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OMITTED VARIABLE BIAS AND CROSS SECTION REGRESSION by Thomas M. Stoker July 1983 WP #1460-83
Thomas M. Stoker is Assistant Professor, Sloan School of Management, Massachusetts Institute of Technology, Cambridge, MA 02139. The author
wished to thank A. Deaton, T. Gorman, J. Hausmann, J. Heckman, D. Jorgenson,, A. Lewbel, J. Muellbauer, J. Powell and J. Roteinberg for helpful comments on this and related work. the author. All errors, etc., remain the responsibility of
This paper reinterprets and explains the standard omitted variable bias formula in the context of cross section regression when the true model underlying behavior is unknown and possibly nonlinear. The vehicle employed
to analyze cross section regression in this case is the macroeconomic interpretation of cross section OLS coefficients established in Stoker (1982a). The exposition begins by indicating precisely the distributional assumptions underlying a correctly specified linear cross section regression equation when the true model is nonlinear and possibly unknown. By considering
the case of too many regressors, we show that the omitted variable bias formula reflects constraints in distribution movement, which alternatively allow the bias formula to be derived as a total derivative formula among macroeconomic effects. By considering the case of too few regressors, we
show that the macroeconomic impact of the omitted variables can be measured by their partial contribution to the variance of the dependent variable in a cross section regression. Some practical implications of these results
are discussed and an illustrative example is given.
Introduction The purpose of this paper is to reinterpret and explain standard omitted
variable bias formulae in the context of cross section regression when the true model underlying behavior is unknown and possible nonlinear. The vehicle
employed to analyze cross section regression in this case is the macroeconomic interpretation of cross section OLS coefficients established in Stoker (1982a). The omitted variable bias formula is a very useful tool for judging the impact on regression analysis of omitting important influences on behavior which are not observed in the data set. In small sample form, the bias
formula was developed and popularized by Thiel (1957, 1971), and has been used extensively in empirical research. The bias interpretation of the
formula, however, relies exclusively on the assumed linearity of the included and omitted variables in the equation modeling the dependent variable. The formula itself has an empirical counterpart which holds an identity among computed OLS regression coefficients from equations with different subsets of regressors. 2 The question of interest here is whether this
regression coefficient relationship can be interpreted when the behavioral model is general and possibly unknown. A macroeconomic interpretation for
cross section OLS coefficients in this case was established by Stoker (1982a). In this paper we will extend the interpretation to the standard omitted variable bias formula. The precise issue addressed can be described in more detail as follows. Stoker (1982a) established that OLS slope coefficients obtained from regressing a dependent variable y on predictor variables X computed using cross section data will consistently estimate the effects of changing mean X, E(X) on mean y, E(y), provided that the X distribution varies through time via the
exponential family form.
This latter condition is of interest because it
implies no testable restrictions on the cross section data, and in particular does not rely on a particular functional form of the relationship between y and X. But suppose that X is partitioned as X = (X1,X 2). The above result
can also be applied to say that the OLS coefficients of y on X1 consistently estimate the effects of changing E(X1) on E(y). In this paper, we will explain
exactly how the assumptions underlying the macroeconomic interpretations of these two regressions differ. In so doing, we obtain a general interpretation
of the omitted variable bias formula, which connects the coefficients of these two regressions. The results of the paper shift the misspecification question from the behavioral model to the assumptions which control the way the population distribution evolves through time. of the predictor distribution are If the driving variables (to be defined) , then the proper macroeconomic effects The bias
are estimated by the cross section regression of y on X1 only.
formula connecting these coefficients to those of regression of y on X 1 and X2 just reflects the induced effect of E(X1 ) on E(X2). The development
for changes in the joint income - family size distribution, the cross section OLS coefficients of food on income and family size consistently estimate the effects of changing average income and average family size on average food.
