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Econometrics
Endogenous Regressors
and
Instrumental Variables
Anna Donina
Lecture 7
Endogeneity Problem
• An endogenous variable is one that is correlated with u
• An exogenous variable is one that is uncorrelated with u
• Intuition behind 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
• In IV regression, we focus on the case that X is endogenous and there is
an instrument, Z, which is exogenous.
Digression on terminology: “Endogenous” literally means “determined
within the system.” If X is jointly determined with Y, then a regression of
Y on X is subject to simultaneous causality bias. But this definition of
endogeneity is too narrow because IV regression can be used to address
OV bias and errors-in-variable bias. Thus we use the broader definition
of endogeneity above.
Endogeneity Problem
• Omitted variable bias from a variable that is correlated with X but is
unobserved and for which there are inadequate control variables;
• Selection bias: an unobservable characteristic has influence on both
dependent and independent variables;
• Measurement error bias (X is measured with error)
• Simultaneous causality bias (X causes Y, Y causes X);
All three problems cause X to be endogenous, E(u|X) ≠ 0
Selection Bias
• Very similar to omitted variable bias;
• We suppose there is some unobservable characteristic that influences
both the level of the dependent variable y and of the explanatory
variable x;
• This unobservable characteristic forms part of the error term ε, causing
cov(ε, x)≠0 (in the same manner as an omitted variable);
• Example: surveying only non-smoking mothers when inferring the
impact of the number of prenatal visits on the birth weight of children.
• Smoking affects both the number of prenatal visits and the birth
weight
Simultaneity
• Occurs in models where variables are jointly determined:
• Intuitively: change in y1i will cause a change in y2i, which in turn
will cause y1i to change again
• Technically:
y1i = α0 + α1y2i + ε1i
y2i = β0 + β1y1i + ε2i
• The endogeneity problem is endemic in social sciences/economics
• In many cases important personal variables cannot be observed
(examples?)
• These are often correlated with observed explanatory
information
• In addition, measurement error may also lead to endogeneity
• Solutions to endogeneity problems:
• Proxy variables method for omitted regressors
• Fixed effects methods if: 1) panel data is available, 2) endogeneity is
time-constant, and 3) regressors are not time-constant
• Instrumental variables method (IV)
• IV is the most well-known method to address endogeneity
problems
Endogeneity Problem
• Answer to the situation when Cov(x, ε) ≠0
• Instrumental variable (or instrument) should be a variable z
such that
1. z is uncorrelated with the error term: Cov(z, ε) = 0
2. z is correlated with the explanatory variable x: Cov(x, z) ≠ 0
• Intuition behind instrumental variables approach:
• project the endogenous variable x on the instrument z;
• this projection is uncorrelated with the error term and can
be used as an explanatory variable instead of x
Instrumental Variables (IV)
Yi = β0 + β1Xi + ui
• IV regression breaks X into two parts: a part that might be
correlated with u, and a part that is not. By isolating the part
that is not correlated with u, it is possible to estimate β1.
• This is done using an instrumental variable, Zi, which is
correlated with Xi but uncorrelated with ui.
Instrumental Variables
• Properties of IV with a poor instrumental variable
• IV may be much more inconsistent than OLS if the
instrumental variable is not completely exogenous
and only weakly related to
• Variance of IV estimator is always (!) greater than
variance of OLS estimator!
IV worse than OLS if: e.g.
There is no problem if the
instrumental variable is really
exogenous. If not, the asymptotic bias
will be the larger the weaker the
correlation with x.
Instrumental Variables
• IV estimation in the multiple regression model
• Conditions for instrumental variable
• 1) Does not appear in regression equation
• 2) Is uncorrelated with error term
• 3) Is partially correlated with endogenous explanatory
variable
endogenous exogenous variables
This is the so called
„reduced form regression“
In a regression of the
endogenous explanatory
variable on all exogenous
variables, the instrumental
variable must have a nonzero
coefficient.
Instrumental Variables
Two Stage Least Squares: 2SLS
As it sounds, TSLS has two stages – two regressions:
1. Isolate the part of X that is uncorrelated with u by regressing
X on Z using OLS:
Xi = π0 + π1Zi + vi (1)
• Because Zi is uncorrelated with ui, π0 + π1Zi is uncorrelated
with ui. We don’t know π0 or π1 but we have estimated them,
so…
• Compute the predicted values of Xi,
2. Replace Xi by in the regression of interest:
regress Y on using OLS:
Yi = β0 + β1 + ui (2)ˆ
iX
ˆ
iX
ˆ
iX
Two Stage Least Squares: 2SLS
• Because is uncorrelated with ui, the first least squares
assumption holds for regression (2). (This requires n to be
large so that π0 and π1 are precisely estimated.)
• Thus, in large samples, β1 can be estimated by OLS using
regression (2)
• The resulting estimator is called the Two Stage Least Squares
(TSLS) estimator, .
ˆ
iX
1
ˆTSLS

Two Stage Least Squares: 2SLS
Suppose Zi, satisfies the two conditions for a valid instrument:
1.Instrument relevance: corr(Zi, Xi) ≠ 0
2.Instrument exogeneity: corr(Zi, ui) = 0
Two-stage least squares:
Stage 1: Regress Xi on Zi (including an intercept), obtain the
predicted values,
Stage 2: Regress Yi on (including an intercept); the
coefficient on is the TSLS estimator, .
is a consistent estimator of β1.
ˆ
iX
ˆ
iX
ˆ
iX 1
ˆTSLS

1
ˆTSLS

• Estimating the impact of education on the number of
children for a sample of women in Botswana (OLS)
Example
• 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
• 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 - the 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 the number of children (family background etc.)
Example: Intuition behind the IV
Example: 2SLS
Example: 2SLS
• Compare the estimates:
• OLS:
• 2SLS
Example
• Why does Two Stage Least Squares work?
• All variables in the second stage regression are exogenous
because endogenous variable has been replaced by a
prediction based on only exogenous information;
• By using the prediction based on exogenous information,
endog. variable is purged of its endogenous part (the part
that is related to the error term)
• Properties of Two Stage Least Squares
• The standard errors from the OLS second stage regression
are wrong. However, it is not difficult to compute correct
standard errors.
• If there is one endogenous variable and one instrument then
2SLS = IV
• The 2SLS estimation can also be used if there is more than
one endogenous variable and at least as many instruments
Two Stage Least Squares: 2SLS
Statistical properties of 2SLS/IV-estimation
• Under assumptions completely analogous to OLS, but
conditioning on .. rather than on , 2SLS/IV is
consistent and asymptotically normal
• 2SLS/IV is typically much less precise because there is
more multicollinearity and less explanatory variation in
the second stage regression
• Corrections for heteroscedasticity analogous to OLS
• 2SLS/IV easily extends to time series and panel data
situations
Two Stage Least Squares: 2SLS