Book Econometric Analysis of Cross Section and Panel Data By Wooldridge - Chapter 2 pdf

Although it is not always explicitly stated, the goal of most applied econometric studies is to estimate or test hypotheses about the pectation of one variable—called the explained varia

2.1 The Role of Conditional Expectations in Econometrics

As we suggested in Section 1.1, the conditional expectation plays a crucial role

in modern econometric analysis Although it is not always explicitly stated, the goal

of most applied econometric studies is to estimate or test hypotheses about the pectation of one variable—called the explained variable, the dependent variable, theregressand, or the response variable, and usually denoted y—conditional on a set ofexplanatory variables, independent variables, regressors, control variables, or covari-ates, usually denoted x¼ ðx1; x2; ; xKÞ

ex-A substantial portion of research in econometric methodology can be interpreted

as finding ways to estimate conditional expectations in the numerous settings thatarise in economic applications As we briefly discussed in Section 1.1, most of thetime we are interested in conditional expectations that allow us to infer causalityfrom one or more explanatory variables to the response variable In the setup fromSection 1.1, we are interested in the e¤ect of a variable w on the expected value of

y, holding fixed a vector of controls, c The conditional expectation of interest is

Eð y j w; cÞ, which we will call a structural conditional expectation If we can collectdata on y, w, and c in a random sample from the underlying population of interest,then it is fairly straightforward to estimate Eð y j w; cÞ—especially if we are willing tomake an assumption about its functional form—in which case the e¤ect of w on

Eð y j w; cÞ, holding c fixed, is easily estimated

Unfortunately, complications often arise in the collection and analysis of economicdata because of the nonexperimental nature of economics Observations on economicvariables can contain measurement error, or they are sometimes properly viewed asthe outcome of a simultaneous process Sometimes we cannot obtain a randomsample from the population, which may not allow us to estimate Eð y j w; cÞ Perhapsthe most prevalent problem is that some variables we would like to control for (ele-ments of c) cannot be observed In each of these cases there is a conditional expec-tation (CE) of interest, but it generally involves variables for which the econometriciancannot collect data or requires an experiment that cannot be carried out

Under additional assumptions—generally called identification assumptions—wecan sometimes recover the structural conditional expectation originally of interest,even if we cannot observe all of the desired controls, or if we only observe equilib-rium outcomes of variables As we will see throughout this text, the details di¤erdepending on the context, but the notion of conditional expectation is fundamental

In addition to providing a unified setting for interpreting economic models, the CEoperator is useful as a tool for manipulating structural equations into estimableequations In the next section we give an overview of the important features of the

conditional expectations operator The appendix to this chapter contains a more tensive list of properties.

ex-2.2 Features of Conditional Expectations

2.2.1 Definition and Examples

Let y be a random variable, which we refer to in this section as the explained variable,

and let x 1ðx1; x2; ; xKÞ be a 1
K random vector of explanatory variables IfEðj yjÞ < y, then there is a function, say m: RK! R, such that

Eð y j x1; x2; ; xKÞ ¼ mðx1; x2; ; xKÞ ð2:1Þ

or Eð y j xÞ ¼ mðxÞ The function mðxÞ determines how the average value of y changes

as elements of x change For example, if y is wage and x contains various individualcharacteristics, such as education, experience, and IQ, then Eðwage j educ; exper; IQÞ

is the average value of wage for the given values of educ, exper, and IQ Technically,

we should distinguish Eð y j xÞ—which is a random variable because x is a randomvector defined in the population—from the conditional expectation when x takes on

a particular value, such as x0: Eð y j x ¼ x0Þ Making this distinction soon becomescumbersome and, in most cases, is not overly important; for the most part we avoid

it When discussing probabilistic features of Eð y j xÞ, x is necessarily viewed as arandom variable

