ECO 344 - Applied Econometrics II

Chapter 15

Muhammad Salman Khalid

School of Economics & Social Sciences

April 24, 2025

Salman (IBA) Chapter 15 1 / 24

Outline

1 Introduction to IV Estimation

2 IV Using Two-Staged Least Squares

3 Testing for Endogeneity and Over identifying Restriction

Salman (IBA) Chapter 15 2 / 24

 Introduction to IV Estimation

Salman (IBA) Chapter 15 3 / 24

Understanding Endogeneity

Definition:
Endogeneity arises when an explanatory variable is correlated with the
error term in a regression model.
This violates a key OLS assumption: E(µ|x) = E(µ) = 0.

Salman (IBA) Chapter 15 4 / 24

Understanding Endogeneity

Definition:
Endogeneity arises when an explanatory variable is correlated with the
error term in a regression model.
This violates a key OLS assumption: E(µ|x) = E(µ) = 0.

Common Sources of Endogeneity:

1 Omitted Variable Bias: A relevant variable is left out and is
correlated with both the included explanatory variable and the
dependent variable.

Salman (IBA) Chapter 15 4 / 24

Understanding Endogeneity

Definition:
Endogeneity arises when an explanatory variable is correlated with the
error term in a regression model.
This violates a key OLS assumption: E(µ|x) = E(µ) = 0.

Common Sources of Endogeneity:

1 Omitted Variable Bias: A relevant variable is left out and is
correlated with both the included explanatory variable and the
dependent variable.
2 Measurement Error: Errors in measuring an independent variable
lead to a a correlation with the error term.

Salman (IBA) Chapter 15 4 / 24

Understanding Endogeneity

Definition:
Endogeneity arises when an explanatory variable is correlated with the
error term in a regression model.
This violates a key OLS assumption: E(µ|x) = E(µ) = 0.

Common Sources of Endogeneity:

1 Omitted Variable Bias: A relevant variable is left out and is
correlated with both the included explanatory variable and the
dependent variable.
2 Measurement Error: Errors in measuring an independent variable
lead to a a correlation with the error term.
3 Simultaneity / Reverse Causality: The dependent variable affects
the independent variable, causing a feedback loop.

Salman (IBA) Chapter 15 4 / 24

Instrumental Variables (IV) Method
The Problem:
Suppose the model is: y = β0 + β1 x + u
If Cov(x, u) ̸= 0, OLS is inconsistent

Salman (IBA) Chapter 15 5 / 24

Instrumental Variables (IV) Method
The Problem:
Suppose the model is: y = β0 + β1 x + u
If Cov(x, u) ̸= 0, OLS is inconsistent

The IV Solution:
Find an instrument z for x that satisfies:
1 Exogeneity: Cov(z, u) = 0
2 Relevance: Cov(z, x) ̸= 0
Then z can be used to obtain a consistent estimator of β1

Salman (IBA) Chapter 15 5 / 24

Instrumental Variables (IV) Method
The Problem:
Suppose the model is: y = β0 + β1 x + u
If Cov(x, u) ̸= 0, OLS is inconsistent

The IV Solution:
Find an instrument z for x that satisfies:
1 Exogeneity: Cov(z, u) = 0
2 Relevance: Cov(z, x) ̸= 0
Then z can be used to obtain a consistent estimator of β1

Interpretation:
Exogeneity (15.4): z must not directly affect y other than through x
Relevance (15.5): z must explain variation in x

Salman (IBA) Chapter 15 5 / 24

Instrumental Variables (IV) Method
The Problem:
Suppose the model is: y = β0 + β1 x + u
If Cov(x, u) ̸= 0, OLS is inconsistent

The IV Solution:
Find an instrument z for x that satisfies:
1 Exogeneity: Cov(z, u) = 0
2 Relevance: Cov(z, x) ̸= 0
Then z can be used to obtain a consistent estimator of β1

