Gradient Metrics for Artificial _2020_Lec4.pdf

Lecture 5: Instrumental Variables
Applied Micro-Econometrics,Fall 2020
Zhaopeng Qu
Nanjing University
10/29/2020
Zhaopeng Qu (Nanjing University) Lecture 5: Instrumental Variables 10/29/2020 1 / 99

1 Review Previous Lecture of Internal Validity
2 Instrumental Variable Method
3 Checking Instrument Validity
4 Instrumental Variable for multiple regression
5 Review the last lecture
6 IV with Heterogeneous Causal Effects
7 Some Practical Guides by Angrist and Pischke(2012)
8 An good example: Long live Keju

Review Previous Lecture of Internal Validity
Section 1

Threatens to Internal Validity
Three endogenous in OLS regression are:
Omitted Variable Bias(a variable that is correlated with X but is
unobserved)
Simultaneity or reverse causality Bias (X causes Y,Y causes X)
Errors-in-Variables Bias (X is measured with error)
One easy way to deal with these endogeneity is using Instrumental
Variable method.

Instrumental Variable Method
Section 2

Introduction
The earliest application involved attempts to estimate demand and
supply curve for product.
A simple but diﬀicult question: How to find the supply or demand
curves?
Diﬀiculty: We can only observe intersections of supply and demand,
yielding pairs.
Solution: Wright(1928) use variables that appear in one equation to
shift this equation and trace out the other.
The variables that do the shifting came to be known as Instrumental
Variables method.
It is well-known that IV can address the problems of omitted variable
bias, measurement error and reverse causality problems.

Terminology: endogeneity and exogeneity
An endogenous variable is one that both we are interested in and is
correlated with u.
An exogenous variable is one that is uncorrelated with u.
Historical note: “Endogenous” literally means “determined within the
system,” that is, a variable that is jointly determined with Y, that is,
a variable subject to simultaneous causality.
However, this definition is narrow and IV regression can be used to
address OVB and errors-in-variable bias, not just to simultaneous
causality bias.

Instrumental variables: 1 endogenous regressor & 1
instrument
suppose a simple OLS regression like previous equation
𝑌𝑖 = 𝛽0 + 𝛽1𝑋𝑖 + 𝑢𝑖
Because 𝐸[𝑢𝑖|𝑋𝑖] ≠ 0, then we can use an instrumental variable(𝑍𝑖)
to obtain an consistent estimate of coeﬀicient.
Intuitively, we want to split 𝑋𝑖 into two parts:
1 part that is correlated with the error term.
2 part that is uncorrelated with the error term.
If we can isolate the variation in 𝑋𝑖 that is uncorrelated with 𝑢𝑖,then
we can use this part to obtain a consistent estimate of the causal
effect of 𝑋𝑖 on 𝑌𝑖.

Instrumental variables: 1 endogenous regressor & 1
instrument
An instrumental variable 𝑍𝑖 must satisfy the following 2 properties:
1 Instrumental relevance: 𝑍𝑖 should be correlated with the casual
variable of interest, 𝑋𝑖 (endogenous variable),thus
𝐶𝑜𝑣(𝑋𝑖, 𝑍𝑖) ≠ 0
.
2 Instumental exogeneity: 𝑍𝑖 is as good as randomly assigned and 𝑍𝑖
only affect on 𝑌𝑖 through 𝑋𝑖 affecting 𝑌𝑖 channel.
𝐶𝑜𝑣(𝑍𝑖, 𝑢𝑖) = 0

IV estimator：Jargon
Our simple OLS regression: Causal relationship of interest
First-Stage regression: regress endogenous variable on IV
𝑋𝑖 = 𝜋0 + 𝜋1𝑍𝑖 + 𝑣𝑖
Reduced-Form: regress outcome variable on IV
𝑌𝑖 = 𝛿0 + 𝛿1𝑍𝑖 + 𝑒𝑖

IV estimator：Two Steps Least Square (2SLS)
We can estimate the causal effect of 𝑋𝑖 on 𝑌𝑖 in two steps
1 First stage: Regress 𝑋𝑖 on 𝑍𝑖 & obtain predicted values of ̂
𝑋𝑖,if
𝐶𝑜𝑣(𝑍𝑖, 𝑢𝑖) = 0, then ̂
𝑋𝑖 contains variation in 𝑋𝑖 that is uncorrelated
with 𝑢𝑖
̂
𝑋𝑖 = ̂
𝜋0 + ̂
𝜋1𝑍𝑖
.
2 Second stage: Regress 𝑌𝑖 on ̂
𝑋𝑖 to obtain the Two Stage Least
Squares estimator ̂
𝛽2𝑆𝐿𝑆
̂
𝛽2𝑆𝐿𝑆 =
∑(𝑌𝑖 − ̄
𝑌 )( ̂
𝑋𝑖 − ̂
𝑋)
∑( ̂
𝑋𝑖 − ̂
𝑋)2

we substitute
̂
𝑋𝑖 − ̂
𝑋 = ̂
𝜋1(𝑍𝑖 − ̄
𝑍)
then we could obtain
̂
𝛽2𝑆𝐿𝑆 =
∑(𝑌𝑖 − ̄
𝑌 )( ̂
𝑋𝑖 − ̂
𝑋)
∑( ̂
𝑋𝑖 − ̂
𝑋)2

we substitute
̂
𝑋𝑖 − ̂
𝑋 = ̂
𝜋1(𝑍𝑖 − ̄
𝑍)
̂
𝛽2𝑆𝐿𝑆 =
∑(𝑌𝑖 − ̄
𝑌 )( ̂
𝑋𝑖 − ̂
𝑋)
∑( ̂
𝑋𝑖 − ̂
𝑋)2
=
∑(𝑌𝑖 − ̄
𝑌 ) ̂
𝜋1(𝑍𝑖 − 𝑍)
∑ ̂
𝜋2
1(𝑍𝑖 − 𝑍)2

we substitute
̂
𝑋𝑖 − ̂
𝑋 = ̂
𝜋1(𝑍𝑖 − ̄
𝑍)
̂
𝛽2𝑆𝐿𝑆 =
∑(𝑌𝑖 − ̄
𝑌 )( ̂
𝑋𝑖 − ̂
𝑋)
∑( ̂
𝑋𝑖 − ̂
𝑋)2
=
∑(𝑌𝑖 − ̄
𝑌 ) ̂
∑ ̂
𝜋2
1(𝑍𝑖 − 𝑍)2
=
1
̂
𝜋1
∑(𝑌𝑖 − ̄
𝑌 )(𝑍𝑖 − 𝑍)
∑(𝑍𝑖 − 𝑍)2

we substitute
̂
𝑋𝑖 − ̂
𝑋 = ̂
𝜋1(𝑍𝑖 − ̄
𝑍)
̂
𝛽2𝑆𝐿𝑆 =
∑(𝑌𝑖 − ̄
𝑌 )( ̂
𝑋𝑖 − ̂
𝑋)
∑( ̂
𝑋𝑖 − ̂
𝑋)2
=
∑(𝑌𝑖 − ̄
𝑌 ) ̂
∑ ̂
𝜋2
1(𝑍𝑖 − 𝑍)2
=
1
̂
𝜋1
∑(𝑌𝑖 − ̄
𝑌 )(𝑍𝑖 − 𝑍)
∑(𝑍𝑖 − 𝑍)2
=
∑(𝑍𝑖 − 𝑍)2
∑(𝑋𝑖 − 𝑋)(𝑍𝑖 − 𝑍)
×
∑(𝑌𝑖 − ̄
𝑌 )(𝑍𝑖 − 𝑍)
∑(𝑍𝑖 − 𝑍)2

