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ECMT2150 Instrumental Variable Estimation

创建日期：2024-05-28 18:11:17

所属分类：数学mathematics

标签： ECMT2150

Instrumental Variable Estimation

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ECMT2150 – Lecture 10

Topics Today
Week 10
Review Endogeneity
Instrumental Variable Estimation
References/Required reading: Chp 15 & A few pages from
Cameron & Trivedi (Sections 4.8.1 and 4.8.2)
Review Endogeneity
Endogeneity is a major challenge in social sciences incl.
economics:
– With observational data:
• Omitted variables: In many cases important characteristics cannot
be observed AND these are often correlated with observed
explanatory information.
• Simultaneity: two or more variables are simultaneously determined
(in market equilibrium)
• Measurement error: variables are measured with error
• Endogeneity = violation of Assumption MLR.4/ZCM
– There is a correlation between the error u and the regressor(s)
– Changes in x are associated not only with with changes in y but also
with changes in u
Endogeneity
3
What can we do about endogeneity?
Given omitted variable bias (or unobserved heterogeneity),
x and u will be correlated so OLS is biased & inconsistent.
Options:
• Ignore it: Live with the biased estimates (perhaps sign the
bias)
• Proxy variables:
– Solve the omitted variable problem by replacing the unobserved
variable with a proxy
– Can’t always find a good proxy
• Instrumental variables
• Panel methods
Instrumental Variable (IV) Estimation
Instrumental Variables
Faced with omitted variable bias, IV is one of our options.
The IV approach
• leaves the unobserved variable in the error term
• estimates the model parameters consistently using some
additional information =
a new variable = the instrument = z
• the instrument z must satisfy 2 key assumptions:
– changes in z are associated with changes in x
– z is exogenous to u
+ z does not belong in the model
• the instrument is used to generate only exogenous
variation in our explanatory variable of interest (x)
– We use the instrument to isolate or extract the exogenous
component of x
Instrumental Variables
OLS, x is exogenous

