Appendix III. Advanced Econometrics Methods

• Pages: 415-430

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APPENDIX III

ADVANCED ECONOMETRIC METHODS

A. Introduction

This appendix introduces the technical aspects of advanced

econometric techniques, in particular, the logit model, AIDS, and the

price cost model, all of which are described generally in Chapters XI and

XII. In addition, this appendix sets forth the equations used by the

Department of Justice in its analysis of the competitive effects of

WorldCom’s proposed acquisition of Sprint—a case described generally

in Appendix I.

B. Maximum-Likelihood Estimation, The Logit Model

1. Introduction

In the past decade, econometric analyses have relied more frequently

on the use of maximum-likelihood methods to estimate demand systems

and demand parameters. This appendix offers a brief introduction to the

maximum-likelihood approach, which offers a more general approach to

parameter estimation than least squares. Section 2 introduces the

concept of maximum-likelihood estimation. Section 3 introduces the

logit model. In Section 4, maximum-likelihood methods are applied to

the estimation of the basic logit model.1

2. Maximum-Likelihood Estimation

Maximum likelihood estimation focuses on the fact that different

populations generate different samples; any one sample being scrutinized

is more likely to have come from some populations than from others.

1. This material is drawn from ROBERT S. PINDYCK & DANIEL L.

RUBINFELD, ECONOMETRIC MODELS AND ECONOMIC FORECASTS (4th ed.

1998).

Econometrics in Antitrust

416

For example, if one were sampling coin tosses and a sample mean of .5

were obtained (representing half heads and half tails), the most likely

population from which the sample was drawn would be a population

with a mean of .5. Figure 1 illustrates a more general case in which a

sample (X1, X2, ... X8) is known to be drawn from a normal population

with given variance but unknown mean. Assume that observations come

from either distribution A or distribution B. If the true population were

B, the probability that we would have obtained the sample shown would

be quite small. However, if the true population were A, the probability

would be substantially larger. Thus, the observations select the

population A as the most likely to have generated the observed data.

Figure 1

Maximum-likelihood Estimation

We define the maximum-likelihood estimator of a parameter

β

as

the value of β

Ⱡ

which would most likely generate the observed sample

observations Y1, Y2,...,YN. In general, if Yi is normally distributed and

each of the Ys is drawn independently, the maximum-likelihood

estimator maximizes

p(Y1) p (Y2) ... p(YN)

where each p represents a probability associated with the normal

distribution. Thus, the calculated maximum-likelihood estimate is a

function of the particular sample of Y values chosen. A different sample

would result in a different maximum-likelihood estimate.