Which gives the instrumental variable estimator
̂
𝛽2𝑆𝐿𝑆 =
∑(𝑌𝑖 − ̄
𝑌 )(𝑍𝑖 − ̄
𝑍)
∑(𝑋𝑖 − ̄
𝑋)(𝑍𝑖 − ̄
𝑍)
=
𝑠𝑍𝑌
𝑠𝑍𝑋
The TSLS estimator of 𝛽1 is the ratio of the sample covariance
between 𝑍 and 𝑌 to the sample covariance between 𝑍 and 𝑋.
If 𝑍𝑖 = 𝑋𝑖, then
̂
𝛽2𝑆𝐿𝑆 = ̂
𝛽𝑜𝑙𝑠

Statistical propertise of 2SLS estimator: Unbiasedness
Consider 𝐸[ ̂
𝛽𝐼𝑉 ]
𝐸[ ̂
𝛽2𝑆𝐿𝑆] = 𝐸[
∑(𝑌𝑖 − ̄
𝑌 )(𝑍𝑖 − ̄
𝑍)
∑(𝑋𝑖 − ̄
𝑋)(𝑍𝑖 − ̄
𝑍)
]

Consider 𝐸[ ̂
𝛽𝐼𝑉 ]
𝐸[ ̂
∑(𝑌𝑖 − ̄
𝑌 )(𝑍𝑖 − ̄
𝑍)
∑(𝑋𝑖 − ̄
𝑋)(𝑍𝑖 − ̄
𝑍)
]
= 𝐸[
∑[(𝛽0 + 𝛽1𝑋𝑖 + 𝑢𝑖) − (𝛽0 + 𝛽1
̄
𝑋 + ̄
𝑢)](𝑍𝑖 − ̄
𝑍)
∑(𝑋𝑖 − ̄
𝑋)(𝑍𝑖 − ̄
𝑍)
]

Consider 𝐸[ ̂
𝛽𝐼𝑉 ]
𝐸[ ̂
∑(𝑌𝑖 − ̄
𝑌 )(𝑍𝑖 − ̄
𝑍)
∑(𝑋𝑖 − ̄
𝑋)(𝑍𝑖 − ̄
𝑍)
]
= 𝐸[
∑[(𝛽0 + 𝛽1𝑋𝑖 + 𝑢𝑖) − (𝛽0 + 𝛽1
̄
𝑋 + ̄
𝑢)](𝑍𝑖 − ̄
𝑍)
∑(𝑋𝑖 − ̄
𝑋)(𝑍𝑖 − ̄
𝑍)
]
= 𝐸[
∑ 𝛽1(𝑋𝑖 − ̄
𝑋)(𝑍𝑖 − ̄
𝑍) + ∑(𝑢𝑖 − ̄
𝑢)(𝑍𝑖 − ̄
𝑍)
∑(𝑋𝑖 − ̄
𝑋)(𝑍𝑖 − ̄
𝑍)
]

Consider 𝐸[ ̂
𝛽𝐼𝑉 ]
𝐸[ ̂
∑(𝑌𝑖 − ̄
𝑌 )(𝑍𝑖 − ̄
𝑍)
∑(𝑋𝑖 − ̄
𝑋)(𝑍𝑖 − ̄
𝑍)
]
= 𝐸[
∑[(𝛽0 + 𝛽1𝑋𝑖 + 𝑢𝑖) − (𝛽0 + 𝛽1
̄
𝑋 + ̄
𝑢)](𝑍𝑖 − ̄
𝑍)
∑(𝑋𝑖 − ̄
𝑋)(𝑍𝑖 − ̄
𝑍)
]
= 𝐸[
∑ 𝛽1(𝑋𝑖 − ̄
𝑋)(𝑍𝑖 − ̄
𝑍) + ∑(𝑢𝑖 − ̄
𝑢)(𝑍𝑖 − ̄
𝑍)
∑(𝑋𝑖 − ̄
𝑋)(𝑍𝑖 − ̄
𝑍)
]
= 𝛽1 + 𝐸[
∑(𝑢𝑖 − ̄
𝑢)(𝑍𝑖 − ̄
𝑍)
∑(𝑋𝑖 − ̄
𝑋)(𝑍𝑖 − ̄
𝑍)
]

Consider 𝐸[ ̂
𝛽𝐼𝑉 ]
𝐸[ ̂
∑(𝑌𝑖 − ̄
𝑌 )(𝑍𝑖 − ̄
𝑍)
∑(𝑋𝑖 − ̄
𝑋)(𝑍𝑖 − ̄
𝑍)
]
= 𝐸[
∑[(𝛽0 + 𝛽1𝑋𝑖 + 𝑢𝑖) − (𝛽0 + 𝛽1
̄
𝑋 + ̄
𝑢)](𝑍𝑖 − ̄
𝑍)
∑(𝑋𝑖 − ̄
𝑋)(𝑍𝑖 − ̄
𝑍)
]
= 𝐸[
∑ 𝛽1(𝑋𝑖 − ̄
𝑋)(𝑍𝑖 − ̄
𝑍) + ∑(𝑢𝑖 − ̄
𝑢)(𝑍𝑖 − ̄
𝑍)
∑(𝑋𝑖 − ̄
𝑋)(𝑍𝑖 − ̄
𝑍)
]
= 𝛽1 + 𝐸[
∑(𝑢𝑖 − ̄
𝑢)(𝑍𝑖 − ̄
𝑍)
∑(𝑋𝑖 − ̄
𝑋)(𝑍𝑖 − ̄
𝑍)
]
= 𝛽1 + 𝐸[
∑ 𝑢𝑖(𝑍𝑖 − ̄
𝑍)
∑(𝑋𝑖 − ̄
𝑋)(𝑍𝑖 − ̄
𝑍)
]

Because instrument exogeneity implies 𝐶𝑜𝑣(𝑍𝑖, 𝑢𝑖) = 0,but not
𝐸[𝑢𝑖|𝑍𝑖, 𝑋𝑖] = 0,then
𝐸[
∑ 𝑢𝑖(𝑍𝑖 − ̄
𝑍)
∑(𝑋𝑖 − ̄
𝑋)(𝑍𝑖 − ̄
𝑍)
] = 𝐸[
∑ 𝐸[𝑢𝑖|𝑋𝑖, 𝑍𝑖](𝑍𝑖 − ̄
𝑍)
∑(𝑋𝑖 − ̄
𝑋)(𝑍𝑖 − ̄
𝑍)
] ≠ 0
Then we have
𝐸[ ̂
𝛽2𝑆𝐿𝑆] ≠ 𝛽1
It means that 2SLS estimator is biased.