Now, average family size may be correctly excluded from an average food equation if family size has a zero cross section food regression coefficient or if the conditional distribution of family size given income is constant through time. This latter condition says that average food is a function only of average income, with average family size having no independent effect. It is for checking this In
latter possibility that the omitted variable bias calculations are useful.
particular, the estimated coefficients of the auxiliary equation of family size regressed on income indicates the effect of average income changes on average family size. If two or more time series observations on average income and average
family size are consistent with the estimated effects, then omitting average family size from the average food model is suggested. If the estimated effects bear
no relation to the time series patterns of average income and average family size, and if family size has a nonzero cross section regression coefficient in a food equation, then average family size has an independent influence on average food demand. We begin with the notation, a discussion of the omitted variable bias formula and a review of the OLS coefficient results of Stoker (1982a). In
Section 3 we consider the case of too many regressors in the cross section equation, and present the alternative derivation of the omitted variable bias formulae using macroeconomic derivatives. In Section 4 we consider the case
of too few regressors , indicating the macroeconomic analogue of coefficient bias. In Section 5 we present an algebraic example, and in Section 6
discuss some related work.
Notation and Background Results 2.1 Individual Models and Cross Section Data
All of our results will concern interpretations of OLS regression coefficients computed with cross section data observed at a particular time period, say t = t
Denote by y a dependent variable of interest, and by X
an M vector of predictor variables, observations on these variables
The cross section data consists of K
Yk, Xk, k=l. , K, which are assumed to Moreover,
represent a random sample from a distribution with density P (y,X). the entire population at t = t
of (say) N observations is assumed to be a 4 -.andom sample from:the same distribution with N >> K. The following r assumption characterizes the cross section structure. ASSUMPTION 1: The means, variances and covariances of y and
x exist, and the variance-covariance matrix of X is nonsingular and positive definite. The conditional distribution
of y given X exists,. with density qo(ytX), as does the mean of y given X, denoted E(ylX) F(X).
:. For the purpose of considering omitted variables, we suppose that X is partitioned into an M 1 vector X 1 and an M 2 vector X 2 as X' = (X1' where M , X2 )
+ M2 M. Denote the means of y and X by E (y) - pY and 0 1,.2 EX E-i * (9 (, 2), the variance of y by oy, the variance-covariance matrix of X yby. matrix of X-;by
0 rXX Z12
and the covariance matrix between y and X as
o = ZX = [
when the notation corresponds to the partitioning of X.
The overall density
P (yjX) which underlies the cross section can be factored as Po(YIX) =
(X) is the marginal distribution of X.
P (X), where po(X) is the marginal distribution of X.
The conditional density q(yX) corresponds to the true econometric model relating y and X for individual observations. In standard practice, in order
to study the relationship between y and X, one would spedify a behavioral model y = f (X,u), where u represents unobserved individual heterogenteity and
y a set of parameters to be estimated, together with the stochastic distribution of u given X, say with density q(uIX). Combining the behavioral function We
and the heterogeneity distribution gives the conditional density q (yIX). assume y is equal to its true value, and thus suppress it in the notation. For concreteness, consider the example given in the introduction, where y denotes the demand for food by individual families, X 1 income and X 2 family size.
The true Engel curve with family size is represented by E(ylX) = F(X),
and q(ylX) reflects the Engel curve together with the stochastic specification of the deviation y - F(X). If the true behavioral function was linear with
additive disturbance - i.e., y = f (X,u) = yO
y 2 X2 + u - and the
distribution of u conditional on X was normal with mean 0 and variance a , then qo(yX) denotes a normal distribution with mean F(X) = 2 and variance a. o + Y 1 'X1 + y2'X
Alternatively, the framework will accomodate many other
standard econometric modeling situations - for example if y takes on only a finite number of values and behavior is described by a discrete choice model, then q(ylX) gives the choice probabilities for each of the possible values of y given X, which could be of the probit or logit form with appropriate specification of the distribution of unobserved individual influences on the choice process. All of the exposition is concerned with interpretations of regressions performed using the cross section data, represented as The regression of y on X is
yk = a
= a y.12 +Xlkby y.1(2) + 1k
X'by 2k y.2(1)
;. wh'ere by 1 y.12
-(byl( ), b: 2 ( 1 )) are computed using ordinary least squares (OLS) 2 Y.1(2)'.y.2(1) We denote the
and 'the notation reflects the partitioning of X' = (X1 , X2 ).
large sample (probability.limit) values of the statistics from this regression as' in
plim E k/K = a - E0(ZO )-1 E * yy Xy X X
= y.12 Also of interest is the regression of y on X1 only, which we denote as
+w Xlkby. + k y1 k = 1,. , k.
