MULTIPI,E MINIMA IN THE klSTIMATIONOF MODELS WITH AUTOREGRESSIVE DISTURBANCES

and San Kmenta

No. 46 May 1990

330.0182

U58wno.46

MULTIPLE MINIMA IN THE ESTIMATIONOF MODELS WITH AUTOREGRESSIVE DISTURBANCES*

Howard DORANUniversity of New England

and

Jan KMENTAUniversity of Michigan

AbstractIn this paper we show that the problem of multiple minima obtained by using the search

procedure in the context of the Cochrane-Orcutt transformation disappears when theobservation set is extended to include the first observation, as proposed by Prais-Winsten.

*Kmenta’s contribution was supported by the Alexander-von-Humboldt Foundation.

Address for correspondence

Howard DoranDepartment of Economics

University of MichiganAnn Arbor, MI 48109-1220

1. Introduction

We consider, without a loss of generality, the following simple regression model with

autoregressive disturbances:

Yt-- a+/~Xt, + ~t’ t= 1,2 .....n,

= P~t-1 + ut, I~,l < 1,

where all the usual definitions and assumptions apply. We also assume that ut is normally

distributed.

To remove the autoregressive ~t’ one can apply the following transformation:

Yt = aWt + /~Xt + ut

where, for t = l,

Yt =Yt -P,

and, for t = 2,3 .....n,

Yt = Yt - PYt-1’

Wt= -p, = -p,

When the first observation, (Y~, W~, X~), is dropped, the transformation is called Cochrane-

Orcutt (C-O); when it is included, the transformation is known as Prais-Winsten (P-W).

The transformed equation is usually estimated in one of two ways.

(a) Iterative procedure

Starting with the least squares estimates of the untransformed equation, the residuals are

used to obtain an initial estimate of p. This estimate is used to transform the original equation

and to obtain the second-stage estimates of a and/~, and so on. The procedure is repeated

until convergence.

From Huzurbazar (1948) and Oberhofer and Kmenta (1974) it follows that this procedure

converges and the resulting estimator is unique and consistent regardless of whether the C-O

or P-W transformations are used.

2

(b) Search procedure..

Suggested originally by Hi]dreth and Lu (1960), the sum-of-squared-errors (SSE) is

computed as a function of p and the chosen estimates a, fl and p are those that correspond to

minimum SSE. This minimum is located by searching over p in the range I P I < 1. It is with

this method that the phenomenon of multiple minima has been documented, always in the

context of the C-O transformation.

2. Multiple minima_

The first to raise this question were Hildreth and Lu (1960), who provided an artificial,

five-observation example of the existence of double minima of SSE. Another example,

involving a more realistic model and data, was provided by Dufour et al. (1980). The issue was

also more extensively treated by Oxley and Roberts (1986) who used a lagged dependent

variable model. It should be pointed out. though, that in this case the iterative C-O estimator

is inconsistent, since the starting least squares estimator is inconsistent (unless p = 0).

In the example of Hildreth and Lu (1960) the authors found dual minima of SSE at p =

-0.9 and p = 0.3, while Dufour et al. (1980) reported minima at ~ 0.3289 and ~ = 0.9318.

We have recomputed both sets of estimates, using double precision and confirmed these dual

minima. Thus the existence of multiple minima in small samples cannot be ruled out when

the C-O transformation is used.

In this paper we examine the possibility of the existence of multiple minima when using

the search procedure with the P-W transformation. To this end we reestimated the

parameters of the models of Hildreth and Lu (1960) and Dufour et aI. (1980), using the

authors’ respective data sets but including the first observation (YI’ WI’ and Xl). The results

turned out to be rather startling=, in bot___h_h case___ss the dual minima of SSE completely disappear.

The unique minimum in the Hildreth and Lu case occurs at p = -0.99, and in the Dufour et

aI. case at p = 0.3. (The latter is shown in Figure 1.) When using the full maximum

hkehhood procedure that allows for the appropriate Jacobian, the results turned out to be

similar. In the case of Hildreth and Lu, the likehhood function peaked at p = - 0.78 and in

the case of Dufour et al, at p = 0.315. These unique minima correspond to the estimates of p

obtained by the itemtive procedure. Since the importance of the first observation diminishes

as the sample size increases, our results are consistent with the claim that the occurrence of

multiple minimum of the SSE curve (or multiple maxima of the likelihood function) will

asymptotically disappear.

3. An explanation

During our analysis of both the Hildreth-Lu (1960) and the Dufour et al. (1980) data sets,

two features emerged. First, as emphasized above, when the transformed first observation was

included, the dual minima problem disappeared. Second, when the intercept a was omitted

from the model, the same thing happened even when the first observation was omitted. As

dropping the first transformed observation converts the variable Wt into a column of

constants, there is the strong suggestion that the occurrence of dual minima is associated with

the presence of a constant term in the transformed model.

Let us suppose now that the values of the dependent variable Yt in the sample can be

adequately described by

Yt = ~ + 6Yt_l+ vt

where 161 < 1. As the search approaches p = 6, the transformed dependent variable Yt will

be given by

Yt = "~ + vt"

When the weighted first observation is omitted, SSE = (1 - R2) ~] (vt - ~)2, where R2

refers to the regression of Y on W and X , wher~ when it is included,

SSE = (1 - R2) I](vt+ ~f) .

^2As ~(vt - ~)2 ~_ F~vt _ .~) , end would o~en he ver~ much smaller, omission of the firs~

observation could result in a drop in SSE as p becomes close to 3.

A Box and Jenkins (1976) diagnostic analysis of the Dufour et al. (1980) dependent

variable Yt clearly indicated first-order autoregressive characteristics, with ~ = 0.77. It is in

the neighborhood of p = 0.7 that SSE descends to a second minimum (see Figure 1.).

Our general explanation is that if the dependent variable closely approximates a first-ordex

autoregressive process, dropping the first weighted observation (i.e., using the C-O

transformation) may well induce a spurious minimum in SSE. Our results supersede the

recommendation of Dufour et al. (1980, p.46) "to combine a search routine...with the

Cochrane-Orcurt Procedure~ by the recommendation always to replace the C-O transformation

by the P-V~ transformation that requires the inclusion of the first observation in the

observation set.

5

REFERENCES

Box, G.E.P. and G.M. Jenkins, 1976, Time Series Analysis (Holden-Day, San Francisco).

Dufour, Jean-Marie, Marc J.I. Gandry and Tran Cong Liem, 1980, The Cochraue-Orcuttprocedure: Numerical examples of multiple admissible minima, Economics Letters 6,43-48.

Hildreth, C. and J.Y. Lu, 1960, Demand relations with autocorrelated disturbances, TechnicalBulletin 276, Department of agricultural economics (Michigan State University, EastLansing, MI).

Huzurbazar, V.S., 1984, The likelihood equation, consistency and the maxima of the likelihoodfunction, Annals of Eugenics 14, 185--200.

Oberhofer, W. and J. Kmenta, 1974, A general procedure for obtaining maximum likelihoodestimates in generalized regression models, Econometrica, 579-590.

Oxley, L.T. and C.J. Roberts, 1986, Multiple minima and the Cochrane-Orcutt technique:Some initial Monte-Carlo results, Economics Letters, 247-250.

>fULTIPLE MINIMA IN THE STIMATION OF MODELS WITHAUTOREGRESSIVE DISTURBANCES

by Howard Doran and Jan Kmenta

0.42

0.40

0.36

O.32

0.300.0

0.6VALUE OF RHO

Fig. 1

1.0

C-OP-W

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