Quantitative Techniques for Competition and Antitrust Analysis_3 docx

Because of the supply behavior of thefringe, if they are able to supply whomever they so desire at any given price p, thedominant firm will face the residual demand curve: Ddominant.p/ D

5 Multiproduct, multiplant, price-setting monopolist:

Single-product monopolists will act to set marginal revenue equal to marginal cost

In those cases, since the monopoly problem is a single-agent problem in a singleproduct’s price or quantity, our analysis can progress in a relatively straightforwardmanner In particular, note that single-agent, single-product problems give us a singleequation (first-order condition) to solve In contrast, even a single agent’s optimiza-tion problem in the more complex multiplant or multiproduct settings generates anoptimization problem is multidimensional In such single-agent problems, we willhave as many equations to solve as we have choice variables In simple cases wecan solve these problems analytically, while, more generally, for any given demandand cost specification the monopoly problem is typically relatively straightforward

to solve on a computer using optimization routines

Naturally, in general, monopolies may choose strategic variables other than priceand quantity For example, if a single-product monopolist chooses both price andadvertising levels, it solves the problem maxp;a.p
c/D.p; a/, which yields theusual first-order condition with respect to prices,

apD.p; a/ D

41 For an empirical application, see Ward (1975).

firm = Dmarket− Sfringe

MC of dominant firmSupply from fringe

Figure 1.23. Deriving the residual demand curve

1.3.3.2 The Dominant-Firm Model

The dominant-firm model supposes that there is a monopoly (or collection of firmsacting as a cartel) which is nonetheless constrained to some extent by a competitivefringe The central assumption of the model is that the fringe acts in a nonstrategicmanner We follow convention and develop the model within the context of a price-setting, single-product monopoly Dominant-firm models analogous to each of thecases studied above are similarly easily developed

If firms which are part of the competitive fringe act as price-takers, they willdecide how much to supply at any given price p We will denote the supply fromthe fringe at any given price p as Sfringe.p/ Because of the supply behavior of thefringe, if they are able to supply whomever they so desire at any given price p, thedominant firm will face the residual demand curve:

Ddominant.p/ D Dmarket.p/
Sfringe.p/:

Figure 1.23 illustrates the market demand, fringe supply, and resulting firm demand curve We have drawn the figure under the assumption that (i) there

dominant-is a sufficiently high price p1such that the fringe is willing to supply the wholemarket demand at that price leaving zero residual demand for the dominant firmand (ii) there is analogously a sufficiently low price p2below which the fringe isentirely unwilling to supply

Given the dominant firm’s residual demand curve, analysis of the dominant-firmmodel becomes entirely analogous to a monopoly model where the monopolist facesthe residual demand curve, Ddominant.p/ Thus our dominant firm will set prices so

that the quantity supplied will equate the marginal revenue to its marginal cost ofsupply That level of output is denoted Qdominantin figure 1.23 The resulting pricewill be pand fringe supply at that price is Sfringe.p/ D Qfringeso that total supply(and total demand) are

QtotalD QdominantC QfringeD Sfringe.p/ C Ddominant.p/ D Dmarket.p/:

A little algebra gives us a useful expression for understanding the role of the fringe

in this model Specifically, the dominant firm’s own-price elasticity of demand can

Sharedom



fringesupply;where  indicates a price elasticity That is, the dominant firm’s demand curve—theresidual demand curve—depends on (i) the market elasticity of demand, (ii) thefringe elasticity of supply, and also (iii) the market shares of the dominant firm andthe fringe Remembering that demand elasticities are negative and supply elasticitiespositive, this formula suggests intuitively that the dominant firm will therefore face

a relatively elastic demand curve when market demand is elastic or when marketdemand is inelastic but the supply elasticity of the competitive fringe is large andthe fringe is of significant size

42 Recall from your favorite mathematics textbook that for any suitably differentiable function f x/

1.4 Conclusions

 Empirical analysis is best founded on economic theory Doing so requires

a good understanding of each of the determinants of market outcomes:the nature of demand, technological determinants of production and costs,regulations, and firm’s objectives

 Demand functions are important in empirical analysis in antitrust The ticity of demand will be an important determinant of the profitability of priceincreases and the implication of those price increases for both consumer andtotal welfare

elas- The nature of technology in an industry, as embodied in production andcost functions, is a second driver of the structure of markets For example,economies of scale can drive concentration in an industry while economies

of scope can encourage firms to produce multiple goods within a single firm.Information about the nature of technology in an industry can be retrievedfrom input and output data (via production functions) but also from cost, out-put and input price data (via cost functions) or alternatively data on inputchoices and input prices (via input demand functions.)