The large sample values of these statistics are denoted as in: plim b K-_> y.l - I I __ pl-m ay_1 _.i a _ (F )-1E0
( ly II
The slope regression coefficients of (2,3) and (2,5) are connected by the identity
bb 1 =b = by.1(2) +B 2.1 by.2(1) where B2.1 is the M 1 x M 2 regression matrix of OLS coefficients of the auxiliary
X2k = A2. 1
+ XB2.1 +
The version of (2.7) relating the large sample values of the coefficients is
Byl = P y.1(2) 3-B + B2.1 13y.2 (1 ) where B2. 1 = (Z1) 12 = plim B2.1
The Omitted Variable Bias Formula
The standard omitted variable bias formula is an equation explaining the small sample expectation of b y.1 when the true behavioral model pecifies y as
a linear function of X1 and X2 with additive residual.
The equation is formally
quite similar to (2.7) and (2.9), and we introduce it separately here for later comparison with our general development. We begin by assuming that q(yjX) is a distribution with mean E(y1X) = F(X) o + Y'Xl +Y2 X2, or equivalently that the true behavioral model is
2'X 2 + u
where u has zero expectation conditional on X. verify thata y.12 notation. Yo' y.1(2) = and
In this case, it is easy to 21 using our previous
The omitted variable bias formula is derived by inserting for Yk into the OLS formula defining b This yields
2,10) evaluated expectation.
of (2,5) and taking its
E(by.llX data) = y1 + B2.1 Y 2.
where B2.1 is defined as the OLS coefficients of (2.8) and
X data" denotes (2.11) is the
that the expectation is taken conditional on Xlk,X 2 k,k=l,.,K. omitted variable bias formula.
The practical usefulness of this formula can be illustrated using our previous example. Suppose y is food expenditure, X 1 is income, X2 is family (2.11)
size and (2.10) is the true demand equation, with yl and y2 positive.
says if one regresses food y on income X 1 only (omitting family size X2) that by 1 will on average overestimate (underestimate) the income effect Y if the regression coefficient B2. 1 of family size X 2 on income X 1 is positive (negative). The magnitude of the bias E(by. lX data) 1
depends on the size
of the true family size effect
2 and the amount of the correlation between
family size and income in the data. Perhaps better use of the equation (2.11) occurs when X2 is not observed in the data. Suppose for instance that y and X 1 are as above, but X 2 now
represents an unobserved variable, say the amount of gambling done by each family. If we suppose that gambling has a negative effect on food expenditure,
Y2 < 0, then (2.11) says that b.1 will on average overestimate (underestimate) the true income effect if income and amount of gambling are negatively (positively) correlated in the sample. If the analyst has outside information
that gambling activity is weakly correlated with income level, then (2.11) provides an argument for robustness, namely that b the true income effect y1. 1 will on average equal
In our general framework, where we relax the linear model assumption (2.10), it is difficult to characterize the small sample properties of b y.' so we lose the omitted variable bias formula (2.11) as a tool for analysis. We will instead concentrate on interpreting (2.9), the large sample version of (2.7) and (2,11). For this task, we must first review the macroeconomic
interpretation of cross section regression coefficients, which characterizes the large sample values and
Macroeconomic Effects and Regression Coefficients
The results of Stoker (1982a) (reviewed below) establish that cross section OLS regression coefficients consistently estimate the macroeconomic effects of changing the mean of X on the mean of y. In this section we review the We then
exponential family assumptions which are sufficient for the result.
provide an immediate proof of the result for the exponential family case. In order to discuss a relationship between the mean of y and the mean of X for a general behavioral model q(yX), we must specify precisely how the population density P (y,X) changes through time. We assume that the behavioral
model q (9X) is stable through time, so that we can focus attention on how the marginal X density Po(X) varies. In this paper we will employ a particular
structure for the X distribution, known as exponential family structure, which is introduced through the following assumptions: 5
The La Place Transform of Po(X):
L(TI) = eC exists for
exp (H'X) dX
() in a convex open neighborhood of the origin in R
The exponential family generated by p (X) with
driving variables X is the family defined by
p (XlH) = C(n) Po(X) exp (IX) where iEr and C(TH) is defined via (2.12).