Interpretation:
Exogeneity (15.4): z must not directly affect y other than through x
Relevance (15.5): z must explain variation in x

Caveat:
Exogeneity cannot be directly tested — it relies on theory or external
justification.
Salman (IBA) Chapter 15 5 / 24
Testing Instrument Relevance
Goal: Ensure that the instrument z is relevant, i.e., Cov(z, x) ̸= 0

Salman (IBA) Chapter 15 6 / 24

Testing Instrument Relevance
Goal: Ensure that the instrument z is relevant, i.e., Cov(z, x) ̸= 0

First Stage Regression:

x = π0 + π1 z + v

Salman (IBA) Chapter 15 6 / 24

Testing Instrument Relevance
Goal: Ensure that the instrument z is relevant, i.e., Cov(z, x) ̸= 0

First Stage Regression:

x = π0 + π1 z + v
Hypothesis Test:
H0 : π1 = 0 (instrument is irrelevant)
H1 : π1 ̸= 0
Use a t-test to evaluate π1 from the regression

Salman (IBA) Chapter 15 6 / 24

Testing Instrument Relevance
Goal: Ensure that the instrument z is relevant, i.e., Cov(z, x) ̸= 0

First Stage Regression:

x = π0 + π1 z + v
Hypothesis Test:
H0 : π1 = 0 (instrument is irrelevant)
H1 : π1 ̸= 0
Use a t-test to evaluate π1 from the regression

Interpretation:
If H0 is rejected at a small significance level (e.g., 1%), then z is
statistically relevant.
Otherwise, z may be a weak instrument, which can lead to biased
IV estimates.

Salman (IBA) Chapter 15 6 / 24

Testing Instrument Relevance
Goal: Ensure that the instrument z is relevant, i.e., Cov(z, x) ̸= 0

First Stage Regression:

x = π0 + π1 z + v
Hypothesis Test:
H0 : π1 = 0 (instrument is irrelevant)
H1 : π1 ̸= 0
Use a t-test to evaluate π1 from the regression

Interpretation:
If H0 is rejected at a small significance level (e.g., 1%), then z is
statistically relevant.
Otherwise, z may be a weak instrument, which can lead to biased
IV estimates.
Note: The rule of thumb is to check that the F statistics for the joint
significance test are greater than 10.
Salman (IBA) Chapter 15 6 / 24
IV Estimation

Sample IV Estimator:
Pn
IV (zi − z̄)(yi − ȳ ) Cov(z, y )
β̂ = Pi=1
n = (15.10)
i=1 (zi − z̄)(xi − x̄) Cov(z, x)

β̂0 = ȳ − β̂1 x̄

Salman (IBA) Chapter 15 7 / 24

IV Estimation

Sample IV Estimator:
Pn
IV (zi − z̄)(yi − ȳ ) Cov(z, y )
β̂ = Pi=1
n = (15.10)
i=1 (zi − z̄)(xi − x̄) Cov(z, x)

β̂0 = ȳ − β̂1 x̄

Notes:
When z = x, IV reduces to OLS.
IV estimator is consistent, but generally not unbiased.
Large sample sizes improve the reliability of the IV estimator.

Salman (IBA) Chapter 15 7 / 24

Inference with the IV Estimator
Large-Sample Distribution:
Under homoskedasticity and the IV assumptions:
E(u 2 |z) = σ 2 (15.11)

Salman (IBA) Chapter 15 8 / 24

Inference with the IV Estimator
Large-Sample Distribution:
Under homoskedasticity and the IV assumptions:
E(u 2 |z) = σ 2 (15.11)
Then, β̂ IV is approximately normal with:
σ2
Asymptotic Var(β̂1IV ) = (15.12)
n · σx2 · Rx,z
2

Salman (IBA) Chapter 15 8 / 24

Inference with the IV Estimator
Large-Sample Distribution:
Under homoskedasticity and the IV assumptions:
E(u 2 |z) = σ 2 (15.11)
Then, β̂ IV is approximately normal with:
σ2
Asymptotic Var(β̂1IV ) = (15.12)
n · σx2 · Rx,z
2