Statistical propertise of 2SLS estimator: Consistent
We have a simple regression 𝑌𝑖 = 𝛽0 + 𝛽1𝑋𝑖 + 𝑢𝑖 and take a
covariance of 𝑌𝑖 and 𝑍𝑖
𝐶𝑜𝑣(𝑍𝑖, 𝑌𝑖) = 𝐶𝑜𝑣[𝑍𝑖, (𝛽0 + 𝛽1𝑋𝑖 + 𝑢𝑖)]
= 𝐶𝑜𝑣(𝑍𝑖, 𝛽0) + 𝛽1𝐶𝑜𝑣(𝑍𝑖, 𝑋𝑖) + 𝐶𝑜𝑣(𝑍𝑖, 𝑢𝑖)
= 𝛽1𝐶𝑜𝑣(𝑍𝑖, 𝑋𝑖)
Thus if the instrument is valid,
𝛽1 =
𝐶𝑜𝑣(𝑍𝑖, 𝑌𝑖)
𝐶𝑜𝑣(𝑍𝑖, 𝑋𝑖)
The population coeﬀicient is the ratio of the population covariance
between 𝑍 and 𝑌 to the popualtion covariance between 𝑍 and 𝑋.

Statistical propertise of 2SLS estimator: Consistent
As discussed in Section 3.7,the sample covariance is a consistent
estimator of the population covariance, thus 𝑠𝑍𝑌
𝑝
−
→ 𝐶𝑜𝑣(𝑍𝑖, 𝑌𝑖) and
𝑠𝑍𝑋
𝑝
−
→ 𝐶𝑜𝑣(𝑍𝑖, 𝑋𝑖)
Then the TSLS estimator is consistent.
̂
𝛽2𝑆𝐿𝑆 =
𝑠𝑍𝑌
𝑠𝑍𝑋
𝑝
−
→
= 𝛽1

Statistical propertise of 2SLS : sampling distribution
Similar to the expression for the OLS estimator in Equation
(4.30,page 183 in S.W)
̂
𝛽2𝑆𝐿𝑆 =
∑(𝑌𝑖 − ̄
𝑌 )(𝑍𝑖 − ̄
𝑍)
∑(𝑋𝑖 − ̄
𝑋)(𝑍𝑖 − ̄
𝑍)

(4.30,page 183 in S.W)
̂
𝛽2𝑆𝐿𝑆 =
∑(𝑌𝑖 − ̄
𝑌 )(𝑍𝑖 − ̄
𝑍)
∑(𝑋𝑖 − ̄
𝑋)(𝑍𝑖 − ̄
𝑍)
=
∑[(𝛽0 + 𝛽1𝑋𝑖 + 𝑢𝑖) − (𝛽0 + 𝛽1
̄
𝑋 + ̄
𝑢)](𝑍𝑖 − ̄
𝑍)
∑(𝑋𝑖 − ̄
𝑋)(𝑍𝑖 − ̄
𝑍)

(4.30,page 183 in S.W)
̂
𝛽2𝑆𝐿𝑆 =
∑(𝑌𝑖 − ̄
𝑌 )(𝑍𝑖 − ̄
𝑍)
∑(𝑋𝑖 − ̄
𝑋)(𝑍𝑖 − ̄
𝑍)
=
∑[(𝛽0 + 𝛽1𝑋𝑖 + 𝑢𝑖) − (𝛽0 + 𝛽1
̄
𝑋 + ̄
𝑢)](𝑍𝑖 − ̄
𝑍)
∑(𝑋𝑖 − ̄
𝑋)(𝑍𝑖 − ̄
𝑍)
=
∑ 𝛽1(𝑋𝑖 − ̄
𝑋)(𝑍𝑖 − ̄
𝑍) + ∑(𝑢𝑖 − ̄
𝑢)(𝑍𝑖 − ̄
𝑍)
∑(𝑋𝑖 − ̄
𝑋)(𝑍𝑖 − ̄
𝑍)

(4.30,page 183 in S.W)
̂
𝛽2𝑆𝐿𝑆 =
∑(𝑌𝑖 − ̄
𝑌 )(𝑍𝑖 − ̄
𝑍)
∑(𝑋𝑖 − ̄
𝑋)(𝑍𝑖 − ̄
𝑍)
=
∑[(𝛽0 + 𝛽1𝑋𝑖 + 𝑢𝑖) − (𝛽0 + 𝛽1
̄
𝑋 + ̄
𝑢)](𝑍𝑖 − ̄
𝑍)
∑(𝑋𝑖 − ̄
𝑋)(𝑍𝑖 − ̄
𝑍)
=
∑ 𝛽1(𝑋𝑖 − ̄
𝑋)(𝑍𝑖 − ̄
𝑍) + ∑(𝑢𝑖 − ̄
𝑢)(𝑍𝑖 − ̄
𝑍)
∑(𝑋𝑖 − ̄
𝑋)(𝑍𝑖 − ̄
𝑍)
= 𝛽1 +
∑(𝑢𝑖 − ̄
𝑢)(𝑍𝑖 − ̄
𝑍)
∑(𝑋𝑖 − ̄
𝑋)(𝑍𝑖 − ̄
𝑍)

(4.30,page 183 in S.W)
̂
𝛽2𝑆𝐿𝑆 =
∑(𝑌𝑖 − ̄
𝑌 )(𝑍𝑖 − ̄
𝑍)
∑(𝑋𝑖 − ̄
𝑋)(𝑍𝑖 − ̄
𝑍)
=
∑[(𝛽0 + 𝛽1𝑋𝑖 + 𝑢𝑖) − (𝛽0 + 𝛽1
̄
𝑋 + ̄
𝑢)](𝑍𝑖 − ̄
𝑍)
∑(𝑋𝑖 − ̄
𝑋)(𝑍𝑖 − ̄
𝑍)
=
∑ 𝛽1(𝑋𝑖 − ̄
𝑋)(𝑍𝑖 − ̄
𝑍) + ∑(𝑢𝑖 − ̄
𝑢)(𝑍𝑖 − ̄
𝑍)
∑(𝑋𝑖 − ̄
𝑋)(𝑍𝑖 − ̄
𝑍)
= 𝛽1 +
∑(𝑢𝑖 − ̄
𝑢)(𝑍𝑖 − ̄
𝑍)
∑(𝑋𝑖 − ̄
𝑋)(𝑍𝑖 − ̄
𝑍)
= 𝛽1 +
1
𝑛 ∑ 𝑢𝑖(𝑍𝑖 − ̄
𝑍)
1
𝑛 ∑(𝑋𝑖 − ̄
𝑋)(𝑍𝑖 − ̄
𝑍)

