LECTURE 7
Introduction to Econometrics  Endogeneity
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November 26, 2021
Dali Laxton

A LITTLE REVISION: OLS CLASSICAL ASSUMPTIONS
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1.Linearity: the regression model is linear in the parameters  (coefficients)
2.Random sampling: the data is a random sample drawn  from the population and each data point
follows the  population equation
3.No perfect collinearity: the values of explanatory variables  are not all the same and no
explanatory variable is a  perfect linear function of any other explanatory variable(s)
4.Zero conditional mean: values of explanatory variables  must contain no information about the
mean of the  unobserved factors - explanatory variables are  uncorrelated with the error term
5.Homoskedasticity: the error term has a constant variance
6.Normality of the error term: the error term is normally  distributed

ON PREVIOUS LECTURES
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►We discussed what happens if some of the assumptions are  violated
►Linearity of coefficients and no perfect multicollinearity  are essential for the definition of
OLS estimator
►Zero mean of the error term is always ensured by the  inclusion of intercept
►Normality of the error term is needed for statistical  inference, but it can be shown that if the
number of  observations is sufficiently high, the OLS estimate will  have asymptotically normal
distribution even if the  stochastic error term is not normal
►Heteroskedasticity leads to incorrect statistical inference,  but we have studied tests to detect
it and techniques to  overcome this problem

ON TODAY’S LECTURE
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►The assumption of no correlation between explanatory  variables and the error term is crucial
►Variables that are correlated with the error term are called
endogenous variables (as opposed to exogenous variables)
►We will show that the estimated coefficients of endogenous  variables are inconsistent and biased
►We will explain in which situations we may encounter  endogenous variables
►We will define the concept of instrumental variables
►We will derive the 2SLS technique to deal with  endogeneity

ENDOGENOUS VARIABLES
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►Notation:
►Intuition behind the bias:
If an explanatory variable x and the error term ε are  correlated with each other, the OLS estimate
attributes to x  some of the variation in y that actually came form the error  term ε
►Example: Analysis of household consumption patterns
 Households with lower income may indicate higher  consumption (because of shame)
►Leads to inconsistent estimates

GRAPHICAL REPRESENTATION
X
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True model
Estimated model

INCONSISTENCY OF ESTIMATES
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►We can express
►We assume that there exists a finite matrix Q so that
n
−→
1 X'X n→∞ Q
►It can be shown that
►This implies:

TYPICAL CASES OF ENDOGENEITY
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1.Omitted variable bias
 An explanatory variable is omitted from the equation and  makes part of the error term
2.Selection bias
 An unobservable characteristic has influence on both  dependent and explanatory variables
3.Simultaneity
 The causal relationship between the dependent variable  and the explanatory variable goes in both
directions
4.Measurement error
 Some of the variables are measured with error
►In all 4 cases, the sign of the bias is given by the sign of
Cov(εi, xi)

OMITTED VARIABLE BIAS
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►Studied on lecture 5
►True model: yi = βxi + γzi + ui
►Model as it looks when we omit variable z:
►This gives
►It can be remedied by including the variable in question,  but sometimes we do not have data for
it
►We can include some proxies for such variable, but this  may not reduce the bias completely and
some endogeneity  remains in the equation

SELECTION BIAS
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Smoking affects both the number of prenatal visits and the birth weight

SIMULTANEITY
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►Occurs in models where variables are jointly determined
y1i  y2i
=   α0 + α1y2i + ε1i
=   β0 + β1y1i + ε2i
►Intuitively: change in y1i will cause a change in y2i, which  will in turn cause y1i to change
again
►Technically:

SIMULTANEITY
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►Example:
QDi
QSi  QDi
=
=
=
α0 + α1Pi + α2Ii + ε1i
β0 + β1Pi + ε2i
QSi
QD
. . .
quantity demanded
QS
. . .
quantity supplied
P
. . .
price
I
. . .
income
where
►Endogeneity of price: it is determined from the interaction  of supply and demand

MEASUREMENT ERROR I
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►True regression model:
►Estimated regression:
ui = εi + νi and so
yi = β0 + β1xi + εi
y∗i = β0 + β1xi + ui where

MEASUREMENT ERROR II
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►Classical measurement error in the explanatory variable
x∗i = xi + νi where Cov(νi, xi) = 0
►True regression model:
yi = β0 + β1xi + εi