Estimating the Standard Error:

Estimate residuals: ûi = yi − β̂0 − β̂1 xi
Residual variance:
n
2 1 X 2
σ̂ = ûi
n−2
i=1
Standard error: s
σ̂ 2
SE(β̂1IV ) = 2
(15.13)
SSTx · Rx,z
Salman (IBA) Chapter 15 8 / 24
IV vs OLS Variance

σ2 σ2
Var(β̂1OLS ) = , Var(β̂1IV ) = 2
SSTx SSTx · Rx,z

Salman (IBA) Chapter 15 9 / 24

IV vs OLS Variance

σ2 σ2
Var(β̂1OLS ) = , Var(β̂1IV ) = 2
SSTx SSTx · Rx,z
Key Insights:
IV is less efficient than OLS (variance is always larger).
2 ) is crucial:
The strength of z (i.e., Rx,z
2
If Rx,z is small ⇒ IV estimator is high variance.
2
If Rx,z is near 1 ⇒ IV approaches OLS efficiency.

Salman (IBA) Chapter 15 9 / 24

IV vs OLS Variance

σ2 σ2
Var(β̂1OLS ) = , Var(β̂1IV ) = 2
SSTx SSTx · Rx,z
Key Insights:
IV is less efficient than OLS (variance is always larger).
2 ) is crucial:
The strength of z (i.e., Rx,z
2
If Rx,z is small ⇒ IV estimator is high variance.
2
If Rx,z is near 1 ⇒ IV approaches OLS efficiency.
Use IV only when necessary — i.e., when x is endogenous.

Salman (IBA) Chapter 15 9 / 24

IV Estimation with Weak Instruments
Asymptotic Inconsistency of IV Estimator:

Corr(z, u) σu
plim(β̂1IV ) = β1 + · (15.19)
Corr(z, x) σx

Salman (IBA) Chapter 15 10 / 24

IV Estimation with Weak Instruments
Asymptotic Inconsistency of IV Estimator:

Corr(z, u) σu
plim(β̂1IV ) = β1 + · (15.19)
Corr(z, x) σx

Contrast with OLS:

σu
plim(β̂1OLS ) = β1 + Corr(x, u) · (15.20)
σx

Salman (IBA) Chapter 15 10 / 24

IV Estimation with Weak Instruments
Asymptotic Inconsistency of IV Estimator:

Corr(z, u) σu
plim(β̂1IV ) = β1 + · (15.19)
Corr(z, x) σx

Contrast with OLS:

σu
plim(β̂1OLS ) = β1 + Corr(x, u) · (15.20)
σx

Key Implications:
IV can be more inconsistent than OLS if Corr(z, x) is small—even
when Corr(z, u) is small.

Salman (IBA) Chapter 15 10 / 24

IV Estimation with Weak Instruments
Asymptotic Inconsistency of IV Estimator:

Corr(z, u) σu
plim(β̂1IV ) = β1 + · (15.19)
Corr(z, x) σx

Contrast with OLS:

σu
plim(β̂1OLS ) = β1 + Corr(x, u) · (15.20)
σx

Key Implications:
IV can be more inconsistent than OLS if Corr(z, x) is small—even
when Corr(z, u) is small.
Weak correlation between z and x magnifies inconsistency due to
Corr(z, u) ̸= 0.
Weak Instruments → unreliable IV estimates and misleading
inference.
Salman (IBA) Chapter 15 10 / 24
R-Squared in IV Estimation

Formula under IV estimation:

SSRIV X
R2 = 1 − , where SSRIV = (yi − ŷiIV )2
SST

Salman (IBA) Chapter 15 11 / 24

R-Squared in IV Estimation

Formula under IV estimation:

SSRIV X
R2 = 1 − , where SSRIV = (yi − ŷiIV )2
SST

Key Differences from OLS:

Can be negative: if IV residuals are large, SSR may exceed SST.