Statistical propertise of 2SLS: sampling distribution
Large sample: ̄
𝑍 ≅ 𝜇𝑧. Let 𝑞𝑖 = (𝑍𝑖 − 𝜇𝑍)𝑢𝑖,then the numerator
1
𝑛
∑ 𝑢𝑖(𝑍𝑖 − ̄
𝑍) ≅
1
𝑛
∑ 𝑞𝑖 = ̄
𝑞
Because 𝐶𝑜𝑣(𝑍𝑖, 𝑢𝑖) = 0 and 𝐸(𝑢𝑖)=0,so
𝐶𝑜𝑣(𝑍𝑖 − 𝜇𝑍, 𝑢𝑖) = 𝐸[(𝑍𝑖 − 𝜇𝑍)𝑢𝑖] = 𝐸(𝑞𝑖) = 0

In addition,the variance of 𝑞𝑖 is 𝜎2
𝑞 = 𝑉 𝑎𝑟[(𝑍𝑖 − 𝜇𝑍)𝑢𝑖].
We also have
𝑉 𝑎𝑟( ̄
𝑞) = 𝜎2
̄
𝑞 =
𝜎2
𝑞
𝑛
=
1
𝑛
𝑉 𝑎𝑟[(𝑍𝑖 − 𝜇𝑍)𝑢𝑖]
By the C.L.T.(central limit theorem) in large sample,
̄
𝑞
𝜎2
̄
𝑞
𝑑
−
→ 𝑁(0, 1)

Because the sample covariance is consistent for the population
covariance,thus 𝑠𝑋𝑌
𝑝
−
→ 𝐶𝑜𝑣(𝑋𝑖, 𝑌𝑖), then we obtain
̂
𝛽2𝑆𝐿𝑆 ≅ 𝛽1 +
̄
𝑞
In addition,because ̄
𝑞
𝑑
−
→ 𝑁(0, 𝜎2
̄
𝑞),then we have
̄
𝑞
𝑑
−
→ 𝑁(0,
𝜎2
̄
𝑞
[𝐶𝑜𝑣(𝑍𝑖, 𝑋𝑖)]2
)

At last, so in large samples ̂
𝛽2𝑆𝐿𝑆 is approximately distributed
̂
𝛽2𝑆𝐿𝑆
𝑑
−
→ 𝑁(𝛽, 𝜎2
̂
𝛽2𝑆𝐿𝑆
)
Where
𝜎2
̂
𝛽2𝑆𝐿𝑆
=
𝜎2
̄
𝑞
[𝐶𝑜𝑣(𝑍𝑖, 𝑋𝑖)]2
=
1
𝑛
𝑉 𝑎𝑟[(𝑍𝑖 − 𝜇𝑍)𝑢𝑖]
𝐶𝑜𝑣[(𝑍𝑖, 𝑋𝑖)]2
(12.8)

Statistical propertise of 2SLS: Statistical Inference
The variance ̂
𝛽2𝑆𝐿𝑆 can be estimated by estimating the variance and
covariance terms appearing in Equation （12.8),thus
𝑆𝐸( ̂
𝛽2𝑆𝐿𝑆) = √
1
𝑛 ∑(𝑍𝑖 − 𝜇𝑍)2 ̂
𝑢2
𝑖
𝑛( 1
𝑛 ∑(𝑍𝑖 − 𝜇𝑍)𝑋𝑖)2
Then the square root of the estimate of 𝜎2
̂
𝛽2𝑆𝐿𝑆
, thus the standard
error of the IV estimator, which is a little bit complicated.
Fortunately,this is done automatically in TSLS regression commands
in econometric software packages.
Because ̂
𝛽2𝑆𝐿𝑆 is normally distributed in large samples, hypothesis
tests about 𝛽 can be performed by computing the t-statistic,and a
95% large-sample confidence interval is given by
̂
𝛽2𝑆𝐿𝑆 ± 1.96𝑆𝐸( ̂
𝛽2𝑆𝐿𝑆)

Application: Angrist and Krueger(1991)
Angrist, Joshua D. and Alan B. Krueger. 1991. “Does Compulsory
School Attendance Affect Schooling and Earnings?” The Quarterly
Journal of Economics 106 (4):pp979–1014.
They use quarter of birth as an instrument for education to estimate
the returns to schooling.

Why is the Quarter of Birth?
In most of the U.S. must attend school until age 16 (at least during
1938-1967)
Age when starting school depends on birthday, so grade when can
legally drop out depends on birthday by compulsory schooling laws.

Is Schooling related to Quarter of Birth?(Assumption 1)

Angrist and Krueger(1991): The First Stage
Does quarter of birth affect education?
Regress education outcomes on quarter of birth dummy variables:
𝑆𝑖𝑗𝑐 = 𝛼 + 𝛽1𝑄1𝑖𝑐 + 𝛽2𝑄2𝑖𝑐 + 𝛽3𝑄3𝑖𝑐 + 𝜖𝑖𝑗𝑐
where individual 𝑖, cohort 𝑐, education outcome 𝑆, birth quarter 𝑄𝑗
It is the first stage regression

Angrist and Krueger(1991): The First Stage
It shows that 𝑄𝑗 does impact education outcomes such as total years
of education and high school graduation.

Angrist and Krueger(1991): exogeneity
Due to compulsory schooling laws?
Indirect evidence: on post-secondary outcomes that are not expected
to be affected by compulsory schooling laws.

Angrist and Krueger(1991): Reduced form
Is Earnings related to Quarter of Birth?

Angrist and Krueger(1991): OLS v.s IV
IV Estimates

Angrist and Krueger(1991): OLS v.s IV with covariates

Checking Instrument Validity
Section 3

Assumption #1 Instrument Relevance
Instrumental strategy that seems very robust.
But how to understand that Angrist and Krueger(1991) IV’s result
larger than that of OLS?
Bound et al(1995) prove that when instruments have limited
explanatory power over endogenous variable,
1.IV is biased towards OLS in finite samples. 2.May happen even on
very large sample

Recall 2SLS: a simple OLS regression equation is
Get the predict value from the first stage
̂
𝑋𝑖 = ̂
𝜋0 + ̂
𝜋1𝑍𝑖
Running the second stage regression
𝑌𝑖 = 𝛽0 + 𝛽1
̂
𝑋𝑖 + 𝑢𝑖
So following the OLS formula in large sample, we can obtain
̂
𝛽1
𝑝
−
→ 𝛽1 +
𝐶𝑜𝑣( ̂
𝑋, 𝑢)
𝑉 𝑎𝑟( ̂
𝑋)

An 2SLS version of OVB
̂
𝛽2𝑆𝐿𝑆
𝑝
−
→ 𝛽 +
𝐶𝑜𝑣( ̂
𝑋, 𝑢)
𝑉 𝑎𝑟( ̂
𝑋)
= 𝛽 +
𝐶𝑜𝑣( ̂
𝜋0 + ̂
𝜋1𝑍, 𝑢)
𝑉 𝑎𝑟( ̂
𝜋0 + ̂
𝜋1𝑍)
= 𝛽 +
̂
𝜋1𝐶𝑜𝑣(𝑍, 𝑢)
̂
𝜋2
1𝑉 𝑎𝑟( ̂
𝑍)
= 𝛽 +
𝑉 𝑎𝑟(𝑍)
𝐶𝑜𝑣(𝑍, 𝑋)
𝐶𝑜𝑣(𝑍, 𝑢)
𝑉 𝑎𝑟(𝑍)
= 𝛽 +

Weak Instruments
Assumption 1: Instrument Relevance
𝐶𝑜𝑣(𝑋𝑖, 𝑍𝑖) ≠ 0
.
Intuition: the more the variation in 𝑋 is explained by the instruments,
thus the more information is available for use in IV regression
On the contrary, instruments explain little of variation in 𝑋 are called
Weak Instruments, thus there is a very weak correlation between
𝑋(endogenous variable) and 𝑍(IV).
Because
̂
𝛽2𝑆𝐿𝑆
𝑝
−
→ 𝛽 +
So if 𝐶𝑜𝑣(𝑍, 𝑋) = 0,thus 𝑋 and 𝑍 is irrelevant,the bias will
approximate to infinity.