As given the exponential family form is a standard distribution form known to statistics, which encompasses virtually all of the "textbook" distribution forms, such as Poisson, gamma, beta, multivariate normal and lognormal distributions among others, found by appropriate specification of the generating distribution and driving variables. parameters Notice for our purposes that the natural = 0
serve to index movements in the X distribution,with the cross section density p (X) = p (XIO). For each time period t
We formalize this as
ASSUMPTION 3A: H t
to , there exists
such that the marginal X density at time t is given via the
exponential family form with driving variables X and parameter Rt; i.e., Pt(X) = p (XIlt) of (2.12). The joint distribution
of y and X at time t has density Pt(y,X) = qo(YJX) p (XITt). Assumption 3A provides sufficient structure to determine the means of y and X as functions of the natural parameters direct integration, we have of the X distribution. By
y q(yfX) p*(Xfn) dX -
X p (Xn)dX
= 0 we have the cross section values of vY = ~ (0) and
0 = H(O).
The probldm with (2,14-15) is inconvenience, for it is not clear how to behaviorally interpret the natural parameters E. To overcome this, we
--I_._XII.-l---l^ -Ill_--l_l____-X--XII__ ___
reparameterize the X distribution by - = E(X), and derive the relation between E(y) = Y H() and E(X) = induced by (2.14-15). This is possible because
of (2,15) is invertible, and so we can redefine p (XII) as
= p (XIH-i())
and derive the (macroeconomic) aggregate function between
Of course, we have pY =
(p ) for the cross section parameter values. is invertible, note that7
Parenthetically, to see that H()
is the gradient operator) is invertible locally at n = 0 if and This matrix is
only if its differential (Jacobian) matrix is nonsingular. easily seen to be the covariance matrix of X via
which is assumed nonsingular at The aggregate function py =
= 0, the cross section value. (p) represents the model of macroeconomic
behavior in our framework, corresponding with the individual behavioral model qo(ylX) and Assumption 3A on the X distribution. p = E(X) on P by D. =
The macroeconomic effects of
E(y) are defined as the first derivatives of (p), denoted
Our results are concerned with the value of these derivatives at , the cross section parameter values.
As a final bit of background notation, it is useful to introduce formulae which capture the local behavior of the expectations (2.14), (2,15) at are related to changes in X, d, = 0.
For (2.15) we have that changes in 11, d, at 11= 0 as in
which is obvious from (2.19). changes in Y, d
Similarly, for (2.14), it is easy to show that at = 0 as in
are related to d
= E dI Xy
We refer to (2.20) and (2.21) as the "local equations" corresponding to (2.15) aId (2.14) respectively. The local equations provide very convenient methods for manipulating derivatives of expectations in our framework. For an illustration, we provide
an immediate proof of the result of Stoker (1982a) that cross section (OLS) coefficients always consistently estimate macroeconomic effects under exponential family structure on the distribution of X. (2.20) and insert into (2.21) as To see this, invert
and so plim b
12= y.1 2
, the macroeconomic effects.
manipulations just reflect application of the chain rule to the aggregate function (2.17).
Before proceeding to discuss omitted variables, it is useful to point out some salient aspects of the development preceding the OLS coefficient result (2.22). First, the result holds for a virtually arbitrary behavioral
model q (yX) and cross section distribution p(X), which are restricted by only the innocuous Assumptions and 2. Second, the driving variables X of
the exponential family play an important role, as they constitute the proper regressors in the cross section equation whose coefficients consistently estimate the macroeconomic effects. Elaboration of this relation is what
permits analysis of omitted variables and specification error, to which we now turn.
Too Many Regressors The result of Stoker (1982a) reviewed above provides a macroeconomic
interpretation of the OLS coefficients of any cross section regression performed, under the corresponding set of distribution movement assumptions. In this
section we consider the case where the regression (2.5) of y on X 1 is the correct one for estimating macroeconomic effects, as opposed to the regression (2.3) of y on X 1 and X2. The local equations (2.20), (2.21) and (2.22) are derived under the structure where the latter regression (2.3) is appropriate. The equations (2.20)
and (2.21) rewritten to reflect the X' = (X1 ',X2 ') partitioning are
o =1 dl + Z12d 1
o 0 dI 22 2
dpY =Edn + lyd 1
d 2 yl
where to X
1' = (1', and X2.
