Econ 388 R. Butler 2014 revisions lecture 21
I. Endogeneous Variables
The assumption E (  | X )  0 is very important in the regression setup, and assumes,
among other things, that none of our right hand sign regressors are correlated with the
error in the regression model. If this is true, the variables are said to be exogenous in the
equation. Then regressors that are correlated with the error are said to be endogenous.
There are three common reasons to suspect endogeneity in the model, and instrumental
variable estimation can address them all. Hence, chapter 15 is very important in the
course.
Three reasons when E (  | X )  0 :
1.Omitted Variables. Section 15.1 treats this very well and provides a nice introduction
to instrumental variable estimation: What to do when one of the regressors are correlated
with the error? The textbook case given there has the true log(wage) regression depend
on education and ability, where ability is unobserved. If we ignore ability in the
specification, it becomes part of the error term, and if it is correlated with education, then
this induces an omitted variable bias and our estimate of the impact of education on
wages will be inconsistent. We can still consistently estimate the effect of education on
wages, however, when we can find an instrumental variable that is correlated with
education and uncorrelated with ability (see the discussion in 15.1 for possible candidates
for an instrumental variable in this case).
2. measurement error. Suppose we want to measure the effect of back injury severity on
the cost of back pain treatments, but our self reported measures of back pain (RolandMorris scale, the back-pain index, and the leg-pain index) are known to be measured with
error. Then if use the Roland Morris index, our estimate of its impact will be bias
towards zero (this is called attenuation bias). Let RM* be the true level of pain, and RM
be the measured level of pain (given by the Roland-Morris scale):
RM=RM* + 
 is independent of RM* (that is, the measurement error,  , is not systematic), but RM
obviously depends on  . However, RM* should be in the equation, but RM is there
instead. Hence the true equation is
Costs  0  1 RM *  X 2   , but we estimate
𝐶𝑜𝑠𝑡𝑠 = 𝛽0 + 𝛽1 𝑅𝑀 + 𝑋𝛽2 + (𝜇 − 𝛽𝜀)
Hence, the error in this last estimable equation is, by construction, correlated with RM
(through  ), even though, by assumption,  is uncorrelated with X , RM*, and  . What
can be done in this case is to use the other (imperfectly measured) indices of back pain,
the back-pain index (“On a score of 0 to 100, how do you rank your back pain?”) and the
leg-pain index, as instruments for RM. This indeed, will fix the attenuation bias.
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3. Simultaneity. When one of the right hand side variables are jointly determined along
with the outcome variable, y. This is more difficult, more tricky to get right, because we
must be able to imagine a time when we could vary this right hand side variable
exogeneously, even if in the data we have the y and r.h.s. endogenous variable were
determined together. More on this below.
II. INSTRUMENTAL VARIABLE ESTIMATION
Review from lecture 10:
If you take the standard model and pre-multiply both sides by X’ you go from:
Y  X  
to X ' Y  X ' X  X ' 
In the standard model, X '  is unknown (  is unknown), but it was zero expectation and
its plim is zero as well. If this is the case then we X ' Y  X ' X which we can solve for
 . Note that this procedure will also work—more generally--with an appropriately
dimensioned Z matrix, which we can also pre-multiply the standard model by to get
Z ' Y  Z ' X  Z ' 
Z will work in this role (given it has the right size so that Z ' X can be inverted) as long as
two conditions are met
in “small” samples: 1) Z is correlated with X, and 2) Z is uncorrelated with 
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in “large” samples: 1) plim [ Z ' X ] 1   ZX1 , and 2) plim [ Z '  ]  0 .
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n
What is the best Z? The one that is most correlated with the Xs but uncorrelated with (or
un-plim-ed with) the error term. If the model assumptions of the classical regression
model hold, then this best Z is X. Although g(.)-transforms of X would work too (as
elements of the Z matrix, see Wooldridge chapter 5), they wouldn’t be quite as efficient
as X itself (among unbiased or consistent estimators, these alternative estimators
wouldn’t have as small variance as the OLS estimators have).
But sometimes (like when we have simultaneous equation problems), it may not be the
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case that X '  has zero expectation, nor will plim [ X '  ] =0 . But there may be
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instrumental variable matrix Z consisting of n-rows that does have zero expectation, or at
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least is such that plim [ Z '  ]  0 . Then these instrumental variables could be used to
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get consistent estimates of  . From the results above, the instrumental variables
estimator would be
ˆ IV  (Z ' X ) 1 Z 'Y
And the “instrumental” variable estimator of our model would be consistent since
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1
1
plim ˆ IV  plim [ Z ' X ] 1 plim[ Z ' ( X '    )]
n
n
1
1
=   plim [ Z ' X ] 1 plim [ Z '  ]
n
n
= 
If the Zs are as well behaved as the Xs, then we can apply central limit theorem results to
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n ( ˆ IV   )  [ Z ' X ]1[
Z' ]
n
n
to get the asymptotic distribution of the estimator: the instrumental variable estimator will
be asymptotically normally distributed with mean  and an asymptotic covariance that
can be estimated as
s2 ( Z ' X ) 1 ( Z ' Z )( X ' Z ) 1
(Note that is the generalization of the result in chapter 15 of Wooldridge; see also below
under the 2SLS).