Weak Instruments: How to test weak instruments ?
We should therefore always check whether an instrument is relevant
enough.
Compute the first stage F-statistic provide a measure of the in
formation content contained in the instruments.
Stock and Yogo(2005) showed that
𝐸(𝛽2𝑆𝐿𝑆) − 𝛽 ≅
𝐸(𝛽𝑜𝑙𝑠) − 𝛽
𝐸(𝐹) − 1
𝐸(𝐹) is the expectation of the first stage F-statistics.And if
𝐸(𝐹) = 10,the bias of 2SLS, relative to the bias of OLS,is
approximately 1
9 , which is small enough to be acceptable.
A Rule of Thumb: if F-statistic exceeds 10,then don’t need worry
about too much.

Angrist and Krueger(1991): Why IV over OLS?
In Angrist and Krueger(1991),despite large samples sizes, the
F-statistics for a test of the joint statistical significance of the
excluded exogenous variables in the first-stage regression are not over
2.

Wrap up
If the correlation between the instruments and the endogenous
variable is small, then even the enormous sample sizes do not
guarantee that quantitatively important finite sample biases will be
eliminated from IV estimates.
The first assumption of IV method, thus relevance of IV, can be
justified by the F-statistic in the first stage.
Potential Solutions
If you have many IVs, some are strong, some are weak. Then discard
weak ones.
If you only have an weak IV, then find other more stronger IV(easy to
say, very hard to do)
Employing other estimator(LIML) other than 2SLS methods.

Assumption #2 Instrument Exogeneity
If the instruments are not exogenous, then TSLS is inconsistent.
After all, the idea of instrumental variables regression is that the
instrument contains information about variation in 𝑋𝑖 that is
unrelated to the error term 𝑢𝑖.
Can we statistically test the assumption that the instruments are
exogenous?
Answer: In most case,NO.
Assessing whether the instruments are exogenous necessarily requires
making an expert judgment based on personal knowledge and expert
opinion of the application.(“讲好故事”)
In some case,you can test partially,thus overidentification test.

Assumption #2 Instrument Exogeneity
Terminology: The relationship between the number of
instruments(𝑚) and the number of endogenous regressors(𝑘)
exactly(just) identified:𝑚 = 𝑘
overidentified 𝑚 > 𝑘
underidentified 𝑚 < 𝑘
when the coeﬀicients are just identified, you can’t do a formal
statistical test of the hypothesis that the instruments are in fact
exogenous.
If, however, there are more instruments than endogenous regressors,
then there is a statistical tool that can be helpful in this process: the
so-called test of overidentifying restrictions.

Overidentification-test:Intuition
Suppose there are two valid instruments: 𝑍1 𝑍2(you are very lucky.)
Then you could compute two separate TSLS estimates.
Intuitively,if these 2 TSLS estimates are very different from each
other, then something must be wrong: one or the other (or both) of
the instruments must be invalid.
The overidentifying restrictions test makes this comparison in a
statistically precise way.

Overidentification test:
Our model is a multiple regression
𝑌𝑖 = 𝛽0+𝛽1𝑋1,𝑖+𝛽2𝑋2,𝑖+...+𝛽𝑘𝑋𝑘,𝑖+𝛽𝑘+1𝑊1,𝑖+...+𝛽𝑘+𝑟𝑊𝑟,𝑖+𝑢𝑖
(12.13)
Where
𝑌𝑖 is the dependent variable
𝑋1, 𝑋2, ...𝑋𝑘 are 𝐾 endogenous regressors
𝑊1, 𝑋2, ...𝑊𝑟 are the additional exogenous variables
we have 𝑚 instruments,𝑍1, 𝑍2, ...𝑍𝑚,instrumental variables
𝑢𝑖 is the regression error term.

A set of m instruments,𝑍1, 𝑍2, ...𝑍𝑚
then 2sls regression
𝑌𝑖 = 𝛽0+𝛽1
̂
𝑋1,𝑖+𝛽2
̂
𝑋2,𝑖+...+𝛽𝑘
̂
𝑋𝑘,𝑖+𝛽𝑘+1𝑊1,𝑖+...+𝛽𝑘+𝑟𝑊𝑟,𝑖+𝑢𝑖
(12.13)
then we can get the predict value of ̂
𝑢𝑖
𝑇𝑆𝐿𝑆

Let
̂
𝑢𝑇𝑆𝐿𝑆
𝑖 = 𝛿0 + 𝛿1𝑍1𝑖 + ... + 𝛿𝑚𝑍𝑚𝑖 + 𝛿𝑚+1𝑊1,𝑖 + ... + 𝛿𝑚+𝑟𝑊𝑟𝑖 + 𝑒𝑖
Let 𝐹 denote the homoskedasticity-only F-statistic testing the
hypothesis that 𝛿0 = ... = 𝛿𝑚 = 0
Then the overidentifying restrictions test statistic is 𝐽 = 𝑚𝐹
Under the null hypothesis that all the instruments are exogenous,
𝐽
𝑑
−
→ 𝜒2
𝑚−𝑘
Where 𝑚 − 𝑘 is the “degree of over-identification,” that is, the
number of instruments minus the number of endogenous regressors.

Application: Demand for Cigarettes
A serious public health issue: huge externalities
One policy tool is to tax cigarettes so heavily that current smokers
cut back and potential new smokers are discouraged from taking up
the habit.
Precisely how big a tax hike is needed to make a dent in cigarette
consumption?
For example, what would the after-tax sales price of cigarettes need
to be to achieve a 20% reduction in cigarette consumption?
The answer to this question depends on the elasticity of demand for
cigarettes.

Because of the interactions between supply and demand, the elasticity
of demand for cigarettes cannot be estimated consistently by an OLS
regression of log quantity on log price.
Using annual data for the 48 contiguous U.S. states for in 1995,we
therefore use TSLS to estimate the elasticity of demand for cigarettes.
The instrumental variable, 𝑆𝑎𝑙𝑒𝑠𝑇𝑎𝑥𝑖, is the portion of the tax on
cigarettes arising from the general sales tax,measured in dollars per
pack.
Cigarette consumption, 𝑄𝑐𝑖𝑔𝑎𝑟𝑒𝑡𝑡𝑒𝑠
𝑖 , is the number of packs of
cigarettes sold per capita in the state,
and the price, 𝑃𝑐𝑖𝑔𝑎𝑟𝑒𝑡𝑡𝑒𝑠
𝑖 ,is the average real price per pack of
cigarettes including all taxes.

