference is that after we compute the residual sum of squares RSS(p) for each p, we choose the value of p for which {TIT) log RSS(p) - (1/2) log (1 - p) is minimum. If the number of observations is large, the latter term will be small compared to the former, and the ML procedure and the Hildreth-Lu procedure will give almost the same results.

Illustrative Example

Consider the data in Table 3.11 and the estimation of the production function log Z = a -I- (3, log L, + p, log , + M

The OLS estimation gave a DW statistic of 0.86, suggesting significant positive autocorrelation. Assuming that the errors were AR(1), two stimation procedures were used: the Hildreth-Lu grid search and the iterative Cochrane-Orcutt (C-O). The other procedures we have described can also be tried, but this is left as an exercise.

The Hildreth-Lu procedure gave p = 0.77. The iterative C-O procedure gave p = 0.80. The DW test statistic implies that p = (l/2)(2 - 0.86) = 0.57.

The estimates of the parameters (with the standard errors in parentheses) were as follows:

In this example the parameter estimates given by Hildreth-Lu and the iterative C-O procedures are pretty close to each other. Correcting for the auto-