OLS, x is endogenous

= 0 + 1 +
IV, x is endogenous, z is an instrumental variablez

Instrumental Variables
More generally:
• Consider the model
y = β0 + β1x+ u
But we believe MLR.4 does not hold, that is,
Cov(x,u) ≠ 0, i.e. E(u|x) ≠ 0
To get consistent estimators of β0, β1 we require additional
information in the form of a new variable, z, that satisfies 2
conditions
Definition of a instrumental variable (z)
In a regression with an endogenous variable (x):
z does not belong in the regression and,
1) z is highly correlated with the endogenous variable (x) (is
relevant)
is large
2) z is uncorrelated with u (is exogenous)
Instrumental Variables
9
Instrument
Relevance
Instrument
Validity
Otherwise it would
be in the model
already
Instrumental Variables
Example: Wages and the return on education, with unobserved ability
• We wish to estimate the return to education:
Consider the population model
log(wage) = β0 + β1educ + β2ability + e (1)
but suppose we do not observe ability, and we use the simple regression
model:
log(wage) = β0 + β1educ + u (2)
⇒ ability is now absorbed into the error term for model (2)
⇒ OLS applied to (2) will give us a biased and inconsistent estimator for β1 if
educ and ability are correlated (and β2 ≠ 0).
⇒ We can still use model (2) IF we can find an instrumental variable for educ
Instrumental Variables
Example: Wages and the return on education, with unobserved ability
• An IV for education must be
(1) RELEVANT: correlated with education
(2) VALID: uncorrelated with ability
• Potential instruments for education?
⇒ Last digit of Medicare number?
• it is random, so satisfies condition (2), but not condition (1).
⇒ IQ score?
• It’s a good proxy variable for ability, but would be a poor instrument for education
• a good proxy means that it is highly correlated with ability, which violates the
second condition for an IV for educ (as ability is part of the error term).
⇒ family background variables
• often used by labour economists as IVs for education
• Examples:
- mother’s or father’s education - commonly found to be positively related to
children’s education ... though it may also be correlated with child’s ability.
- number of siblings while growing up
⇒ state compulsory schooling laws
IV Example 1
Returns to Education
• Example: Father‘s education as an IV for education
OLS:
IV:
Is the education of the father a good IV?
1) Doesn‘t appear as regressor
2) Sign. correlated with educ
3) Uncorrelated w/ the error (?)
The estimated return to
education decreases (which
is to be expected)
It is also much less
precisely estimated
Return to education
likely overestimated
12
So how does IV work?
OLS – deriving the OLS estimator using exogeneity
Before looking at IV, let review the OLS estimator when x is
exogenous:
The OLS estimator = sample covariance / sample variance:
̂1 = � ,� = 1 − 1∑ − ̅)( − ̅1
− 1∑ − ̅ 2 = ∑ − ̅)( − ̅∑ − ̅ 2
and assume
Exogeneity
= population parameter
OLS – deriving the OLS estimator using exogeneity
Before looking at IV, let review the OLS estimator when x is
exogenous:
The OLS estimator = ̂1 = � ,�
And the OLS estimator is consistent, as long as the data are such that sample
variances and covariances converge to their population counterparts as n →∞,
i.e. if a LLN holds.
⇒ Bottom Line: OLS will be consistent if, and only if, exogeneity holds.
Exogeneity
• To see how the IV works, assume we have an instrumental
variable (Z) that meets the previous conditions:
... even though
IV-estimator:
Z is exogenous,
i.e. uncorrelated
with u
Z is correlated with the
explanatory variable &
implies this correlation is
non-zero
Instrumental Variables
Properties of the IV Estimator
• It can be shown that the IV estimator is consistent and
asymptotically normal.
• Again, similar to the OLS case, the IV estimator can be decomposed
as follow
• With , = 0, we have that, as → ∞, the numerator
in is zero => ̂1
= 1
• IV estimators can have substantial bias in small samples …
therefore want large samples. (see p 465)
17
Properties of IV Estimators
Our 2 IV Conditions
• We cannot test for condition (2) (Instrument Validity)
– so we need to use economic theory and intuition to decide if assuming
Cov(z,u) = 0 is reasonable.
• We can test for condition (1), i.e. if Cov(z,x) ≠ 0 holds:
To do this:
– Run the regression
x = γ0 + γ1z + v
and then test:
H0: γ1 = 0
against
H1 :γ1 ≠ 0
18
Sometimes called
the “first-stage”
regression
For instrument relevance – we want γ1 to
be large and highly significant
F-stat>10 is a usual rule of thumb
What happen if the IV is invalid or weak?
IV may be much more inconsistent than OLS if the IV is not
completely exogenous and only weakly related to x.
IV worse than OLS if: e.g.
There is no problem if
the instrumental
variable is really
exogenous.
If not, the
inconsistency will be
the larger the weaker
the correlation with x.
Invalid/Weak Instrumental Variables
19
is small