Salman (IBA) Chapter 15 11 / 24

R-Squared in IV Estimation

Formula under IV estimation:

SSRIV X
R2 = 1 − , where SSRIV = (yi − ŷiIV )2
SST

Key Differences from OLS:

Can be negative: if IV residuals are large, SSR may exceed SST.
No meaningful interpretation:
Cannot be used to:
Assess model fit.
Perform F -tests for joint significance.
We will use Wald statistics for the joint testing. (Not dependent on
R2

Salman (IBA) Chapter 15 11 / 24

 IV Using Two-Staged Least Squares

Salman (IBA) Chapter 15 12 / 24

The Structural Equation

Model:
y = β0 + β1 x1∗ + β2 x2 + u1

x1∗ : endogenous regressor (correlated with u1 )

x2 : included exogenous regressor
Problem: OLS is inconsistent if Cov(x1∗ , u1 ) ̸= 0

Salman (IBA) Chapter 15 13 / 24

The Structural Equation

Model:
y = β0 + β1 x1∗ + β2 x2 + u1

x1∗ : endogenous regressor (correlated with u1 )

x2 : included exogenous regressor
Problem: OLS is inconsistent if Cov(x1∗ , u1 ) ̸= 0
We can come up with z1 such that:
1 Relevance: Cov(z1 , x1∗ ) ̸= 0
2 Exogeneity: Cov(z1 , u1 ) = 0
z1 will be the valid IV if the above conditions are satisfied.

Salman (IBA) Chapter 15 13 / 24

The Structural Equation

Model:
y = β0 + β1 x1∗ + β2 x2 + u1

x1∗ : endogenous regressor (correlated with u1 )

x2 : included exogenous regressor
Problem: OLS is inconsistent if Cov(x1∗ , u1 ) ̸= 0
We can come up with z1 such that:
1 Relevance: Cov(z1 , x1∗ ) ̸= 0
2 Exogeneity: Cov(z1 , u1 ) = 0
z1 will be the valid IV if the above conditions are satisfied.
These assumptions cannot be tested jointly, but relevance can be tested
during first-stage regression.

Salman (IBA) Chapter 15 13 / 24

Two Staged Least Squares

Regress endogenous variable on instruments and controls:

x1∗ = π0 + π1 z1 + π2 x2 + v2

Salman (IBA) Chapter 15 14 / 24

Two Staged Least Squares

Regress endogenous variable on instruments and controls:

x1∗ = π0 + π1 z1 + π2 x2 + v2

Save fitted values: xˆ1∗ = π̂0 + π̂1 z1 + πˆ2 x2

Interpretation: xˆ1∗ is the exogenous part of x1∗ - that is, the part
uncorrelated with u1
Note: The exogenous variables must also be included in the regression of
the first stage.

Salman (IBA) Chapter 15 14 / 24

Two Staged Least Squares

Regress endogenous variable on instruments and controls:

x1∗ = π0 + π1 z1 + π2 x2 + v2

Save fitted values: xˆ1∗ = π̂0 + π̂1 z1 + πˆ2 x2

Interpretation: xˆ1∗ is the exogenous part of x1∗ - that is, the part
uncorrelated with u1
Note: The exogenous variables must also be included in the regression of
the first stage.
Stage 2: Plug into Structural Equation

y = β0 + β1 xˆ1∗ + β2 x2 + µ

Salman (IBA) Chapter 15 14 / 24

Two Staged Least Squares

Regress endogenous variable on instruments and controls:

x1∗ = π0 + π1 z1 + π2 x2 + v2

Save fitted values: xˆ1∗ = π̂0 + π̂1 z1 + πˆ2 x2

Interpretation: xˆ1∗ is the exogenous part of x1∗ - that is, the part
uncorrelated with u1
Note: The exogenous variables must also be included in the regression of
the first stage.
Stage 2: Plug into Structural Equation

y = β0 + β1 xˆ1∗ + β2 x2 + µ

Estimates are consistent if z1 is valid instrument.