Because Eð y j xÞ is an expectation, it can be obtained from the conditional density

of y given x by integration, summation, or a combination of the two (depending onthe nature of y) It follows that the conditional expectation operator has the samelinearity properties as the unconditional expectation operator, and several additionalproperties that are consequences of the randomness of mðxÞ Some of the statements

we make are proven in the appendix, but general proofs of other assertions requiremeasure-theoretic probabability You are referred to Billingsley (1979) for a detailedtreatment

Most often in econometrics a model for a conditional expectation is specified todepend on a finite set of parameters, which gives a parametric model of Eð y j xÞ Thisconsiderably narrows the list of possible candidates for mðxÞ

Example 2.1: For K ¼ 2 explanatory variables, consider the following examples ofconditional expectations:

Eð y j x1; x2Þ ¼ b0þ b1x1þ b2x2þ b3x22 ð2:3Þ

Eð y j x1; x2Þ ¼ b0þ b1x1þ b2x2þ b3x1x2 ð2:4Þ

Eð y j x1; x2Þ ¼ exp½b0þ b1logðx1Þ þ b2x2; y b 0; x1>0 ð2:5ÞThe model in equation (2.2) is linear in the explanatory variables x1and x2 Equation(2.3) is an example of a conditional expectation nonlinear in x2, although it is linear

in x1 As we will review shortly, from a statistical perspective, equations (2.2) and(2.3) can be treated in the same framework because they are linear in the parameters

bj The fact that equation (2.3) is nonlinear in x has important implications forinterpreting the bj, but not for estimating them Equation (2.4) falls into this sameclass: it is nonlinear in x¼ ðx1; x2Þ but linear in the bj

Equation (2.5) di¤ers fundamentally from the first three examples in that it is anonlinear function of the parameters bj, as well as of the xj Nonlinearity in theparameters has implications for estimating the bj; we will see how to estimate suchmodels when we cover nonlinear methods in Part III For now, you should note thatequation (2.5) is reasonable only if y b 0

2.2.2 Partial E¤ects, Elasticities, and Semielasticities

If y and x are related in a deterministic fashion, say y¼ f ðxÞ, then we are ofteninterested in how y changes when elements of x change In a stochastic setting wecannot assume that y¼ f ðxÞ for some known function and observable vector x be-cause there are always unobserved factors a¤ecting y Nevertheless, we can define thepartial e¤ects of the xj on the conditional expectation Eð y j xÞ Assuming that mðÞ

is appropriately di¤erentiable and xj is a continuous variable, the partial derivativeqmðxÞ=qxj allows us to approximate the marginal change in Eð y j xÞ when xj isincreased by a small amount, holding x1; ; xj1; xjþ1; xK constant:

DEð y j xÞ AqmðxÞ

qxj

 Dxj; holding x1; ; xj1; xjþ1; xK fixed ð2:6ÞThe partial derivative of Eð y j xÞ with respect to xj is usually called the partial e¤ect

of xj on Eð y j xÞ (or, to be somewhat imprecise, the partial e¤ect of xj on y) preting the magnitudes of coe‰cients in parametric models usually comes from theapproximation in equation (2.6)

Inter-If xj is a discrete variable (such as a binary variable), partial e¤ects are computed

by comparing Eð y j xÞ at di¤erent settings of xj(for example, zero and one when xjisbinary), holding other variables fixed

Example 2.1 (continued): In equation (2.2) we have

This latter expression gives the elasticity its interpretation as the approximate centage change in Eð y j xÞ when xj increases by 1 percent.

per-Example 2.1 (continued): In equations (2.2) to (2.5), most elasticities are not stant For example, in equation (2.2), the elasticity of Eð y j xÞ with respect to x1 is

con-ðb1x1Þ=ðb0þ b1x1þ b2x2Þ, which clearly depends on x1 and x2 However, in tion (2.5) the elasticity with respect to x1 is constant and equal to b1

equa-How does equation (2.10) compare with the definition of elasticity from a modellinear in the natural logarithms? If y > 0 and xj>0, we could define the elasticity asqE½logð yÞ j x

where u has zero mean and is independent of ðx1; x2Þ, then the elasticity of y withrespect to x1 is b1 using either definition of elasticity If Eðu j xÞ ¼ 0 but u and x arenot independent, the definitions are generally di¤erent