We consider quantity and price changes that occur over 10-year
periods.
Dependent variable:
Δ𝑙𝑛(𝑄𝑐𝑖𝑔𝑎𝑟𝑒𝑡𝑡𝑒𝑠
𝑖 ) = 𝑙𝑛(𝑄𝑐𝑖𝑔𝑎𝑟𝑒𝑡𝑡𝑒𝑠
𝑖,1995 ) − 𝑙𝑛(𝑄𝑐𝑖𝑔𝑎𝑟𝑒𝑡𝑡𝑒𝑠
𝑖,1985 )
Independent variable:
Δ𝑙𝑛(𝑃𝑐𝑖𝑔𝑎𝑟𝑒𝑡𝑡𝑒𝑠
𝑖 ) = 𝑙𝑛(𝑃𝑐𝑖𝑔𝑎𝑟𝑒𝑡𝑡𝑒𝑠
𝑖,1995 ) − 𝑙𝑛(𝑃𝑐𝑖𝑔𝑎𝑟𝑒𝑡𝑡𝑒𝑠
𝑖,1985 )
Control variable:
Δ𝑙𝑛(𝐼𝑛𝑐𝑖) = 𝑙𝑛(𝐼𝑛𝑐𝑖,1995) − 𝑙𝑛(𝐼𝑛𝑐𝑖,1985)

Two instruments
1 the change in the sales tax over 10 years,
Δ𝑆𝑎𝑙𝑒𝑠𝑇𝑎𝑥𝑖 = 𝑆𝑎𝑙𝑒𝑠𝑇𝑎𝑥𝑖,1995 − 𝑆𝑎𝑙𝑒𝑠𝑇𝑎𝑥𝑖,1985
2 the change in the cigarette-specific tax over 10 years
Δ𝐶𝑖𝑔𝑇𝑎𝑥𝑖 = 𝐶𝑖𝑔𝑇𝑎𝑥𝑖,1995 − 𝐶𝑖𝑔𝑇𝑎𝑥𝑖,1985

The first stage

Over-identifying J-test reject the null hypothesis that both the
instruments are exogenous at the 5% significant
level(𝑝 − 𝑣𝑎𝑙𝑢𝑒 = 0.026)

The reason the J-statistic rejects the null hypothesis that both
instruments are exogenous is that the two instruments produce rather
different estimated coeﬀicients.
The J-statistic rejection means that the regression in column (3) is
based on invalid instruments (the instrument exogeneity condition
fails).

The J-statistic rejection says that at least one of the instruments is
endogenous, so there are three logical possibilities
The sales tax is exogenous but the cigarette-specific tax is not, in
which case the column (1) regression is reliable;
the cigarette-specific tax is exogenous but the sales tax is not, so the
column (2) regression is reliable;
or neither tax is exogenous, so neither regression is reliable. The
statistical evidence cannot tell us which possibility is correct, so we
must use our judgement.

We think that the case for the exogeneity of the general sales tax is
stronger than that for the cigarette-specific tax.
because the political process can link changes in the cigarette-specific
tax to changes in the cigarette market and smoking policy.
if smoking decreases in a state because it falls out of fashion, there
will be fewer smokers and a weakened lobby against cigarettespecific
tax increases, which in turn could lead to higher cigarette-specific
taxes.

So the result that use the cigarette-only tax as an instrument and
adopting the price elasticity estimated using the general sales tax as
an instrument is more reliable.
The estimate of -0.94 indicates that cigarette consumption is
somewhat elastic:An increase in price of 1% leads to a decrease in
consumption of 0.94%.

Instrumental Variable for multiple regression
Section 4

IV for multiple regression(Key Concept 12.1)
Our model is a multiple regression
𝑌𝑖 = 𝛽0+𝛽1𝑋1,𝑖+𝛽2𝑋2,𝑖+...+𝛽𝑘𝑋𝑘,𝑖+𝛽𝑘+1𝑊1,𝑖+...+𝛽𝑘+𝑟𝑊𝑟,𝑖+𝑢𝑖
(12.13)
Where
𝑌𝑖 is the dependent variable
𝑋1, 𝑋2, ...𝑋𝑘 are 𝐾 endogenous regressors
𝑊1, 𝑋2, ...𝑊𝑟 are the additional exogenous variables
we have 𝑚 instruments,𝑍1, 𝑍2, ...𝑍𝑚,instrumental variables
𝑢𝑖 is the regression error term.

Two Conditions for Valid Instruments
A set of m instruments ,𝑍1, 𝑍2, ...𝑍𝑚 must satisfy the following two
conditions to be valid:
1 Instrument Relevance:
In general,let ̂
𝑋∗
1𝑖 be the predicted value of 𝑋1𝑖 from the population
regression of 𝑋1𝑖 on the instruments (𝑍) and the included exogenous
regressors (𝑊), and let “1” denote the constant regressor that takes on
the value 1 for all observations. Then ( ̂
𝑋∗
1𝑖, ..., ̂
𝑋∗
𝑘𝑖, 𝑊1, 𝑋2, ...𝑊𝑟, 1)
are not perfectly multicollinear.
If there is only one X, then for the previous condition to hold, at least
one 𝑍 must have a non-zero coeﬀicient in the population regression of
𝑋 on the 𝑍 and the 𝑊.
2 Instrument Exogeneity
The instruments are uncorrelated with the error term,
𝐶𝑜𝑣(𝑍1𝑖, 𝑢𝑖) = 0, ..., 𝐶𝑜𝑣(𝑍𝑚𝑖, 𝑢𝑖) = 0

The IV Regression Assumptions(Key Concept 12.4)
The variables and errors in the IV regression model in Key Concept
12.1 satisfy the following:
1 𝐸(𝑢𝑖|𝑊1𝑖, ..., 𝑊𝑟𝑖) = 0
2 (𝑋1𝑖, ..., 𝑋𝑘𝑖, 𝑊1𝑖, ..., 𝑊𝑟𝑖, 𝑍1𝑖, ..., 𝑍𝑚𝑖, 𝑌 𝑖) are i.i.d. draws from
their joint distribution;
3 Large outliers are unlikely: The 𝑋,𝑊,𝑍, and 𝑌 have nonzero finite
fourth moments;
4 The two conditions for a valid instrument hold.
Under the IV regression assumptions,the TSLS estimator is consistent
and normally distributed in large samples.
Because the sampling distribution of the TSLS estimator is normal in
large samples,the general procedures for statistical inference
(hypothesis tests and confidence intervals) in regression models
extend to TSLS regression.