 To model competitive interaction, one must make a behavioral assumptionabout firms and an assumption about the nature of equilibrium Generally, weassume firms wish to maximize their own profits, and we assume Nash equi-librium The equilibrium assumption resolves the tensions otherwise inherent

in a collection of firms each pursuing their own objectives One must alsochoose the dimension(s) of competition by which we mean defining the vari-ables that firms choose and respond to Those variables are generally prices orquantity but can also include, for example, quality, advertising, or investment

in research and development

 The two baseline models used in antitrust are quantity- and price-setting els otherwise known as Cournot and (differentiated product) Bertrand modelsrespectively Quantity-setting competition is normally used to describe indus-tries where firms choose how much of a homogeneous product to produce.Competition where firms set prices in markets with differentiated or brandedproducts is often modeled using the differentiated product Bertrand model.That said, these two models should not be considered as the only modelsavailable to fit the facts of an investigation; they are not

mod- An environment of perfect competition with price-taking firms produces themost efficient outcome both in terms of consumer welfare and productionefficiency However, such models are typically at best a theoretical abstrac-tion and therefore they should be treated cautiously and certainly should notsystematically be used as a benchmark for the level of competition that canrealistically be implemented in practice

Econometrics Review

Throughout this book we discuss the merits of various empirical tools that can beused by competition authorities This chapter aims to provide important backgroundmaterial for much of that discussion Our aim in this chapter is not to replicate thecontent of an econometrics text Rather we give an informal introduction to the toolsmost commonly used in competition cases and then go on to discuss the often practi-cal difficulties that arise in the application of econometrics in a competition context.Particular emphasis is given to the issue of identification of causality Where appro-priate, we refer the reader to more formal treatments in mainstream econometricstextbooks.1

Multiple regression is increasingly common in reports of competition cases injurisdictions across the world Like any single piece of evidence, a regression analy-sis initially performed in an office late at night can easily surge forward and end

up becoming the focus of a case Once under the spotlight of intense scrutiny,regression results are sometimes invalidated Sometimes, it is the data Outliers oroddities that are not picked up by an analyst reveal the analysis was performedusing incorrect data Sometimes the econometric methodology used is proven toprovide good estimates only under extremely restrictive and unreasonable assump-tions And sometimes the analysis performed proves—once under the spotlight—to

be very sensitive in a way that reveals the evidence is unreliable An important part

of the analyst’s job is therefore to clearly disclose the assumptions and sensitivities

at the outset so that the correct amount of weight is placed on that piece of metric evidence by decision makers Sometimes the appropriate amount of weightwill be a great, on other occasions it will be very little

econo-In this chapter we first discuss multiple regression including the techniques known

as ordinary least squares and nonlinear least squares Next we discuss the importantissue of identification, particularly in the presence of endogeneity Specifically, weconsider the role of fixed-effects estimators, instrumental variable estimators, and

“natural” experiments The chapter concludes with a discussion of best practice

1 A very nice discussion of basic regression analysis applied to competition policy can be found in Fisher (1980, 1986) and Finkelstein and Levenbach (1983) For more general econometrics texts, see, for example, Greene (2007) and Wooldridge (2007) And for an advanced and more technical but succinct discussion of the econometric theory, see, for example, White (2001).

in econometric projects The aim in doing so is, in particular, to help avoid thedisastrous scenario wherein late in an investigation serious flaws in econometricanalysis are discovered.