2') is partitioned into the natural parameters corresponding Equation (2.22), which established that plim by.12 =
is written in partitioned form as
As noted at the end of Section 2, for the regression coefficients of y on X i from equation (2.5) to consistently estimate macroeconomic effects, we must adopt the corresponding assumption that the X distribution changes via the exponential family with driving variables X 1 only, as in
Pi (XInl) = cl(1l) Po(X) exp (X
= C(H), the latter evaluated at
The parallel assumption
is written out as ASSUMPTION 3B:
For each time period t
such that the marginal distribution of X at time t is
given via the exponential family with driving variables X 1 and parameters Hlt ; i.e., pt(X) = (XIIlt) of (3.4). The
joint distribution of y and X at time t has density
p t (y ,X )
: qo(ylX) Pl(XlHlt)-
Under Assumption 3B, we can compute the mean of y and X1 as functions of II as before
E(y) = Y = c(ni)
) E(X1E(X= U=
H HI(]I ) 1
and find the induced relation between
i (Hr 1)
the pertinant aggregate function for this case. result now says that plim b1 = = a(1()
The OLS coefficient
1 are the
coefficients of y on X1 in equation (2.5).
The result can be verified easily
as above by directly deriving the local equations pertinant to (3.5) and (3.6), which are
o0 = E 1dll
and solving them for the induced local relation between Py and
The large sample omitted variable bias formula (2.9) arises out of the differences between Assumptions 3A and 3B. A moments reflection indicates that 2 is held constant
Assumption 3B is just Assumption 3A with the proviso that at 21 = 0.
This is reflected in the fact that the local equations (3.8), (3.9)
coincide with (3.2) and (3.la) when d must coincide when d By requiring d a function of E1. P2(X
Consequently, (3.10) and (3.2)
Detailing this correspondence yields formula (2.9).
2 = 0, Assumption 3B also structures the mean p = E(X 2) as
By factoring the base density Po(X) into p (X) =
where P 1 O(X) is the marginal distribution of X 1, we have 1X)P1O(X1) 1
E(X ) = 2 = 2
= G ( 1)
or in terms of
The local behavior of G equivalently by setting d found from (3.lb) to be
and G at H1 = 0 can be derived directly as before, or
0 in (3.la-b).
The local behavior of G
Inverting (3.la) and inserting into (3.13) gives the local behavior of G as
12 d 1 = 12 1
di 1 = 0 is
Consequently, equation (3.3) under d
y. dY 1(2)
fy. 2 ( 1)d
y.2 ( B2.ld1 (3.15)
= (y. 1(2)
= 8y1 d 11 1i
establishing the equivalence between (3,3) and (3,10).
This development yields several interpretations of standard specification analysis calculus in the context of a general population model, Equation (3,,12)
points out the macroeconomic interpretation of the auxiliary regressions (2.9) of X2 and X 1 ; namely that B2. 1 consistently estimates the induced effects of on 2. The development (3.13) just says that the overall effect of y.1(2) plus the direct effect Consequently, 1 on 'y Y
under Assumption 3B is the direct effect y.2(1) of 1p 2 on
y multiplied by the induced effect of p1 on p 2.
the large sample omitted variable bias formula (2.9) is just the total derivative of p = (p) with respect to p under the constraint that d
Thus this development can be regarded as an alternative proof of equation (2.9) found by taking macroeconomic derivatives. The question of whether Assumption 3B is correct versus Assumption 3A cannot be decided with cross section data, since each restricts only the way the distribution changes away from the cross section. However, the auxiliary
equation coefficients B2. 1 do provide consistent estimates of the induced 2.1 changes when Assumption 3B holds. due to effects on changes in Consequently, if small changes in p1 and p2 are observed (via time series) ^ 9 which are consistent with B2.1 , then Assumption 3B is not rejected. Moreover, the development shows that including too many variables in a cross section regression is not a problem in our general format. In particular,
if equation (2.3) of y regressed on X1 and X2 is estimated, the coefficients will still estimate the independent effects of pi and p2 on pY. 2
By recognizing 1 , the overall
the dependence of the erroneously included variable means
effect on pY indicated is the same as that estimated by the properly specified equation (2.5) of y on X1 ,
Too Few Regressors In this section we consider the classical omitted variables problem of
omitting pertinant regressors, in the context of a general behavioral model. In the macroeconomic interpretation of cross section regression coefficients, the pertinant regressors correspond with the driving variables of the exponential family. Consequently, here we take that Assumption 3A represents population movements, and consider the implications of performing the regression (2.5) of y on X1, omitting X2. The full impact of distribution movements under Assumption 3A is represented by (3.3), reproduced here as
y2 ( 1 )d12
Changes in p2 are no longer constrained as with Assumption 3B.
the misspecified regression (2.5) cannot adequately estimate all possible distribution effects, and the question becomes how to measure the extent of what b 1 of equation (2.5) misses.