Review the last lecture
Section 5

Instrument Variables:Constant-effect
Instrumental Variable is a useful method to make causal inference. It
can eliminate
Omitted Variable Bias
Measurement Error
Reverse Causality
Two Assumptions
Relevance(Weak Instrument): It can be test by the first stage
regression and F-statistic.
Exogeneity: Can’t be test formally but argue it using professional
knowledges.
Estimation and Inference
When IV satisify these two assumptions,the causal effect of coeﬀicients
of interest,TSLS estimator,𝛽𝑇𝑆𝐿𝑆 can be NOT unbiased but
consistent.
The sampling distribution of the TSLS estimator is also normal in large

IV with Heterogeneous Causal Effects
Section 6

Example: Angrist(1990)
Topic: How does veteran status effect on earnings
Methods: Instrumental Variable
Use the lottery outcome as an instrument for veteran status

Example: Angrist(1990) Background
In the 1960s and 70s young men in the US were at risk of being
drafted for military service in Vietnam.
Fairness concerns led to the institution of a draft lottery in 1970 that
was used to determine priority for conscription.
In each year from 1970 to 1972, random sequence numbers were
randomly assigned to each birth date in cohorts of 19-year-olds.
Men with lottery numbers below a cutoff were eligible for the draft
Men with lottery numbers above the cutoff were not.

Example: Angrist(1990) Instrumental Variables
The instrument(𝑍𝑖) is thus defined as follows:
𝑍𝑖 = 1 if lottery implied individual i would be draft eligible,
𝑍𝑖 = 0 if lottery implied individual i would NOT be draft eligible.
The econometrician observes treatment status(𝐷𝑖) as follows:
𝐷𝑖 = 1 if individual i served in the Vietnam war (veteran)
𝐷𝑖 = 0 if individual i did not serve in the Vietnam war (not veteran)

Example: Angrist(1990): IV’s Relevance and Exogenous
While the lottery didn’t completely determine veteran status, it
certainly mattered: relevance.
The lottery outcome was random and seems reasonable to suppose
that its only effect was on veteran status: exogenous.

Example: Angrist(1990): heterogeneous effects
We can classify individuals according to assignment(Z) an
treatment(X) into four parts

Local Average Treatment Effect(LATE)
So IV estimate only get the X effect on Y on the
subpopulation-compilers.
Angrist and Imbens(1994) called it as Local Average Treatment
Effect(LATE), thus the treatment effect on those that change their
behavior under the instrument.

IV with Heterogeneous Causal Effects: Generalization
If the population is heterogeneous, then the 𝑖𝑡ℎ
individual now has his
or her own causal effect, 𝛽1𝑖,then the population regression equation
can be written
𝑌𝑖 = 𝛽0𝑖 + 𝛽1𝑖𝑋𝑖 + 𝑢𝑖 (13.9)
𝛽1𝑖 is a random variable that,just like 𝑢𝑖, reflects unobserved variation
across individuals.
The average causal effect is the population mean value of the causal
effect,𝐸(𝛽1𝑖) which is the expected causal effect of a randomly
selected member of the population.

OLS with Heterogeneous Causal Effects
If there is heterogeneity in the causal effect and if 𝑋𝑖 is randomly
assigned, then the differences estimator is a consistent estimator of
the average causal effect.
̂
𝛽𝑜𝑙𝑠 =
𝑠𝑋𝑌
𝑠2
𝑋
𝑝
−
→
𝐶𝑜𝑣(𝑌𝑖, 𝑋𝑖)
𝑉 𝑎𝑟(𝑋𝑖)
=
𝐶𝑜𝑣(𝛽0𝑖 + 𝛽1𝑖𝑋𝑖 + 𝑢𝑖, 𝑋𝑖)
=
𝐶𝑜𝑣(𝛽1𝑖𝑋𝑖, 𝑋𝑖)
= 𝐸(𝛽1𝑖)
Thus, if 𝑋𝑖 is randomly assigned, ̂
𝛽1 is a consistent estimator of the
average causal effect 𝐸(𝛽1𝑖).

IV Regression with Heterogeneous Causal Effects
Specifically, suppose that 𝑋𝑖 is related to 𝑍𝑖 by the linear model
𝑋𝑖 = 𝜋0𝑖 + 𝜋1𝑖𝑍𝑖 + 𝑣𝑖
where the coeﬀicients 𝜋0𝑖 and 𝜋1𝑖 vary from one individual to the
next. And it is the first-stage equation of TSLS with the modification
of heterogeneous effect of 𝑍 on 𝑋.

Then TSLS estimator becomes
̂
𝛽2𝑆𝐿𝑆 =
𝑠𝑍𝑌
𝑠𝑍𝑋
𝑝
−
→
𝜎𝑍𝑌
𝜎𝑍𝑋
=
𝐸(𝛽1𝑖𝜋1𝑖)
𝐸(𝜋1𝑖)
Exercise: prove it by yourself (refers to Appendix 13.2)
The TSLS estimator converges in probability to the ratio of the
expected value of the product of 𝛽1𝑖 and 𝜋1𝑖 to the expected value of
𝜋1𝑖.

It is a weighted average of the individual causal effects 𝛽1𝑖, The
weights are 𝜋1𝑖
𝐸(𝜋1𝑖) , which measure the relative degree to which the
instrument influences whether the 𝑖𝑡ℎ individual receives treatment,
In other words,TSLS estimator is a consistent estimator of a weighted
average of the individual causal effects, where the individuals who
receive the most weight are those for whom the instrument is most
influential.

Three special cases:
The treatment effect is the same for all individuals.
𝛽1𝑖 = 𝛽1
The instrument affects each individual equally.
𝜋1𝑖 = 𝜋1
The heterogeneity in the treatment effect and heterogeneity in the
effect of the instrument are uncorrelated.
𝐶𝑜𝑣(𝛽1𝑖𝜋1𝑖) = 0

LATE equals to the ATE: all three cases we have
𝐸(𝛽1𝑖𝜋1𝑖)
𝐸(𝜋1𝑖)
= 𝐸(𝛽1𝑖) = 𝛽1
Aside from these three special cases, in general the local average
treatment effect differs from the average treatment effect.

IV Regression with Heterogeneous Causal Effects:
Implications
Different instruments can identify different parameters because they
estimate the impact on different populations.
The difference arises because each researcher is implicitly estimating a
different weighted average of the individual causal effects in the
population.
Recall: J-test of overidentifying restrictions can reject if the two
instruments estimate different local average treatment effects,even if
both instruments are valid. In general neither estimator is a consistent
estimator of the average causal effect.

In Summary
The IV paradigm provides a powerful and flexible framework for
causal inference.
An alternative to random assignment with a strong claim on internal
validity.
The LATE framework highlights questions of external validity
Can one instrument identify the average effect induced by another
source of variation?
Can we go from average effects on compliers to average effects on the
entire treated population or an unconditional effect?
The answer to these questions is usually: NO, at least without
additional assumptions.