2.1.1 The Principle of Ordinary Least-Squares Regressions

Multiple regression provides a potentially extremely useful statistical tool that canquantify actual effects of multiple causal factors on outcomes of interest In anexperimental context, a causal effect can sometimes be measured in a precise andscientific way, holding everything else constant For example, we might measure theeffect of heat on water temperature On the other hand, budget or time constraintsmight mean we can only use a limited number of experiments so that each experimentmust vary more than one causal factor Multiple regression could then be used toisolate the effects of each variable on the outcomes Unfortunately, economists incompetition authorities cannot typically run experiments in the field It would ofcourse make our life far easier if we could just persuade firms to increase theirprices by 5% and see how many customers they lose; we would be able to learnabout their own-price elasticity of demand relatively easily On the other hand, chiefexecutives and their legal advisors may entirely reasonably suggest that the cost ofsuch an experiment would be overly burdensome on business

More typically, we will have data that have been generated in the normal course

of business On the one hand, such data have a huge advantage: they are real! Firms,for example, will take actions to ameliorate the impact of price increases on demand:they may invest in customer retention strategies, such as marketing efforts aimed

at explaining to their customers the cost factors justifying a price increase; theymight change some other terms of the offer (e.g., how many weeks of a magazinesubscription you get for a given amount) or perform short-term retention advertisingtargeted at the most price-sensitive group of customers If we run an experiment in alab, we will have a “pure” price experiment but it may not tell us about the elasticity of

demand in reality, when real consumers are deciding whether to spend their own realmoney given the firm’s efforts at retaining their business On the other hand, as thisexample suggests, a lot will be going on in the real world, and most importantly none

of it will be under the control of the analyst while much of it may be under the control

of market participants This means that while multiple regression analysis will bepotentially useful in isolating the various causes of demand (prices, advertising,etc.), we will have to be very careful to make sure that the real-world decisionsthat are generating our data do not violate the assumptions needed to justify usingthis tool Multiple regression was, after all, initially designed for understanding datagenerated in experimental contexts

2.1.1.1 Data-Generating Processes and Regression Specifications

The starting point of a regression analysis is the presumption, or at least the esis, that there is a real relationship between two or more variables For instance, weoften believe that there is a relation between price and quantity demanded of a givengood Let us assume that the true population relationship between the price charged,

hypoth-P , and the quantity demanded, Q, of a particular good is given by the followingexpression:2

PiD a0C b0QiC ui;where i indicates different possible observations of reality (perhaps time periods orlocal markets) and the parameters a0and b0take on particular values, for example 5and
2 respectively We will call such an expression our “data-generating process”(DGP) This DGP describes the inverse demand curve as a function of the volume ofsales Q and a time- or market-specific element ui, which is unknown to the analyst.Since it is unknown to the analyst, sometimes it is known as a “shock”; we may call

uia demand shock The shock term includes everything else that may have affectedthe price in that particular instance, but is unknown and hence appears stochastic tothe analyst Regression analysis is based on the idea that if we have data on enoughrealizations of P; Q/, we can learn about the true parameters a0; b0/ of the DGPwithout even observing the uis

If we plot a data set of sample size N , denoted P1; Q1/; P2; Q2/; : : : ; PN; QN/

or more compactly f.Pi; Qi/I i D 1; : : : ; N g, that is generated by our DGP, we willobtain a scatter plot with data spread around the picture An ideal situation forestimating a demand curve is displayed in figure 2.1 The reason we call it ideal willbecome clear later in the chapter but for now note that in this case the true DGP, asillustrated by the plotted observations, seems to correspond to a linear relationship

2 It is perhaps easier to motivate a demand equation by considering the equation to describe the price

P which generates a level of sales Q If Q is stochastic and P is treated as a deterministic “control” variable, then we would write this equation the other way around For the purposes of illustration and since P is usually placed on the y-axis of a classic demand and supply diagram, we present the analysis this way around, that is, in terms of the “inverse” demand curve.