INSTRUMENTAL VARIABLES (IV)
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INSTRUMENTAL VARIABLES
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►Suppose the equation we want to estimate is:
y = Xβ + η
 We can have several instruments for several endogenous  variables - we will use the matrix
notation Z and X
 X denotes endogenous variable(s)
 Z denotes instrumental variable(s)
 Assume that we have at least as many instruments as  endogenous variables

TWO STAGE LEAST SQUARES
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►2SLS is a method of implementing instrumental variables  approach
►
►Consists of two steps:
1.Regress the endogenous variables on the instruments
X = Zδ + ν ,
get predicted values
,
2.Use these predicted values instead of X in the original  equation:

TWO STAGE LEAST SQUARES
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►The estimate is
►This estimate is consistent, but it has higher variance than  OLS (it is not efficient)
►Intuitively:
 Only part of the variation in X that is uncorrelated with the  error term is used for the
estimation.
^
This ensures consistency (X that is uncorrelated with error
term).
because not all variation in X is used.

EXAMPLE
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►Estimating the impact of education on the number of  children for a sample of women in Botswana
►
►OLS:
children
Coef.
Std. Err.
t
P>|t|
[95% Conf.
Interval]
educ
-.0905755
.0059207
-15.30
0.000
-.102183
-.0789679
age
.3324486
.0165495
20.09
0.000
.3000032
.364894
agesq
-.0026308
.0002726
-9.65
0.000
-.0031652
-.0020964
_cons
-4.138307
.2405942
-17.20
0.000
-4.609994
-3.66662
Source
SS
df
MS
Model
12243.0295
3
4081.00985
Residual
9284.14679
4357
2.13085765
Total
21527.1763
4360
4.93742577
Prob > F
=
0.0000
R-squared
=
0.5687
Adj R-squared
=
0.5684
Root MSE
=
1.4597
Number of obs = 4361  F( 3, 4357) = 1915.20

EXAMPLE
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►Education may be endogenous - both education and  number of children may be influenced by some
unobserved socioeconomic factors
 Omitted variable bias: family background is an unobserved  factor that influences both the number
of children and  years of education
►Finding possible instrument:
 Something that explains education
 But is not correlated with the family background
►A dummy variable

EXAMPLE
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►Intuition behind the instrument:
►The first condition - instrument explains education:
 School year in Botswana starts in January
⇒ Thus, women born in the first half of the year start  school when they are at least six and a
half.
 Schooling is compulsory till the age of 15
⇒ Thus, women born in the first half of the year get less  education if they leave school at the
age of 15.
►The second condition - instrument is uncorrelated with the  error term:
 Being born in the first half of the year is uncorrelated with  the unobserved socioeconomic
factors that influence  education and number of children (family background etc.)

EXAMPLE
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EXAMPLE
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Instrumented: educ
Instruments: age agesq frsthalf
children
Coef.
Std. Err.
z
P>|z|
[95% Conf.
Interval]
educ
-.1714989
.0531553
-3.23
0.001
-.2756813
-.0673165
age
.3236052
.0178514
18.13
0.000
.2886171
.3585934
agesq
-.0026723
.0002796
-9.56
0.000
-.0032202
-.0021244
_cons
-3.387805
.5478988
-6.18
0.000
-4.461667
-2.313943
= 0.0000
= 0.5502
= 1.49
Wald chi2(3)  Prob > chi2  R-squared  Root MSE
= 5300.22
Instrumental variables (2SLS) regression
Number of obs =
4361

EXAMPLE
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►Compare the estimates from OLS and 2SLS:
►OLS:
children
Coef.
Std. Err.
t
P>|t|
[95% Conf.
Interval]
educ
-.0905755
.0059207
-15.30
0.000
-.102183
-.0789679
►2SLS:
children
Coef.
Std. Err.
z
P>|z|
[95% Conf.
Interval]
educ
-.1714989
.0531553
-3.23
0.001
-.2756813
-.0673165
►Is the bias reduced by IV?
►Are these results statistically different?

SUMMARY
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►We showed that the estimated coefficients of endogenous  variables are inconsistent and biased
►In which situations we may encounter endogenous  variables
Omitted variable (omitting important variable which is  correlated to independent variable)
 Selection bias (unobserved factors influencing both  dependent and independent variable)
 Simultaneity (causality goes both ways)
 Measurement error (in either dependent or independent  variable)
►We can deal with endogeneity by using instrumental  variables (2SLS technique)