We can find such a measure by again adjusting the parameterization of distribution movements under Assumption 3A. family using the natural parameters considered the mean parameterization We introduced the exponential 2') in (2.12) and then
' = (1,'
') in (2.16). Now we 10 reparameterize locally with the mixed parameter (pl, E2) This is accomplished by manipulating the local equations (3.la) and (3.2) as follows. for dl as Solve (3.la)
and insert into (3.2) as
ly -l (l)11 ly 1
dp I + (C2y F~1 t
1 ly2 ( 1 1)
This equation says that the misspecified regression coefficients of y on X1 consistently estimate the effects of P1 an obvious finding in light of Section 3. arise from changes in changes on Y holding 2 constant,
The remaining distributional effects
2' with their relative importance measured by the This coefficient is easily seen to be
coefficient of H2 in (4.3).
0 - o 11
= Covy (X 2 - B 2.1 X,y) 0 1
y. 2 (1 )
the partial covariance between X2 and y holding X 1 constant. local importance of 2 deviations in the mean of y (given ) is
Consequently, the directly measured
by the independent contribution of X2 to the explanation of y in the true cross section regression (2.3). y.2(1) = y.2(1) 2. By. 2.1 2(1) This covariance can alternatively be written as vkv'/K is the large sample residual y.2(1
where a2. = plim 2.1
variance matrix from the auxiliary regression (2.8) and macroeconomic effect of on y holding p1 constant.
is the true
This analysis, along with analysis of Section 3, provides an alternative justification of some common practice techniques of regression analysis in the context of unknown functional form of the true behavioral model. In particular,
for the purpose of characterizing macroeconomic effects, this work suggests performing relatively large regressions (many X's), and choosing variables via their importance in the explanation of the variance of y. Section 3 says when
the list of regressors is too large, there will exist induced constraints between the means of erroneously included variables and means of the correcty included ones. The local version of these constraints are given as the
large sample omitted variables formula (2.9), which equate the macroeconomic effects on mean y indicated by the properly specified regression to those from the regression with too many regressors. This section shows that
omitting proper variables has an impact on mean y which can be measured by the partial covariance between the omitted variables and y in a cross section regression. Consequently, in a circumstance of unknown functional form, this
analysis suggests including all variables which have large partial impacts, since including too many will be reconciled by the induced constraints. The other suggestion of this development concerns the characterization of distribution movements, say with panel data or aggregate time series. For an
exponential family structure, the candidates of most interest for driving variables are those which exhibit substantial contributions to y in a cross section regression framework.
An Example In this section we add some concreteness to the general development by
displaying the various regression and omitted variable formulae for a specific behavioral function with normally distributed regressors. Suppose that the and X 2
true model gives y as a quadratic function of two scalar variables X and an independent (mean zero) disturbance u as
2 0.1o+ X+ 2 + X + Y2 X X 1 1
2 2+ +(5.1)
If the true form above were known, one would perform a regression including linear and quadratic terms to estimate all the y parameters. Here, we consider
rl1-_ _ _~q
the regressions of y on X
and X2, and y on X 1 with (5,1) as the true model,
Suppose that in the cross section (X,X 2 )' is joint normally distributed with mean
(' ,P )
and covariance matrix
The exponential family with driving variables X 1 and X 2 (Assumption 3A) consists of the normal distributions with varying means covariance matrix Z.
and E(X2) =
The aggregate function relating E(y) = PY to E(X1) =
2 is 12
22 l (() 022) o11) (5.3)
() = Ey2 + = + y 12 ( 1 + -12)
2 +1i + + + Y22 ((2)
Given the model (5.1), the following covariances can be verified
o o y 1 +
0 1y = 1
2cY 2 1
1 o o a 11
p + Y12 (G12o c Orrv~~=0
2y (o 2
2 o + Y222 + 2 11
+ P1022) 12Oo 22
22( 12) 22~ ol2
The cross section OLS coefficients of y on X 1 and X2 from (2.3) consistently estimate -a evaluated at g =
Using the covariance formulae (5.4), this is
directly verified as
Y1 + 2Yllo + Y12o
2Ypo + Y12po
a_, (p (0 I= o
For the regression (2.5) of y on X1 only, we must characterize the exponential family with driving variable X1 in correspondence with Assumption 3B. As can be easily verified, under this family the marginal distribution of X 1 is normal with varying mean distribution p 2 (X 2IX ) 1 and fixed variance a11. The conditional and is given
is stable over time under this assumption
by a normal distribution with mean E(X2 jX1 ) = c + pX1 and
of structural changes in individual behavior on macroeconomic functions is treated in Stoker (1983b).