Figure 2.1. Scatter plot of the data and a “best-fit” line.

between the two variables In the figure, we have also drawn in a “best-fit” line, inthis case the line is fit to the data only by examining the data plot and trying to draw

a straight line through the plotted data by hand

In an experimental context, our explanatory variable Q would often be stochastic—we are able to control it exactly, moving it around to generate the pricevariable However, in a typical economics data set the causal variable (here we aresupposing Q) is stochastic A wonderfully useful result from econometric theorytells us that the fact that Q is stochastic does not, of itself, cause enormous problemsfor our tool kit, though obviously it changes the assumptions we require for ourestimators to be valid More precisely, we will be able to use the technique of OLSregression to estimate the parameters a0; b0/ in the DGP provided (i) we considerthe DGP to be making a conditional statement that, given a value of the quantitydemanded Qiand given a particular “shock” ui, the price Pi is generated by theexpression above, i.e., the DGP, (ii) we make an assumption about the relationshipbetween the two causal stochastic elements of the model, Qi and ui, namely thatgiven knowledge of Qi the expected value of the shock is zero, EŒui j Qi D 0,and (iii) the sequence of pairs Qi; ui/ for i D 1; : : : ; n generate an independentand identically distributed sequence.3The first assumption describes the nature ofthe DGP The second assumption requires that, whatever the level of Q, the averagevalue of the shock ui will always be zero That is, if we see many markets withhigh sales, say of 1 million units per year, the average demand shock will be zeroand similarly if we see many markets with lower sales, say 10,000 units per year,the average demand shock will also be zero The third assumption ensures that we

non-3 Note that the technique does not need to assume that Q and u are fully independent of each other, but rather (i) that observations of the pairs Q 1 ; u 1 /, Q 2 ; u 2 /, and so on are independent of each other and follow the same joint distribution and (ii) satisfy the conditional mean zero assumption, E Œu i j Q i  D 0.

In addition to these three assumptions, there are some more technical “regularity” assumptions that primarily act to make sure all of the quantities needed for our estimator are finite—see your favorite econometrics textbook for the technical details.

obtain more information about the process as our sample size gets bigger, whichhelps, for example, to ensure that sample averages will converge to their populationequivalents.4We describe the technique of OLS more fully bellow Other estimatorswill use different sets of assumptions, in particular, we will see that an alternativeestimation technique, instrumental variable (IV) estimation, will allow us to handlesome situations in which EŒuij Qi ¤ 0.

In most if not all cases, there will be a distinction between the true DGP and themodel that we will estimate This is because our model will normally (at best) onlyapproximate the true DGP Ideally, the model that we estimate includes the true DGP

as one possibility If so, then we can hope to learn the true population parametersgiven enough data For example, suppose the true DGP is Pi D 10
2QiC ui

and the model specification is PiD a
bQiC cQi2C ei Then we will be able toreproduce the DGP by assigning particular values to our model parameters In otherwords, our model is more general than the DGP If on the other hand the true DGPis

Pi D 10
5QiC 2Q2i C ui

and our model is

Pi D a
bQiC ei;then we will never be able to retrieve the true parameters with our model In thiscase, the model is misspecified This observation motivates those econometricianswho favor the general-to-specific modeling approach to model specification (see,for example, Campos et al 2005) Others argue that the approach of specifying verygeneral models means the estimates of the general model will be very poor and as

a result the hypothesis tests used to reduce down to more specific models have anextremely low chance of getting you to the right answer All agree that the DGP isnormally unknown and yet at least some of its properties must be assumed if we are

to evaluate the conditions under which our estimators will work Economists mustmainly rely on economic theory, institutional knowledge, and empirical regularities

to make assumptions about the likely true relationships between variables Whennot enough is known about the form of the DGP, one must be careful to eitherdesign a specification that is flexible enough to avoid misspecified regressions orelse test systematically for evidence of misspecification surviving in the regressionequation

Personally, we have found that there are often only a relatively small number ofreally important factors driving demand patterns and that knowledge of an industry(and its history) can tell you what those important factors are likely to be Byimportant factors we mean those which are driving the dominant features of thedata If those factors can be identified, then picking those to begin with and then

4 The third assumption is often stated using the observed data P i ; Q i / and doing so is equivalent given the DGP For an introduction to the study of the relationships between the data, DGP, and shocks, see the Annex to this chapter (section 2.5).

refining an econometric model in light of specification tests seems to provide areasonably successful approach, although certainly not one immune to criticism.5

Whether you use a specific-to-general modeling approach or vice versa, the greaterthe subtlety in the relationship between demand and its determinants, the better datayou are likely to need to use any econometric techniques