Note: Standarad Errors will also be computed differently as described in
the last section.

Salman (IBA) Chapter 15 14 / 24

Example: Return to Education

Structural Model:

log(wage) = β0 + β1 educ + β2 exper + β3 exper 2 + u

Salman (IBA) Chapter 15 15 / 24

Example: Return to Education

Structural Model:

log(wage) = β0 + β1 educ + β2 exper + β3 exper 2 + u

Instrument: Mother’s education (motheduc)

Salman (IBA) Chapter 15 15 / 24

Example: Return to Education

Structural Model:

log(wage) = β0 + β1 educ + β2 exper + β3 exper 2 + u

Instrument: Mother’s education (motheduc)

Stage 1:
ˆ = πˆ0 + πˆ1 motheduc + πˆ2 eper + πˆ3 exper 2
educ

Salman (IBA) Chapter 15 15 / 24

Example: Return to Education

Structural Model:

log(wage) = β0 + β1 educ + β2 exper + β3 exper 2 + u

Instrument: Mother’s education (motheduc)

Stage 1:
ˆ = πˆ0 + πˆ1 motheduc + πˆ2 eper + πˆ3 exper 2
educ

Stage 2:

[ + β2 exper + β3 exper 2 + µ
log(wage) = β0 + β1 educ

Salman (IBA) Chapter 15 15 / 24

Example: Return to Education

Structural Model:

log(wage) = β0 + β1 educ + β2 exper + β3 exper 2 + u

Instrument: Mother’s education (motheduc)

Stage 1:
ˆ = πˆ0 + πˆ1 motheduc + πˆ2 eper + πˆ3 exper 2
educ

Stage 2:

[ + β2 exper + β3 exper 2 + µ
log(wage) = β0 + β1 educ

β1 will be the consistent estimator.

Salman (IBA) Chapter 15 15 / 24

Multicollinearity in 2SLS Estimation

Asymptotic Variance of 2SLS Estimator:

In multiple regression model:

σ2
Var(β̂12SLS ) ≈
SSTx̂1∗ (1 − R 2ˆ∗ )
x1 ,x2 ,..

Salman (IBA) Chapter 15 16 / 24

Multicollinearity in 2SLS Estimation

Asymptotic Variance of 2SLS Estimator:

In multiple regression model:

σ2
Var(β̂12SLS ) ≈
SSTx̂1∗ (1 − R 2ˆ∗ )
x1 ,x2 ,..

Implications:
x̂1∗ (fitted values from 1st stage) has less variation than x1∗
x̂1∗ is often highly correlated with other exogenous variables
⇒ Larger R22 leads to inflated standard errors

Salman (IBA) Chapter 15 16 / 24

Multicollinearity in 2SLS Estimation

Asymptotic Variance of 2SLS Estimator:

In multiple regression model:

σ2
Var(β̂12SLS ) ≈
SSTx̂1∗ (1 − R 2ˆ∗ )
x1 ,x2 ,..

Implications:
x̂1∗ (fitted values from 1st stage) has less variation than x1∗
x̂1∗ is often highly correlated with other exogenous variables
⇒ Larger R22 leads to inflated standard errors

Multicollinearity is often more severe in 2SLS than OLS, especially with

weak or many instruments.

Salman (IBA) Chapter 15 16 / 24

Using Multiple Instruments

We can use multiple instruments for one endogenous variable.

Salman (IBA) Chapter 15 17 / 24

Using Multiple Instruments

We can use multiple instruments for one endogenous variable.

Efficiency: More valid instrument may increase the F-statistics of
first stage leading to more consistent estimators.

Salman (IBA) Chapter 15 17 / 24

Using Multiple Instruments

We can use multiple instruments for one endogenous variable.