For the most part, little is lost by treating equations (2.10) and (2.11) as the samewhen y > 0 We will view models such as equation (2.12) as constant elasticitymodels of y with respect to x1whenever logð yÞ and logðxjÞ are well defined Defini-tion (2.10) is more general because sometimes it applies even when logð yÞ is notdefined (We will need the general definition of an elasticity in Chapters 16 and 19.)The percentage change in Eð y j xÞ when xjis increased by one unit is approximatedas

2.2.3 The Error Form of Models of Conditional Expectations

When y is a random variable we would like to explain in terms of observable ables x, it is useful to decompose y as

In other words, equations (2.15) and (2.16) are definitional: we can always write y asits conditional expectation, Eð y j xÞ, plus an error term or disturbance term that hasconditional mean zero

The fact that Eðu j xÞ ¼ 0 has the following important implications: (1) EðuÞ ¼ 0;(2) u is uncorrelated with any function of x1; x2; ; xK, and, in particular, u isuncorrelated with each of x1; x2; ; xK That u has zero unconditional expectationfollows as a special case of the law of iterated expectations (LIE ), which we covermore generally in the next subsection Intuitively, it is quite reasonable that Eðu j xÞ ¼

0 implies EðuÞ ¼ 0 The second implication is less obvious but very important Thefact that u is uncorrelated with any function of x is much stronger than merely sayingthat u is uncorrelated with x1; ; xK

As an example, if equation (2.2) holds, then we can write

y¼ b0þ b1x1þ b2x2þ u; Eðu j x1; x2Þ ¼ 0 ð2:17Þand so

But we can say much more: under equation (2.17), u is also uncorrelated with anyother function we might think of, such as x12; x22; x1x2;expðx1Þ, and logðx2

2þ 1Þ Thisfact ensures that we have fully accounted for the e¤ects of x1 and x2on the expectedvalue of y; another way of stating this point is that we have the functional form of

Eð y j xÞ properly specified

If we only assume equation (2.18), then u can be correlated with nonlinear tions of x1and x2, such as quadratics, interactions, and so on If we hope to estimatethe partial e¤ect of each xj on Eð y j xÞ over a broad range of values for x, we wantEðu j xÞ ¼ 0 [In Section 2.3 we discuss the weaker assumption (2.18) and its uses.]Example 2.2: Suppose that housing prices are determined by the simple modelhprice¼ b0þ b1sqrftþ b2distanceþ u;

func-where sqrft is the square footage of the house and distance is distance of the housefrom a city incinerator For b2 to represent qEðhprice j sqrft; distanceÞ=q distance, wemust assume that Eðu j sqrft; distanceÞ ¼ 0

2.2.4 Some Properties of Conditional Expectations

One of the most useful tools for manipulating conditional expectations is the law ofiterated expectations, which we mentioned previously Here we cover the most gen-eral statement needed in this book Suppose that w is a random vector and y is arandom variable Let x be a random vector that is some function of w, say x¼ fðwÞ.(The vector x could simply be a subset of w.) This statement implies that if we knowthe outcome of w, then we know the outcome of x The most general statement of theLIE that we will need is

In other words, if we write m1ðwÞ 1 Eð y j wÞ and m2ðxÞ 1 Eð y j xÞ, we can obtain

m2ðxÞ by computing the expected value of m2ðwÞ given x: m1ðxÞ ¼ E½m1ðwÞ j x.There is another result that looks similar to equation (2.19) but is much simpler toverify Namely,

Note how the positions of x and w have been switched on the right-hand side ofequation (2.20) compared with equation (2.19) The result in equation (2.20) followseasily from the conditional aspect of the expection: since x is a function of w, know-ing w implies knowing x; given that m2ðxÞ ¼ Eð y j xÞ is a function of x, the expectedvalue of m2ðxÞ given w is just m2ðxÞ

