120C
IV Econ Estimation
IV estimation can be extended to the
multiple regression case
Call the model we are interested in
estimating the structural model
Our problem is that one or more of the
variables are endogenous
We need an instrument for each
endogenous variable
Instrumental Variables & 2SLS
Multiple Regression
1
Generalizing the TSLS method
Econ 120C
Econ 120C
2
Generalizing the TSLS method
Econ 120C
The two stages of TSLS are:
Stage 1. Predict X using
1. Adding exogenous regressors to the model.
Suppose now the regression of interest is
Stage 2: Use the predicted values
where W is exogenous.
Why use W in first stage? We want the best possible
prediction/decomposition.
We need to make arguments as to why W is not
correlated with u as we have earlier
3
4
1
Generalizing the TSLS method
Econ 120C
2. More than one instrument.
Multiple Regression IV/2SLS
Econ 120C
3. In general
yi = 0 + 1x1i ++ kxki
+k+1w1i + k+rwri + ui,
where (x1 ,x2 ,, xk ) are k endogenous regressors and (w1
,w2,,wr ) is are r included exogenous regressors
Let (z1 ,z2 ,, z m ) be m instruments, so cov(z j ,u) = 0.
The model is exactly (just) identified when m=k
It is over-identified when m>k
It is under-identified if m<k
For example, suppose k = 1 but m = 0 (no instruments)!
5
Multiple Regression IV/2SLS
Z1
Z2
Zm
W1
W2
Wr
X1
X2
Xk
Econ 120C
6
Multiple Regression IV/2SLS
Econ 120C
1st Stage Regression
Regress each of the X on ALL exogenous variables
(including both Z and W) to get predicted X's.
u
2nd Stage Regression
Regress Y on predicted X's and W's
Y
In STATA,
ivreg y (x1 x2 =z1 z2 z3) w1 w2 w3 w4,r
or
ivreg y w1 w2 w3 w4 (x1 x2 =z1 z2 z3),r
7
8
2
Econ 120C
Z1
Z2
Zm
X1c
X2c
Xkc
W1
W2
Wr
Example: Demand for cigarettes
Econ 120C
ln(Qicigarettes ) = 0 + 1ln( Pi cigarettes ) + 2ln(Incomei) + ui
u
Z1i = general sales taxi
Z2i = cigarette-specific taxi
Endogenous variable: ln( Pi cigarettes ) (one X)
Included exogenous variable: ln(Incomei) (one W)
Instruments (excluded endogenous variables): general sales
tax, cigarette-specific tax (two Zs)
Is the demand elasticity 1 overidentified, exactly identified, or
underidentified?
Y
The Second Stage Regression
9
Example: Cigarette demand, one instrument
10
Econ 120C
Example: Cigarette demand, Two instruments Econ 120C
Y
W
X
Z
. ivreg lpackpc lperinc (lravgprs = rtaxso) if year==1995, r;
IV (2SLS) regression with robust standard errors
Number of obs =
F( 2,
45) =
Prob > F
=
R-squared
=
Root MSE
=
Y
W
X
Z1
Z2
. ivreg lpackpc lperinc (lravgprs = rtaxso rtax) if year==1995, r;
48
8.19
0.0009
0.4189
.18957
-----------------------------------------------------------------------------|
Robust
lpackpc |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------lravgprs | -1.143375
.3723025
-3.07
0.004
-1.893231
-.3935191
lperinc |
.214515
.3117467
0.69
0.495
-.413375
.842405
_cons |
9.430658
1.259392
7.49
0.000
6.894112
11.9672
-----------------------------------------------------------------------------Instrumented: lravgprs
Instruments:
lperinc rtaxso
STATA lists ALL the exogenous regressors
as instruments slightly different
terminology than we have been using
------------------------------------------------------------------------------
Running IV as a single command yields correct SEs
Use , r for heteroskedasticity-robust SEs
11
IV (2SLS) regression with robust standard errors
Number of obs =
F( 2,
45) =
Prob > F
=
R-squared
=
Root MSE
=
48
16.17
0.0000
0.4294
.18786
-----------------------------------------------------------------------------|
Robust
lpackpc |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------lravgprs | -1.277424
.2496099
-5.12
0.000
-1.780164
-.7746837
lperinc |
.2804045
.2538894
1.10
0.275
-.230955
.7917641
_cons |
9.894955
.9592169
10.32
0.000
7.962993
11.82692
-----------------------------------------------------------------------------Instrumented: lravgprs
Instruments:
lperinc rtaxso rtax
STATA lists ALL the exogenous regressors
as instruments slightly different
terminology from what we have been using
------------------------------------------------------------------------------
12
3
Example: Cigarette demand, one instrument
Econ 120C
TSLS estimates, Z = sales tax (m = 1)
ln(Q
cigarettes
i
) = 9.43 1.14 ln( Pi
(1.26) (0.37)
cigarettes
Z1
) + 0.21ln(Incomei)
u
Y
One instrument
ln(Qicigarettes ) = 9.89 1.28 ln( Pi cigarettes ) + 0.28ln(Incomei)
(0.25)
Z1
Z2
Xc
Xd
Econ 120C
(0.31)
TSLS estimates, Z = sales tax, cig-only tax (m = 2)
(0.96) (0.25)
Clean component X1 is larger
Smaller SEs for m = 2. Using 2 instruments gives more
information more as-if random variation.