III. Simultaneity: Structural and Reduced form presentations of a model
Suppose we want to estimate, from the CPS March supplement, a labor supply education
in which the hours of work (Hi) depends on the wage rate (Wi)and non-wage income (Oi).
However, we realize that the observed hours of work for each individual is not only the
result of what they want to do (demand) but also of the opportunities that are offered to
them (supply). Is it important
Alternative Representations
A. Structural Representation (Based on Theory)
Demand: Wi = 
Supply:

1
< 0, 
1
Hi + 
0
+  2Ei +  i
Hi =  1Wi +  0 +  2Oi +  i
2
> 0,  1> 0,  2< 0
where W, H, E and O denote wage
rate. hours worked, educational
attainment, and other (non-wage)
income.  i ,  i are the errors in the
respective demand and supply equations.
We would like to estimate the supply equation, but realize that one of the assumptions is
violated. Namely, Wi and  i are correlated: as  i increases in the supply equation, then
Hi increases in the supply equation as well. But from the demand equation, we see that
higher Hi yields a lower Wi. So changes in Wi are correlated with changes in  i . This
makes OLS estimates inefficient, biased, and inconsistent.
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B.
Reduced Form Representation
The reduced form representation is logically equivalent to the structural
representation and involves expressing each dependent variable as a function of the
independent variables. In the previous example the reduced form representation for the
labor supply equation is obtained by solving for W by substituting the labor demand
expression in the labor supply expression and simplifying:
H i  1 (1 H i   0   2 Ei   i )   0   2 Oi   i , or
H i (1  11 )  (1 0   0 )  1 2 Ei   2Oi  ( i  1 i ) , or
H i   0   1 Ei   2 Oi   * i
(1  0   0 )
(   1 i )
1  2
2
, 1 
, 2 
, and  i*  i
.
(1  1 1 )
(1   1 1 )
(1  1 1 )
(1  1 1 )
A similar derivation for the wage rate yields Wi   0   1 Ei   2 Oi   * i
(    0 )
(  1  i )
 2 1
2
Where  0  0 1
, 2 
, 1 
, and  i*  i
,
(1   1 1 )
(1  1 1 )
(1  1 1 )
(1   1 1 )
where the endogeneous variables (H, W are jointly determined within this supply and
demand framework) are written as a function of all of the exogeneous variables (E, O are
variables determined outside of this supply and demand framework).
where  0 
The reduced form representation is generally used for purposes of obtaining
predictions and the  i s can be interpreted as "multipliers." This can be illustrated by our
example.
Hi =  1Wi +  0 +  2Oi
W1
W0
Wi = 
1
Hi + 
0
+  2Ei
H   2 E
H0
H1
 H   2 E
where “ W0 , H 0 ” is the equilibrium when eduation equals E, and “ W1 , H 1 ” is the
equilibrium when education equals E + E , that is, it is the new equilibrium after
education has increased by E . The reduced form representation expresses the
equilibrium values as a function of E and O. Note that the structural coefficients of the
independent variables indicate the magnitude of shifts of the graphs of the structural
relations; whereas, the reduced form coefficients are useful in characterizing values
associated with the actual observed changes in the equilibrium values (those with the
little bar over them).
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Based upon the restrictions on the signs of  s and  s, can we unambiguously
deduce the signs of the  s?
Identification. : if you can retrieve the structural estimates ( s ,  s ) from the reduced
form parameters (  s ), then model is identified. Reduced form parameters can always be
estimated; structural parameters can not be unambiguously estimated unless there are
identified. (The identification problem is whether there are more than one (  s,  s )
associated with a particular set of  -values for the reduced form equations (for both H
and W).) If there is a unique set of  s,  s associated with the reduced form
coefficients, then the equations are identified. This is a topic discussed in chapter 16 of
Wooldridge, which we will not discuss further in this course.
STATA commands to estimate the above model is as follows:
ivreg h o (w=e); *for the hours model where (rhs endo. vars=omitted exogen. vars);
iverg w e (h=o); *for the wage model;
where the rhs endogeneous (w) is instrumented with e (plus o), as indicated in the
parenthesis (instrumental variables are e,o).
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