Some find a phrase useful for remembering both equations (2.19) and (2.20): ‘‘Thesmaller information set always dominates.’’ Here, x represents less information than

w, since knowing w implies knowing x, but not vice versa We will use equations(2.19) and (2.20) almost routinely throughout the book

For many purposes we need the following special case of the general LIE (2.19) If

x and z are any random vectors, then

or, defining m1ðx; zÞ 1 Eð y j x; zÞ and m2ðxÞ 1 Eð y j xÞ,

For many econometric applications, it is useful to think of m1ðx; zÞ ¼ Eð y j x; zÞ as

a structural conditional expectation, but where z is unobserved If interest lies in

Eð y j x; zÞ, then we want the e¤ects of the xj holding the other elements of x and zfixed If z is not observed, we cannot estimate Eð y j x; zÞ directly Nevertheless, since

y and x are observed, we can generally estimate Eð y j xÞ The question, then, iswhether we can relate Eð y j xÞ to the original expectation of interest (This is a ver-sion of the identification problem in econometrics.) The LIE provides a convenientway for relating the two expectations

Obtaining E½m1ðx; zÞ j x generally requires integrating (or summing) m1ðx; zÞagainst the conditional density of z given x, but in many cases the form of Eð y j x; zÞ

is simple enough not to require explicit integration For example, suppose we beginwith the model

Eð y j x1; x2; zÞ ¼ b0þ b1x1þ b2x2þ b3z ð2:23Þbut where z is unobserved By the LIE, and the linearity of the CE operator,

Eð y j x1; x2Þ ¼ Eðb0þ b1x1þ b2x2þ b3zj x1; x2Þ

¼ b0þ b1x1þ b2x2þ b3Eðz j x1; x2Þ ð2:24ÞNow, if we make an assumption about Eðz j x1; x2Þ, for example, that it is linear in x1

Now suppose equation (2.23) contains an interaction in x1 and z:

Eð y j x1; x2; zÞ ¼ b0þ b1x1þ b2x2þ b3zþ b4x1z ð2:26ÞThen, again by the LIE,

Eð y j x1; x2Þ ¼ b0þ b1x1þ b2x2þ b3Eðz j x1; x2Þ þ b4x1Eðz j x1; x2Þ

If Eðz j x1; x2Þ is again given in equation (2.25), you can show that Eð y j x1; x2Þ hasterms linear in x1 and x2 and, in addition, contains x21 and x1x2 The usefulness ofsuch derivations will become apparent in later chapters

The general form of the LIE has other useful implications Suppose that for some(vector) function fðxÞ and a real-valued function gðÞ, Eð y j xÞ ¼ g½fðxÞ Then

There is another way to state this relationship: If we define z 1 fðxÞ, then Eð y j zÞ ¼gðzÞ The vector z can have smaller or greater dimension than x This fact is illus-trated with the following example

Example 2.3: If a wage equation is

Eðwage j educ; experÞ ¼ b0þ b1educþ b2experþ b3exper2þ b4educexper

then

Eðwage j educ; exper; exper2; educexperÞ

¼ b0þ b1educþ b2experþ b3exper2þ b4educexper:

In other words, once educ and exper have been conditioned on, it is redundant tocondition on exper2and educexper

The conclusion in this example is much more general, and it is helpful for ing models of conditional expectations that are linear in parameters Assume that, forsome functions g1ðxÞ; g2ðxÞ; ; gMðxÞ,

analyz-Eð y j xÞ ¼ b0þ b1g1ðxÞ þ b2g2ðxÞ þ    þ bMgMðxÞ ð2:28ÞThis model allows substantial flexibility, as the explanatory variables can appear inall kinds of nonlinear ways; the key restriction is that the model is linear in the bj If

we define z11g1ðxÞ; ; zM1gMðxÞ, then equation (2.27) implies that

Eð y j z ; z ; ; z Þ ¼ b þ b z þ b z þ    þ b z ð2:29Þ

This equation shows that any conditional expectation linear in parameters can

be written as a conditional expectation linear in parameters and linear in someconditioning variables If we write equation (2.29) in error form as y¼ b0þ b1z1þ

b2z2þ    þ bMzMþ u, then, because Eðu j xÞ ¼ 0 and the zj are functions of x, itfollows that u is uncorrelated with z1; ; zM (and any functions of them) As we willsee in Chapter 4, this result allows us to cover models of the form (2.28) in the sameframework as models linear in the original explanatory variables