Xc
u
Xd
Two instruments
Y
13
14
Instrument relevance
Econ 120C
Econ 120C
Yi = 0 + 1X1i + + kXki + k+1W1i + + k+rWri + ui
Instrument relevance: General case, multiple Xs.
If the Xs are replaced by their clean components
and there is no perfect multicollinearity in the above
regression, then we say the instruments are relevant
Instrument Relevance
15
Multicollinearity occurs when variables are so highly
correlated with each other that it is difficult to come up
with reliable estimates of their individual regression
coefficients. When two variables are highly correlated,
they are basically measuring the same phenomenon or
construct. In other words, when two variables are
highly correlated, they both convey essentially the
same information.
16
4
Instrument relevance
Econ 120C
Z1
Z2
Zm
X1c
X2c
Xkc
W1
W2
Wr
Instrument relevance: Multi-collinearity
Econ 120C
X1c
X1c
u
Y
X2c
W
X2c
W
No multicollinearity between X1c, ,Xkc W1,,Wr
17
Instrument exogeneity:
Econ 120C
18
Checking Instrument Relevance
Econ 120C
Instrument exogeneity:
corr(Z1i,ui) = 0,, corr(Zmi,ui) = 0
We will focus on a single included endogenous regressor:
Yi = 0 + 1Xi + 2W1i + + 1+rWri + ui
What happens if one of these requirements isnt satisfied?
How can you check? What do you do?
If you have multiple instruments, which should you use?
First stage regression:
Xi = 0 + 1Z1i ++ mZmi + m+1W1i ++ m+rWri + vi
The instruments are relevant if at least one of 1,,m are
nonzero.
The instruments are said to be weak if all the 1,,m are
either zero or nearly zero.
Weak instruments explain very little the of variation in X,
beyond that explained by the Ws
19
20
5
Checking Instrument Relevance
Z1
Z2
Zm
X
W1
W2
Wr
Econ 120C
Checking Instrument Relevance
If instruments are weak, the sampling distribution of TSLS and
its t-statistic are not (at all) normal, even with n large.
Consider the simplest case:
Yi = 0 + 1Xi + ui
Xi = 0 + 1Zi + vi
s
The IV estimator is 1TSLS = YZ
s XZ
If cov(X,Z) is zero or small, then sXZ will be small: With weak
instruments, the denominator is nearly zero.
If so, the sampling distribution of 1TSLS (and its t-statistic) is
not well approximated by its large-n normal approximation
u
Y
Holding Ws constant,
at least one of the Zs can change X
21
Checking Instrument Relevance
Econ 120C
Econ 120C
An example: the sampling distribution of the TSLS t-statistic with weak instruments
22
Model
Econ 120C
Yi Xi ui , i 1, 2, . . . , n
0, 1
Dark line =
irrelevant
instruments
Dashed light line
= strong
instruments
0,
Xi iid N , 1 n 100
0
ui 0. 1Xi ei , ei iid N , 1
Zi iid , 1
0
23
24
6
IV estimate in Repeated Experiments
Econ 120C
-600
.8
-400
1
beta_ols
beta_iv
-200
0
1.2
200
1.4
400
OLS estimate in repeated Experiments Econ 120C
0
200
400
xx
600
800
0
1000
200
400
xx
600
800
1000
25
Checking Instrument Relevance
Econ 120C
Why does our trusty normal approximation fail us?
s
= YZ
s XZ
If cov(X,Z) is small, small changes in sXZ (from one sample to
the next) can induce big changes in 1TSLS
Suppose in one sample you calculate sXZ = .00001...
Thus the large-n normal approximation is a poor approximation
to the sampling distribution of 1TSLS
If instruments are weak, the usual methods of inference are
unreliable potentially very unreliable.
TSLS
1
27
26
Checking Instrument Relevance
Econ 120C
But we have data to examine this, namely the first
stage regression.