Some Practical Guides by Angrist and Pischke(2012)
Section 7

Practical Guides
1 Check IV relevance
Always report the first stage and think about whether it makes
sense(Signs and magnitudes)
Always report the F-statistic on the excluded instruments. The
bigger,the better. Don’t forget the rule of thumb.(𝐹 > 10)
2 Check exclusion restriction
The exclusion restriction cannot be tested directly, but it can be
falsified
Run and examine the reduced form(regression of dependent variable on
instruments) and look at the coeﬀicients, t-statistics and F-statistics
for excluded instruments.
Because the reduced form is proportional to the casual effect of interest
and is unbiased(OLS), so we should see the causal relation in the
reduced form.If you can’t see the causal relation in the reduced
form,it’s probably not there

Practical Guides
3 Provide a substantive explanation for observed difference between
2SLS and OLS
How bid is the difference? What does this tell you?
Is the coeﬀicient bigger when theory of endogeneity suggests it should
be smaller? If so, why?
Measurement Error or heterogeneous effects?
4 If you have multiple instruments, report over-identification tests.
Pick your best single instrument and report just-identified estimates
using this one only because just-identified IV is relatively unlikely to be
subject to a weak bias.
Worry if it is substantially different from what you get using multiple
instruments.
Check over-identified 2SLS estimates with LIML. LIML is less than
precise than 2SlS but also less biased. If the results come out similar,
be happy. If not, worry, and try to find stronger instruments.

How to Evaluate IV paper in a simple way?
1 Relevant: The first stage regression
Does the author report the first stage regression?
Does the instrument perform well in the first stage?
Testable: rule of thumb: first stage 𝐹 > 10
2 Exclusion restriction:
Is the instrument exogenous enough?(the random assignment is the
best)
Would you expect a direct effect of Z on Y
Not directly testable: Except when equation is overidentified.
3 What LATE is being estimated?
Whose behavior is affected by the instrument?
Is this the LATE you would want? Is it a quantify of theoretical
interest?
Would other LATEs possible yield different estimates?

An good example: Long live Keju
Section 8

Chen, Kung and MA(2017)
Title: Long Live Keju! The Persistent Effects of China’s Imperial
Examination System.
Topic: Long term persistence of human capital:the effect of Keju
Dependent Variable: education level in 2010
Indepenet Variable: the density of jinshi in the Ming-Qing dynasties
Data: 272 perfectures in jinshi.

The effect of Keju on human capital at present
Run regression
𝑙𝑛𝑌𝑖 = 𝛽𝑙𝑛(𝐾𝑒𝑗𝑢𝑖) + 𝛾1𝑋𝑐
𝑖 + 𝛾2𝑋ℎ
𝑖 + 𝛼𝑝 + 𝑢𝑖
𝑌𝑖: 2010 年 i 地区的平均受教育年限。
𝐾𝑒𝑗𝑢𝑖: 明清时期 i 地区获得进士的人数。
𝑋𝑐
𝑖 : 控制变量（当代）
，包括经济繁荣程度 (夜间灯光)；地理因素：
该地区到海选距离、地形（免于遭受自然灾害）
。
𝑋𝑐
𝑖 : 控制变量（历史）
：
历史经济繁荣程度
基础教育设施
社会和政治影响力

Chen, Kung and MA(2018): OLS

Chen, Kung and MA(2018): Potential Bias
OVB: that are simultaneously associated with both historical jinshi
density and years of schooling today.
For instance, prefectures that had produced more jinshi may be
associated with unobserved (natural or genetic) endowments.

Chen, Kung and MA(2018): Instrumental Variable
IV: Distance to the Printing Ingredients (Pine and Bamboo) as the
Instrumental Variable of Keju
A logic chain:
𝑀𝑜𝑟𝑒; 𝑠𝑢𝑐𝑒𝑒𝑑𝑒𝑑 𝑖𝑛 𝐾𝑒𝑗𝑢 ⟺ 𝑚𝑜𝑟𝑒 𝑟𝑒𝑓𝑒𝑟𝑒𝑛𝑐𝑒𝑠 𝑏𝑜𝑜𝑘𝑠
⟺ 𝑝𝑟𝑖𝑛𝑡 𝑟𝑒𝑓𝑒𝑟𝑒𝑛𝑐𝑒𝑠 𝑏𝑜𝑜𝑘𝑠 𝑖𝑛 𝑐𝑒𝑛𝑡𝑒𝑟𝑠
⟺ 𝑝𝑟𝑖𝑛𝑡 𝑐𝑒𝑛𝑡𝑒𝑟𝑠 𝑙𝑜𝑐𝑎𝑡𝑒𝑠 𝑛𝑒𝑎𝑟𝑏𝑦 𝑠𝑜𝑚𝑒 𝑖𝑛

Chen, Kung and MA(2018): First Stage

Chen, Kung and MA(2018): Reduced-form and 2SLS

Where Do Valid Instruments Come From?
Section 9

Where do we find an IV?
Generally Speaking
“可遇不可求”
Two main approaches
1 Economic Theory/Logics
2 Exogenous Source of Variation in X(natural experiments)

Example 1: Does putting criminals in jail reduce crime?
Run a regression of crime rates(d.v.) on incarceration rates(id.v) by
using annual data at a suitable level of jurisdiction(states) and
covariates (economic conditions)
Simultaneous causality bias: crime rates goes up, more prisoners and
more prisoners,reduced crime.
IV: it must affect the incarceration rate but be unrelated to any of the
unobserved factors that determine the crime rate.
Levitt (1996) suggested that lawsuits aimed at reducing prison
overcrowding could serve as an instrumental variable.
Result: The estimated effect was three times larger than the effect
estimated using OLS.

Where do we find an IV?: Class Size and Test Score
Example 2: Does cutting class sizes increase test scores?
Omited Variable bias: such as parental interest in learning, learning
opportunities outside the classroom, quality of the teachers and
school facilities.
IV: correlated with class size (relevance) but uncorrelated with the
omitted determinants of test performance.
Hoxby (2000) suggested biology. Because of random fluctuations in
timings of births, the size of the incoming kindergarten class varies
from one year to the next.
But potential enrollment also fluctuates because parents with young
children choose to move into an improving school district and out of
one in trouble. She used the deviation of potential enrollment from
its long-term trend as her instrument.
Result: the effect on test scores of class size is small.

1 Institutional Background
Angrist(1990)-draft lottery: Vietnam veterans were randomly
designated based on birth day used to estimate the wage impact of a
shorter work experience.
Acemoglu, Johnson, and Robinson(2001): the dead rate of some
diseases in some areas to estimate the impact of institutions to
economic growth.
Feng et al.(2012) “The Returns to Education in China: Evidence from
the 1986 Compulsory Education Law”.
Li and Zhang(2007),Liu(2012)- “One Child policy”

2 Natural conditions(geography,weather,disaster)
the Rainfall,Hurricane,Earthquake,Tsunami…
the number of Rivers: Hoxby(2000)
Ying Bai and Ruixue Jia(2014)-“keju” and “the number of small
rivers”

3 Economic theory and Economic logic
study the alcohol consumption and income relationship. alcohol price
in a local market may be as a instrument of alcohol consumption.
Angrist & Evans(1998): have same sex or different sex children used
to estimate the impact of an additional birth on women labor supply.

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