We also need to know how the notion of statistical independence relates to tional expectations If u is a random variable independent of the random vector x,then Eðu j xÞ ¼ EðuÞ, so that if EðuÞ ¼ 0 and u and x are independent, then Eðu j xÞ ¼

condi-0 The converse of this is not true: Eðu j xÞ ¼ EðuÞ does not imply statistical pendence between u and x ( just as zero correlation between u and x does not implyindependence)

inde-2.2.5 Average Partial E¤ects

When we explicitly allow the expectation of the response variable, y, to depend onunobservables—usually called unobserved heterogeneity—we must be careful inspecifying the partial e¤ects of interest Suppose that we have in mind the (structural)conditional mean Eð y j x; qÞ ¼ m1ðx; qÞ, where x is a vector of observable explanatoryvariables and q is an unobserved random variable—the unobserved heterogeneity.(We take q to be a scalar for simplicity; the discussion for a vector is essentially thesame.) For continuous xj, the partial e¤ect of immediate interest is

yjðx; qÞ 1 qEð y j x; qÞ=qxj¼ qm1ðx; qÞ=qxj ð2:30Þ(For discrete xj, we would simply look at di¤erences in the regression function for xj

at two di¤erent values, when the other elements of x and q are held fixed.) Because

yjðx; qÞ generally depends on q, we cannot hope to estimate the partial e¤ects acrossmany di¤erent values of q In fact, even if we could estimate yjðx; qÞ for all x and q,

we would generally have little guidance about inserting values of q into the meanfunction In many cases we can make a normalization such as EðqÞ ¼ 0, and estimate

yjðx; 0Þ, but q ¼ 0 typically corresponds to a very small segment of the population.(Technically, q¼ 0 corresponds to no one in the population when q is continuouslydistributed.) Usually of more interest is the partial e¤ect averaged across the popu-lation distribution of q; this is called the average partial e¤ect (APE )

For emphasis, let xo denote a fixed value of the covariates The average partiale¤ect evaluated at xo is

where Eq½   denotes the expectation with respect to q In other words, we simply averagethe partial e¤ect yjðxo; qÞ across the population distribution of q Definition (2.31) holdsfor any population relationship between q and x; in particular, they need not be inde-pendent But remember, in definition (2.31), xo is a nonrandom vector of numbers.For concreteness, assume that q has a continuous distribution with density func-tion gðÞ, so that

One common assumption in nonlinear models with unobserved heterogeneity isthat q and x are independent We will make the weaker assumption that q and x areindependent conditional on a vector of observables, w:

where Dð j Þ denotes conditional distribution (If we take w to be empty, we get thespecial case of independence between q and x.) In many cases, we can interpretequation (2.33) as implying that w is a vector of good proxy variables for q, butequation (2.33) turns out to be fairly widely applicable We also assume that w isredundant or ignorable in the structural expectation

As we will see in subsequent chapters, many econometric methods hinge on beingable to exclude certain variables from the equation of interest, and equation (2.34)makes this assumption precise Of course, if w is empty, then equation (2.34) is trivi-ally true

Under equations (2.33) and (2.34), we can show the following important result,provided that we can interchange a certain integral and partial derivative:

where Ew½   denotes the expectation with respect to the distribution of w Before weverify equation (2.35) for the special case of continuous, scalar q, we must understandits usefulness The point is that the unobserved heterogeneity, q, has disappeared en-tirely, and the conditional expectation Eð y j x; wÞ can be estimated quite generally