2.1.1.2 The Method of Least Squares

Consider the following regression model:

yiD a C bxiC ei:The OLS regression estimator attempts to estimate the effect of the variable x onthe variable y by selecting the values of the parameters a; b/ To do so, OLSassigns the maximum possible explanatory power to the variables that we specify asdeterminants of the outcome and minimizes the effect of the “leftover” component,

ei The value of the “leftover” component depends on our choice of parameters.a; b/ so we can write ei.a; b/ D yi
a
bxi Formally, OLS will choose theparameters a and b to minimize the sum of squared errors, that is, to solve

in its parameters, but the technique can be more generally applied For example,

we may have a model which is not linear in the parameters which states ei.a; b/ D

yi
f xiI a; b/, where, for example, f xiI a; b/ D axb The same “least-squares”approach can be used to estimate the parameters by solving the analogous problem

“nonlinear” least squares (NLLS)

In the basic linear-in-parameters and linear-in-variables model, a given absolutechange in the explanatory variable x will always produce the same absolute change

in the explained variable y For example, if yi D Qiand xi D Pi, where Qiand

Pirepresent the quantity per week and price of a bottle of milk respectively, then anincrease in the price of milk by€0.50 might reduce the amount of milk purchased by,say, two bottles a week The linear-in-parameters and linear-in-variables assumptionimplies that the same quantity reduction holds whether the initial price is€0.75 or

€1.50 Because this assumption may not be realistic in many cases, alternative

5 An example of this approach is examined in more detail in the demand context in chapter 9.

Price

.

Figure 2.2. Estimated residuals in OLS regression

specifications may fit the data better For example, it is common to operate a logtransformation on price and quantity variables so that the constant estimated effect

is measured in terms of percentages, yi D ln Qi and xi D ln Pi In that case,

@ ln Qi=@ ln PiD b while @Qi=@Pi D bQi=Piso that the absolute changes depend

on the level of both quantity demanded and price Such variable transformations donot change the fact that the model is linear in its parameters, and so the modelremains amenable to estimation using OLS

We first discuss the single-variable regression to illustrate some useful conceptsand results of OLS and then generalize the discussion to the multivariate regression.First we introduce some terminology and notation Let Oa; Ob/ be estimates of theparameters a and b The predicted value of yigiven the estimates and a fixed valuefor xiis

O

yiD Oa C Obxi:The difference between the true value yiand the estimated Oyiis the estimated error,

or the residual ei Therefore, we have

eiD yi
Oyi:Figure 2.2 shows the estimated residuals for our inverse demand curve, where

yi D Pi and xi D Qi We see that positive residuals are above the estimatedline and negative residuals are below it OLS estimation of the inverse demandcurve minimizes the total sum of squares of the “vertical” prediction errors.6If themodel nests the true DGP and the parameters of the estimation are exactly right,then the residuals will be exactly the same as the true “errors,” i.e., the true randomshocks that affect our explained variable

6 In contrast, if we estimated this model on the demand curve, we would be minimizing the “horizontal” prediction errors on this graph: imagine rotating the graph in order to flip the axes The assumptions required would be different, since they would require, for instance, that E Œe i j P i  D 0 rather than

E Œeij Qi D 0 and the estimates we obtain will also be different, even if we plot the two lines on the same graph.

Mathematically, finding the OLS estimators involves solving the minimizationproblem:

If the model is linear in the parameters, then the minimization problem is quadratic

in the parameters and hence the first-order conditions are linear in the parameters

As a result, the first-order conditions provide us with a system of linear equations tosolve, one for each parameter Linear systems of equations are typically often easy tosolve analytically In contrast, if we write down a nonlinear (in parameters) model,

we may have to solve the minimization problem numerically, but conceptually theapproach is no different.7

In the two-parameter case, the first normal equation can be solved to give Oa DN

y
Ob Nx, where Ny and Nx denote sample averages, as shown below:

n

X

i D1

xi:The estimated value of the intercept is a function of the other estimated parameterand the average value of the variables in the regression If the estimated parameterO

b is equal to 0 so that our explanatory variables have no explanatory power, then theestimated parameter Oa (and the predicted value of y) is just the average value of thedependent variable

Given the expression for Oa, we can solve

7 Programs such as Matlab and Gauss provide a number of standard tools to allow nonlinear problems

to be solved Solving nonlinear systems of equations can sometimes be very easy in practice, but can also

be very difficult even with the very good computational algorithms now easily accessible to analysts.