Efficiency: More valid instrument may increase the F-statistics of
first stage leading to more consistent estimators.
Only the first stage of the regression will change. In case of two IVs
(z1 & z2 ) and one exogenous variable (x2 ) , the first stage will be:

xˆ1∗ = πˆo + πˆ1 z1 + πˆ2 z2 + πˆ3 x2

Salman (IBA) Chapter 15 17 / 24

Using Multiple Instruments

We can use multiple instruments for one endogenous variable.

Efficiency: More valid instrument may increase the F-statistics of
first stage leading to more consistent estimators.
Only the first stage of the regression will change. In case of two IVs
(z1 & z2 ) and one exogenous variable (x2 ) , the first stage will be:

xˆ1∗ = πˆo + πˆ1 z1 + πˆ2 z2 + πˆ3 x2

The second stage will not change.

y = β0 + β1 xˆ1∗ + β2 x2

Salman (IBA) Chapter 15 17 / 24

2SLS with Multiple Endogenous Variables
Model:
y = β0 + β1 x1∗ + β2 x2∗ + β3 x3 + β4 x4 + β5 x5 + u1

x1∗ , x2∗ are endogenous: possibly correlated with u1

x3 , x4 , x5 are included exogenous variables
Require excluded exogenous instruments (e.g., z1 , z2 ) that are
correlated with x1∗ , x2∗ respectively.

Salman (IBA) Chapter 15 18 / 24

2SLS with Multiple Endogenous Variables
Model:
y = β0 + β1 x1∗ + β2 x2∗ + β3 x3 + β4 x4 + β5 x5 + u1

x1∗ , x2∗ are endogenous: possibly correlated with u1

x3 , x4 , x5 are included exogenous variables
Require excluded exogenous instruments (e.g., z1 , z2 ) that are
correlated with x1∗ , x2∗ respectively.
We need to run two regressions for the first stage for xˆ1∗ and xˆ2∗ .

Salman (IBA) Chapter 15 18 / 24

2SLS with Multiple Endogenous Variables
Model:
y = β0 + β1 x1∗ + β2 x2∗ + β3 x3 + β4 x4 + β5 x5 + u1

x1∗ , x2∗ are endogenous: possibly correlated with u1

x3 , x4 , x5 are included exogenous variables
Require excluded exogenous instruments (e.g., z1 , z2 ) that are
correlated with x1∗ , x2∗ respectively.
We need to run two regressions for the first stage for xˆ1∗ and xˆ2∗ .
Then the second stage would be:

y = β0 + β1 xˆ1∗ + β2 xˆ2∗ + β3 x3 + β4 x4 + β5 x5 + u1

Salman (IBA) Chapter 15 18 / 24

2SLS with Multiple Endogenous Variables
Model:
y = β0 + β1 x1∗ + β2 x2∗ + β3 x3 + β4 x4 + β5 x5 + u1

x1∗ , x2∗ are endogenous: possibly correlated with u1

x3 , x4 , x5 are included exogenous variables
Require excluded exogenous instruments (e.g., z1 , z2 ) that are
correlated with x1∗ , x2∗ respectively.
We need to run two regressions for the first stage for xˆ1∗ and xˆ2∗ .
Then the second stage would be:

y = β0 + β1 xˆ1∗ + β2 xˆ2∗ + β3 x3 + β4 x4 + β5 x5 + u1

However, we need to satisfy an additional condition (Order Condition)

for consistency.
Number of excluded independent exogenous variables ≥ number of
endogenous regressors
Salman (IBA) Chapter 15 18 / 24
Detecting Weak Instrument

For Single Instrumental Variable:

The F-statistics of the first stage (without exogenous variables)
greater than
√ 10 or the t-statistics of the instrumental variable greater
than 3.2 ( 10).

xˆ1∗ = πˆ0 + πˆ1 z1

Salman (IBA) Chapter 15 19 / 24

Detecting Weak Instrument

For Single Instrumental Variable:

The F-statistics of the first stage (without exogenous variables)
greater than
√ 10 or the t-statistics of the instrumental variable greater
than 3.2 ( 10).

xˆ1∗ = πˆ0 + πˆ1 z1

For Multiple Instrumental Variable:
The F-statistics of the first stage (without exogenous variables)
greater than 10.

xˆ1∗ = πˆ0 + πˆ1 z1 + πˆ2 z2 + ....