As with any standard tool, to attempt to cite all references to the omitted variable bias formula would result in a bibliography much longer than this paper. For an introduction to the skillful use of the formula, the work of Zvi Griliches is strongly recommended - some good examples are Griliches (1957,1971) and Griliches and Ringstad (1971). These relationships have a long history, dating back at least to Frisch (1934), and are included as standard material in textbooks on regression analysis - see Kendall and Stuart (1967) and Rao (1973) for example. The terms "predictor variable" and 'regressor" are used interchangeably + bX +. to describe X in the regression y = This feature eliminates sample selection problems from our framework. For a treatment of cross section regression and macroeconomic effects for general movements in the predictor variable distribution, see Stoker (1983a). For standard textbook treatments of the exponential family, see Ferguson (1967) and Lehmann (1959). For modern treatment, see Barndorff Neilson (1978) and Efron (1978). For derivatives of expectations taken over an exponential family, see Stoker (1982a), Lemma 6. In other words, by altering the driving variables of the exponential family, one obtains a different sequence of marginal distributions of X through time, and if F(X) is nonlinear, a different set of macroeconomic effects. 2 and P1 are the means of X 1 and X 2 observed in a time ') period adjacent to the cross section, we would expect 1( 1 under Assumption 3B. In particular, if For a theoretical treatment of mixed parameterizations of exponential families, see Barndorff Neilson (1978). This idea suggests studying the possibility that y itself is a driving variable. This is pursued in Stoker (1983b), and gives rise to an in a general interesting characterization of residual variance a y. functional form framework. These are easily found by differentiating and evaluating the moment generating function of X 1 and X2. 1
Barndorff-Neilson, 0. (1978), Information and Exponential Families in Statistical Theory, Wiley, New York. Efron, B. (1978), "The Geometry of Exponential Families," Annals of Statistics 6, pp. 362 - 376. Ferguson, T.S. (1967), Mathematical Statistics, A Decision Theoretic Approach, Academic Press, New York. Frisch, R. (1934), Statistical Confluence Analysis by Means of Complete Regression Systems, Oslo, Universitets konomiske Institutt. Griliches, Z. (1957), "Specification Bias in Estimates of Production Functions," Journal of Form Economics, 39, 1, pp. 8 - 20. Griliches, Z. (1971), "Hedonic Price Indexes for Automobiles: An Econometric Analysis of Quality Change," chapter 3 of Price Indices and Quality Change, Z. Griliches, ed., Harvard University Press, pp. 55 - 87. Griliches, Z. and V. Ringstad, (1971), Economies of Scale and the Form of the Production Function: An Econometric Study of Norwegian Manufacturing Establishment Data, North Holland, Amsterdam. Kendall, M.G. and A. Stuart (1967), The Advanced Theory of Statistics, Volume 2, Hafner Publishing Co., New York. Lehmann, E.L. (1959), Testing Statistical Hypotheses, Wiley, New York. Stoker, T.M. (1982a), "The Use of Cross Section Data to Characterize Macro Functions," Journal of the American Statistical Association, June, pp. 369 - 380. Stoker, T.M. (1982b), "Completeness, Distribution Restrictions and the Form of Aggregate Functions," MIT Sloan School of Management Working Paper WP 1345-82, August. Stoker, T.M. (1983a), "Aggregation, Efficiency and Cross Section Regression," MIT Sloan School of Management Working Paper No. WP 1453-83, June. Stoker, T.M. (1983b), "Aggregation, Structural Change and Cross Section Regression," draft, July. Theil, H. (1957), "Specification Errors and the Estimation of Economic Relationships," Review of the International Statistical Institute, 25, pp. 41 - 51. Theil, H. (1971), Principles of Econometrics, John Wiley and Sons, Amsterdam.
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