In general this regression is
We might think to look at the t-stats, however all we
are concerned about is the overall predictability of the
Zs, hence we test this with an F statistic
28
7
Checking Instrument Relevance
Econ 120C
We wish to test the hypotheses
Example: Weak Instruments
Econ 120C
Angrist and Krueger (education pays but by how
much? )
log(wage) =a + b* (years of schooling) + error
Notice that the test is only on the coefficients of the Zs, not the
Ws.
How large an F do we need?
Rule of thumb: F>10 means that instruments are not weak.
Years of schooling
Innate Ability
?
Wage
Innate Ability is a confounding factor
29
Example: Weak Instruments
Econ 120C
30
Example: Weak Instruments (cont.)
Econ 120C
A student who is born in November will generally
start 1st grade nearly a year younger than a student
who is born in December.
Angrist and Krueger attempted to instrument for
years of schooling using quarter of birth.
Their reasoning was that compulsory schooling laws
are based on age, yet the age at which students begin
school depends on quarter of birth.
Students are typically allowed to leave as soon as they
reach their 16th birthday
November-born students tend to be in 10th grade when
they reach the end of compulsory schooling and can
choose to drop out.
For example, in California, a student starts school if
he/she turns 6 by Dec 2.
December-born students tend to be in 9th grade when
they can drop out.
31
32
8
Example: Weak Instruments (cont.)
Econ 120C
A Quote
Econ 120C
Give me a LEVER long enough and a place to stand and I
will move the world
Thus, quarter of birth is correlated with years of
schooling for students who drop out at the end of the
compulsory schooling period.
------------ ARCHIMEDES, 280 B.C.
However, this connection is tenuous.
Give me an instrument
that is strong enough
and does not interfere
with the system directly
and I can make causal
inference using
observational data
It turns out the quarter of birth is a weak instrument
and the famous study is not reliable.
Reasons: Not many people dropped out at the end of
the compulsory schooling period and for those who
dropped, the quarter of birth changes the years of
schooling by at most 1 year.
33
Econ 120C
34
Checking Instrument Exogeneity
Econ 120C
What happens if Z is correlated with u ?
Zi Yi
Zi X i u i
IV
Zi X i
Z iX i
Instrument Exogeneity
Zi X i Z iu i
Zi X i
35
1
n
1
n
Zi u i
Zi X i
covZ, u
covZ, X
36
9
Checking Instrument Exogeneity
Econ 120C
cov Z,u
cov Z,X
If cov(Z,u)0, then IV estimator is inconsistent
37
What if the model is over-identified?
Econ 120C
Z1
Z2
Xc
Z2
38
Checking Instrument Exogeneity
Econ 120C
yi = 0 + 1xi+ ui
u
Suppose we have two instruments: z1i and z2i. Then we have
two possible the FS regression:
Xd
(a)xi = 0a + 1az1i+ error
(b) xi = 0b + 1bz2i+error
Y
Z1
Econ 120C
Can we test exogeneity?
When we have exact identification (i.e. number of
instruments equals number of endogenous
regressors) the answer is no.
This is related to the reason we cannot test the
correlation of X and u.
The correlation between Z and the estimated
residuals is zero, i.e.
What happens if Z is correlated with u ?
IV
Checking Instrument Exogeneity
Xc
u
These two FS regressions will lead to two 2SLS estimates.
If both instruments are exogenous, then these two estimates
are expected to be close to each other as both are consistent.
If these two estimates are far apart from each other, then it
would be reasonable to believe one or both IVs are not
exogenous
Xd
Y
39
40
10
Testing Over-identifying Restrictions
Econ 120C
Econ 120C
If we have multiple instruments, it is
possible to test the over-identifying
restrictions to see if some of the
instruments are correlated with the error
Remember: we can test for the exogeneity
ONLY IF the model is over-identified. If
the model is exactly identified, we must rely
on ex-ante expert reasoning.
41
The J Test
The J Test
Econ 120C
Exogeneity of instruments means that they are
uncorrelated with u. This suggests that 2SLS
residual ui2SLS should be approximately
uncorrelated with the instruments.
In a regression:
ui2SLS = 0 + 1z1i ++ mzmi
+m+1w1i ++m+rwri +ei
F-statistic for H0: 1=2=...=m=0
J=m*F follows chi-square distribution with
(m-k) degree of freedom
42
Econ 120C
The test J=mF has a distribution
This is chi-squared with df=m-k
We reject for large values.
Test for OVERIDENTIFICATION. We require that
m>k otherwise J=0 always.
43
11