Salman (IBA) Chapter 15 19 / 24

Detecting Weak Instrument

For Single Instrumental Variable:

The F-statistics of the first stage (without exogenous variables)
greater than
√ 10 or the t-statistics of the instrumental variable greater
than 3.2 ( 10).

xˆ1∗ = πˆ0 + πˆ1 z1

For Multiple Instrumental Variable:
The F-statistics of the first stage (without exogenous variables)
greater than 10.

xˆ1∗ = πˆ0 + πˆ1 z1 + πˆ2 z2 + ....

Incase of Heteroskedasticity at any
√ stage, we might need even stringer
criteria i.e F > 20 and t > 4.47( 20)

Salman (IBA) Chapter 15 19 / 24

Testing for Endogeneity and Over identifying
Restriction

Salman (IBA) Chapter 15 20 / 24

Testing for Endogeneity
Purpose: Determine whether 2SLS is necessary by testing if an
explanatory variable (e.g., y2 ) is endogenous.

Salman (IBA) Chapter 15 21 / 24

Testing for Endogeneity
Purpose: Determine whether 2SLS is necessary by testing if an
explanatory variable (e.g., y2 ) is endogenous.
Steps:
1 Estimate the reduced form for y using all exogenous variables:
2

x1∗ = π0 + π1 z1 + π2 x2 + v2

Salman (IBA) Chapter 15 21 / 24

Testing for Endogeneity
Purpose: Determine whether 2SLS is necessary by testing if an
explanatory variable (e.g., y2 ) is endogenous.
Steps:
1 Estimate the reduced form for y using all exogenous variables:
2

x1∗ = π0 + π1 z1 + π2 x2 + v2

As xˆ1∗ is the exogenous part of x1∗ and v2 is the endogenous part of

x1∗ . We can test if the endogenous part is relevant in the structural
equation.
Therefore, obtain residuals v̂2 .

Salman (IBA) Chapter 15 21 / 24

Testing for Endogeneity
Purpose: Determine whether 2SLS is necessary by testing if an
explanatory variable (e.g., y2 ) is endogenous.
Steps:
1 Estimate the reduced form for y using all exogenous variables:
2

x1∗ = π0 + π1 z1 + π2 x2 + v2

As xˆ1∗ is the exogenous part of x1∗ and v2 is the endogenous part of

x1∗ . We can test if the endogenous part is relevant in the structural
equation.
Therefore, obtain residuals v̂2 .
2 Estimate augmented structural equation:

y = β0 + β1 x1∗ + β2 x2 + δ1 v̂2 + v1
3 Test H0 : δ1 = 0 using a robust t-test.
If v̂2 is significant ⇒ y2 is endogenous ⇒ use 2SLS.
Salman (IBA) Chapter 15 21 / 24
Over identifying Restriction Test
Over-identification
Although, exogeniety generally cannot be tested—unless we have
more instruments than needed.
In that case, we test whether the extra instruments are exogenous ⇒
over identification test.

Salman (IBA) Chapter 15 22 / 24

Over identifying Restriction Test
Over-identification
Although, exogeniety generally cannot be tested—unless we have
more instruments than needed.
In that case, we test whether the extra instruments are exogenous ⇒
over identification test.
Intuition:
With more instruments than endogenous variables, multiple 2SLS
estimates are possible.

Salman (IBA) Chapter 15 22 / 24

Over identifying Restriction Test
Over-identification
Although, exogeniety generally cannot be tested—unless we have
more instruments than needed.
In that case, we test whether the extra instruments are exogenous ⇒
over identification test.
Intuition:
With more instruments than endogenous variables, multiple 2SLS
estimates are possible.
If all instruments are valid, these estimates should only differ by
sampling error.