More generally, we will want to estimate regression equations where the dent variable is explained by a number of explanatory variables For example, salesmay be determined by both price and advertising levels Alternatively, a “second”explanatory variable may be a lower- or higher-order term such as a square root

depen-or squared term meaning that such a specification can account fdepen-or both ple variables and also particular types of nonlinearities in variables Retaining thelinear-in-parameters specification, a multivariate regression equation takes the form:

multi-yi D a C b1x1iC b2x2iC b3x3iC ei:For given parameter values, the predicted value of yifor given estimates and values

of x1i; x2i; x3i/ is

O

yi D Oa C Ob1x1iC Ob2x2i C Ob3x3i

and so the prediction error is ei D yi
Oyi

In this case, the minimization problem is the same as the case with two parametersexcept that it involves more parameters to minimize over:

To find those solutions, however, it is usually easier to use matrix notation, lowing the unifying treatment provided by Anderson (1958) To do so, simply stack

fol-up observations for the regression equation above to define the equivalent matrixexpression

2664

5C

2664

5D

2664

x10

x0 2

::

:

x0

377

5ˇ C

2664

5;

which can in turn be more simply expressed in terms of vectors and matrices as

y D Xˇ C e;

where y is an n  1/ vector and X is an n  k/ matrix of data, while ˇ is the k  1/vector of parameters to be estimated and e is the n  1/ vector of residuals In ourexample, k D 4 as there are four parameters to be estimated

The general OLS minimization problem can be easily solved by using matrixnotation Specifically, note that the OLS minimization problem can be expressedas

The variance of the OLS estimator can be calculated as follows:

VarŒ OˇOLSj X  D EŒ OˇOLS
EŒ OˇOLSj X / OˇOLS
EŒ OˇOLSj X /0j X :Now if we suppose that the DGP is of the form y D Xˇ0C u, then

EŒ OˇOLSj X  D EŒ.X0X /1X0.Xˇ0C u/ j X 

D ˇ0C X0X /1X0EŒu j X 

D ˇ0:Provided EŒu j X  D 0 and since OˇOLS
ˇ0D X0X /1X0u, we have

VarŒ OˇOLSj X  D EŒ.X0X /1X0u X0X /1X0u/0j X 

D X0X /1X0.EŒuu0j X /X.X0X /1:

If the variance is homoskedastic so that EŒuu0 j X  D 2In, then the formulacollapses to the simpler expression,

VarŒ OˇOLSj X  D X0X /12In:

2.1.2 Properties of OLS

Ordinary least squares is a simple and intuitive method to apply, which explains some

of its popularity However, it is also attractive because the estimators it producesexhibit some very desirable properties provided the assumptions it requires hold.Next we briefly review these properties and the conditions necessary for them tohold

2.1.2.1 Unbiasedness

An estimator is unbiased if its expected value is equal to the true value, i.e., if theestimator is “on average” the true value This means that the average of the coefficientestimates over all possible samples of size n, f.Xi; Yi/I i D 1; : : : ; ng, would beequal to the true value of the coefficient Formally,

EŒ Oˇ D ˇ0;where ˇ0is the true parameter of the DGP The unbiasedness property is equivalent tosaying that, on average, OLS estimation will give us the true value of the coefficient.For OLS estimators to be unbiased, a largely sufficient condition8 given the DGP

y D Xˇ0C u is that EŒu j X  D 0, meaning that the real error term must beunrelated to the value of our explanatory variables For instance, if we are explainingthe quantity demanded as a function of price and income, it is necessary that theshocks to the demand be uncorrelated with the level of prices or income

The unbiasedness condition can formally be obtained by applying the law of iterativeexpectations that states that the expected value of a variable is equal to the expectedvalue of the conditional expectation over the whole set of possible values of theconditions Formally, it states that EŒ OˇOLS D EXŒEŒ OˇOLSj X  This allows us towrite the expected value of the OLS estimator as follows:

EŒ OˇOLSj X  D X0X /1X0EŒy j X  D X0X /1X0EŒXˇ0C u j X 

D X0X /1X0Xˇ0C X0X /1X0EŒu j X 

D ˇ0C 0 if EŒu j X  D 0:

In general, unbiasedness is a tougher requirement than consistency, which wediscuss next In particular, while we will typically be able to find estimators forlinear models which are both unbiased and also consistent, many nonlinear modelswill admit estimators which are consistent but not unbiased

8 Strictly, there are in fact other regularity conditions which together suffice In particular, we will require that X0X=n/
1 exists.

2.1.2.2 Consistency

An estimator is a consistent estimator of a parameter if it tends toward the truepopulation value of the parameter as the sample available for estimation gets large.The property of consistency for averages is derived from a “law of large numbers.” Alaw of large numbers provides a set of assumptions under which a statistic converges

to its population equivalent For example, the sample average of a variable willconverge to the true population average as the sample gets big under weak conditions.Somewhat formally, we can write one such law of large numbers as follows If

X1; X2; : : : ; Xnis an independent random sample of variables from a populationwith mean  < 1 and variance 2 < 1 so that EŒXi D  and VarŒXi D 2,then consistency means that as the sample size n gets bigger the sample averageconverges9to the population average:

N

XnD 1n

1
1

nX

0u

:Note that each of the terms in X0X=n and 1=n/X0u are actually just sample aver-ages The former has, as its j kth element, 1=n/Pn

i D1xijxi kwhile the latter has, asits j th element, 1=n/Pn

i D1xijui These are just sample averages which, ing to a “law of large numbers,” will converge to their respective population means

accord-9 Econometrics textbooks will often spend a considerable amount of time defining precisely what we mean by “converge.” The two most common concepts are “convergence in probability” and “almost sure convergence.” These respectively provide the “weak” law of large numbers and the “strong” law of large numbers (SLLN).

10 A random variable which can only take on a finite set of values (technically, has finite support) will have all moments existing Possible exceptions (might) be price data in hyperinflations, where prices can

go off to close to infinity in extreme cases but even there presumably there is a limit on the amount of money that can be printed and also on the number of zeros that can be printed on any piece of paper In contrast, occasionally economic models of real world quantities do not have finite moments For example, Brownian motions are sometimes used in finance as approximations to the real world.

We also require that inverting the matrix X0X=n/1does not cause any problems(e.g., division by zero would be bad) In fact, the OLS estimator will be consistent

if, for a large enough sample,

E.u;x/Œuixij D ExŒ.EujxŒui j xij/xij D ExŒ.0/xij D 0

It should by now be clear that our assumption EŒuij xij D 0 plays a central role

in ensuring OLS is consistent If this assumption is violated, OLS estimation maywell produce estimators that bear no relation to the true value of the parameters ofthe DGP, even if we fortuitously write down a family of models which includes theDGP Unfortunately, this crucial assumption is often violated in real world settings.Among others, causes can include (i) misspecification of models, (ii) measurementerror, and (iii) endogeneity We discuss these problems, and in particular the problem

of endogeneity, later in this chapter

2.1.3 Hypothesis Testing

Econometric estimation produces an estimate of one or more parameters A samplewill provide an estimate, not the population value Hypothesis testing involving aparameter helps us measure the extent to which the estimated outcome is consistentwith a particular assumption about the real magnitude of the effect In terms of aparameter, the hypothesis could be that the parameter takes on a particular value,say 1.11Concretely, hypothesis testing helps us explicitly reject or not reject a givenhypothesis with a specified degree of certainty—or “confidence.” To understandhow this is done, we need to understand the concept of confidence intervals

11 More generally, we can test whether the assumptions required for our model and econometric mator are in fact satisfied In terms of a model, the hypothesis could be that a model is correctly specified (see, for example, any econometric text’s discussion of the RESET test) In terms of an estimator, the hypothesis could be that an efficient estimator that requires strong assumptions is consistent and the strong assumptions are true (see any econometric discussion of the Wu–Durbin–Hausman test).