University of California Berkeley Statistics Questions

IV
IV
Instrumental variables
definition
Terminology
A
regression
Vi
rr
is
said to be exogenous
endogenous
if
if
Cov Vi
Cov
nil
Vi ui
O
0
An instrument Zi needs to satisfy two conditions
Relevance Cov Zi Xi
O
Exogeneity Cov Zinni
Two stage
There
least
are
O
TSLS
squares
or
2515
different IV estimators Here
the most popular
we
focus
on
TS LS
one
TSLS
Stage 1
stage 2
Xi
regress Yi
regress
on
Zi
on
and obtain predicted values
I
Xi
The estimate of
p we get from stage 2 is the TSLS
estimator pits’s It will be shown to be consistent and
asymptotically normal
February 7
Intuition for TSLS
Key idea decompose Xi into
and
part
a
this
we
exogenous
bad part i.e endogenous part
keep only the good part
and then
To do
good part i.e
a
the first stage
use
regression
x
an
Cov To IT Zi ni
IT Cov Zi ai
O
Cov vi
nil
O
We
I
are
part
Ii Iot it Zi
as
where
the OLS estimators of Cito it
estimate
p by
will be approx
Ho Is It
exogeneity
in general
We can estimate the good
to I
by
and
I
regressing
uncorrelated
I IT
Yi
on
with
Xi If
u
since
n
is large
Consistency
of TSLS
The TSLS estimator is
t
pitses
Xi to it
I
Thus
By
and say it SE
MTSLS
first
please verify
54,2
Its
stage estimator is
III
Xi I
Zi Z
If
IEicz 212
pits
????
standard arguments
Sy
Sx
see
?
Zi
I Sy z
Sy I
The
E
a
Plug in
Thus
? Yi TICK
II Yi E
OLS asymptotics
II Hi
II CA
I Coulti Zi
Is Cov Xi Zi
consistency
Tiki z
TI Ci Z
and
of OLS step 1
By CMT
fists
Cor Yi Zi
I
where Cov Xi Zi
CovCXi 2
O
by relevance
Plug in for Yi
Cool
Cou Yi Zi
ftp.Xitui Zi
Corfu 2
O
PiCouCXi 2i
P Cou Xiii
where
Thus
Corfu 2
pitses
ISIS
is
I
0
by
exogeneity
Cou
Yi Zi
P.co
CouCxi 2
consistent
properties
Zi CoXiz
for pi
of covariance
P
ASYMPTOTIC S
OLS
Intro
a
study the properties of
goal
pi
as
as
n
in
large samples
for
Crucial
hypothesis testing
and
constructing confidence
intervals
Asymptotic theory
Asymptotic theory helps
OLS
are
as
not
understand the behavior of
in large samples
so
n
us
when the
normal
data
d
Adensity
Motor
Dwage
To
fix
ideas
we
start by
January 7
analyzing
It
? Yi
is
OLS
function of
a
averages is
understanding
Notation
sample averages Thus
crucial for understanding OLS
We consider settings with
identically distributed
id
independently and
samples
SW
Concept 2.5
key
random variable
For
rum Y
a
Ectilie
does not
we
and
will sometimes write
of
Var ti
depend on
by iid
i
Convergence
Def In
in
probability and Law of Large Numbers LIN
converges
consistent
my
c
assumption
in probability to
for my
My
if the
becomes
to
a density of In
a
my
or
equivalently
probability that
I
Yi
is
is in
arbitrarily close to 1 as
it
Nz hz
we
n
ECYi
the sample size increases the distribution
of the sample average becomes more and
In words
as
around the true
concentrated
more
expected
value
We
In
write
My
LN
If Yi it
unlikely
Convergence
my
n
as
No oil
Y
1
mean
of
o
converges
iid and
then
the
CDF
of 110,04
in probability to
if large outliers
are
Yue my
Central Limit Theorem
converges in distribution
so
with that
CDF
are
in
In
if
in distribution and
Def Actu
wewrite
Isley
to
1
of M In my
CLT
0,07 if
coincides
cumulative distribution function Fyly P Yay
normal distribution with mean O and variance
E
variance
Muc’t my
I
N
0,07 for
January to
convergence
in distribution
CLT
CLT
Suppose
that Yi
Then
as
suggests
n
iid with o dad
M Tn ny d NCO of
it
are
in
so
the following approximation
I
Derivation
if
By
if
CLT
n
is large enough
n
Actu mil IN 0,07
add my yield the result
is large
Divide by Mu and
by the properties of the normal
Why
R
varia
Suppose
EY
Val
In
I
Yi it in
Eiti
Var
I
II of
III
Var y
I
In
Em
are
iid
pp
IIE
iid assumption
by
independence
by
identical distribution assumption
n
iid sample
En
91
I
Issue
Var
But
would like
we
finite
How
so
as n
so
distribution with positive and
a
variance to make inferences
to
scale
in
so
n Var
in
that the variance is
positive
finite
Consider
Var Min
On the other hand
Va
n’t
n
n
ont
for example
n
of
January 12
of
where Oc
consider
as
as
n
so
of
co

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