Salman (IBA) Chapter 15 22 / 24

Over identifying Restriction Test
Over-identification
Although, exogeniety generally cannot be tested—unless we have
more instruments than needed.
In that case, we test whether the extra instruments are exogenous ⇒
over identification test.
Intuition:
With more instruments than endogenous variables, multiple 2SLS
estimates are possible.
If all instruments are valid, these estimates should only differ by
sampling error.
Significant differences suggest that at least one instrument may be
endogenous.

Salman (IBA) Chapter 15 22 / 24

Over identifying Restriction Test
Over-identification
Although, exogeniety generally cannot be tested—unless we have
more instruments than needed.
In that case, we test whether the extra instruments are exogenous ⇒
over identification test.
Intuition:
With more instruments than endogenous variables, multiple 2SLS
estimates are possible.
If all instruments are valid, these estimates should only differ by
sampling error.
Significant differences suggest that at least one instrument may be
endogenous.
However:

Salman (IBA) Chapter 15 22 / 24

Over identifying Restriction Test
Over-identification
Although, exogeniety generally cannot be tested—unless we have
more instruments than needed.
In that case, we test whether the extra instruments are exogenous ⇒
over identification test.
Intuition:
With more instruments than endogenous variables, multiple 2SLS
estimates are possible.
If all instruments are valid, these estimates should only differ by
sampling error.
Significant differences suggest that at least one instrument may be
endogenous.
However:
Rejection implies some instrument is invalid—but not which one.

Salman (IBA) Chapter 15 22 / 24

Over identifying Restriction Test
Over-identification
Although, exogeniety generally cannot be tested—unless we have
more instruments than needed.
In that case, we test whether the extra instruments are exogenous ⇒
over identification test.
Intuition:
With more instruments than endogenous variables, multiple 2SLS
estimates are possible.
If all instruments are valid, these estimates should only differ by
sampling error.
Significant differences suggest that at least one instrument may be
endogenous.
However:
Rejection implies some instrument is invalid—but not which one.
Similar estimates do not guarantee that all instruments are valid—they
may be invalid in the same way.
Salman (IBA) Chapter 15 22 / 24
Regression-Based Test Procedure

Steps:
1 Estimate the structural equation by 2SLS and obtain residuals û1 .

Salman (IBA) Chapter 15 23 / 24

Regression-Based Test Procedure

Steps:
1 Estimate the structural equation by 2SLS and obtain residuals û1 .
2 Regress û1 on all exogenous variables (included + excluded
instruments).

Salman (IBA) Chapter 15 23 / 24

Regression-Based Test Procedure

Steps:
1 Estimate the structural equation by 2SLS and obtain residuals û1 .
2 Regress û1 on all exogenous variables (included + excluded
instruments).
3 Let R12 be the R-squared from this regression.

Salman (IBA) Chapter 15 23 / 24

Regression-Based Test Procedure

Steps:
1 Estimate the structural equation by 2SLS and obtain residuals û1 .
2 Regress û1 on all exogenous variables (included + excluded
instruments).
3 Let R12 be the R-squared from this regression.
4 Compute the test statistic:

nR12 ∼ χ2q , q = (number of instruments)−(number of endogenous varia

Salman (IBA) Chapter 15 23 / 24

Regression-Based Test Procedure

Steps:
1 Estimate the structural equation by 2SLS and obtain residuals û1 .
2 Regress û1 on all exogenous variables (included + excluded
instruments).
3 Let R12 be the R-squared from this regression.
4 Compute the test statistic:

nR12 ∼ χ2q , q = (number of instruments)−(number of endogenous varia

5 Reject H0 if nR12 exceeds critical value ⇒ some IVs are not exogenous.

Salman (IBA) Chapter 15 23 / 24

 THANK YOU
The difference between ordinary and extraordinary is that little ”extra”.

Salman (IBA) Chapter 15 24 / 24