Inference with the IV Estimator
• It is straightforward to estimate the variance of the IV estimator,
and thus the standard error
• Under homoskedasticity [ 2 = 2], we have
̂1 = �2 �2 ,2= �2
,2
where:
– is the total sum of squares of xi
– ,2 is the R2 from regressing xi on zi
• We can use this standard error for constructing t-tests or CI,
under homoskedasticity
OLS vs IV Estimation
• Consider both IV and OLS estimation when x and u are uncorrelated
=> OLS will be unbiased
OLS: ̂1 = �2
IV: ̂1 = �2,2
• Since ,2 < 1, the IV variance is larger than the OLS variance (when
OLS is valid)
⇒ If the ,2 is small (i.e. we have a ‘weak’ instrument) and x and z are
only slightly correlated this will translate into very large IV standard
errors
⇒ The cost of IV estimation when x and u are uncorrelated; variance
of IV estimator is larger, sometimes much larger, than the variance
of the OLS estimator
IV & Multiple Regression
IV and Multiple Regression
• Straightforward to extend IV estimation to the multiple
regression model
• The model we are interested in estimating is known as
the structural model:
y1 = β0 +β1y2 +β2z1 +u1
where the
• y variables represent ‘endogenous’ variables
• the z variables are ‘exogenous’
⇒ the problem is that one or more of the explanatory
variables (y2) are endogenous.
IV and Multiple Regression - Example
• Wages and return on education
• Structural model:
log(wage) = β0 + β1educ + β2exper + u1
Where
• y1 = log(wage)
• y2 = educ
• z1 = exper
Here we are concerned with endogeneity of educ - allow
for educ to be correlated with u1
IV and Multiple Regression
Simple Case:
If we estimate the structural model
1 = 0 + 12 + 21 + 1
by OLS, all the coefficient estimates, �, will be biased and
inconsistent
• we need an IV for each endogenous variable
⇒Let z2 be the instrument for y2
⇒That is, Cov(z2,u1) = 0, and Cov(z2,y2) ≠ 0
IV and Multiple Regression
General Case
Structural model:
And for our instrument zk we require:
• Cov(zk,u1) = 0, and
• Cov(zk,y2) ≠ 0
We confirm relevance (also known as identification) using the first
stage regression:
endogenous exogenous variables
πk must be large and significant
Notice – all the exogenous variables
are included in the first stage
IV and Multiple Regression
• Let’s think a bit more about the first stage equation
• Notice
– the endogenous variable, y2, is expressed in terms of
exogenous variables only
– the IV estimator requires that πk ≠ 0 ⇒ y2 and zk are
correlated after controlling for z1, …, zk-1
– this is known as an “identification condition” and it is
easy to test:
• F-test on πk with the F-stat>10
IV and Multiple Regression
• It is straightforward to add many more exogenous
explanatory variables to the structural model:
– these additional explanatory variables must then also
enter the first stage, and the identification condition must
still be satisfied
• Intuitively, the instrument zk extracts a component of the
variation in y2 that is uncorrelated with the other
exogenous variables and uncorrelated with the error
term u1.
Testing for Endogeneity
Testing for Endogeneity
• Since OLS is preferred to IV if we do not have an endogeneity
problem, then we’d like to be able to test for endogeneity.
• If we do not have endogeneity, both OLS and IV are consistent.
⇒ Idea of Hausman test is to see if the estimates from OLS and IV
are different.
⇒Good idea to always compute OLS and IV estimates and see if
the estimates are practically similar or different
⇒To test whether any differences are significant, there is a
simple testing procedure
30
Testing for Endogeneity
Structural model
First stage regression:
Variable y2 is exogenous if and only if v2 is uncorrelated with u1,
i.e. if the parameter 1 is zero in the regression:
31
Variable that is suspected to be
endogenous
Testing for Endogeneity
To implement the test:
• we need the predicted residuals from estimating the first stage
by OLS, �2
• Then use them as an additional regressor in the structural
model:
Test equation:
1 = 0 + 12 + 21 + ⋯+ −1 + 1 �2 +
Hypotheses:
0: 1 = 0 …
1:1 ≠ 0 …
Conclusions:
• We conclude that 2 is endogeneous is the null hypothesis is
rejected, i.e. if the parameter 1 is significantly different from zero.
• If we do not reject the null, this is not proof of exogeneity.

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EMATM0051 COMP220 BLAW1002 E20 TCP8001 ECON61001 EART40005 ECONM1022 MATH 365 MATH 367 ELEC6218 COMP 5812M EDUC 5061M ELEC ENG 1101 equation MATH6174 ST334 C4 EC9570 A33648 EBUS603 ECON47815 stata WFMS0001 PEM102 INF6060 ionically ELECTRICAL MATH 375 MMW 12 MKTG860 ECON60401 ALY6010 SOST20131 Introduction COM3524 MME2 GGR305 JNB 538 EECS101 COMM5030 IEOR E4601 2B CSC336 EN2315 LAW 432 CSC263 MATH2001 OLET1143 CSCB63 SIT772 COMM1170 STAT 441 MS930 INFT3050 GLBH0030 Stat 120B MAST10007 MGEC61 Machine GGR 252H1S ECON7030 Financial Marketing MGMT30019 ECON5120 SDSC 8009 CSC165H1S DSCI 551 3081W ACM41000 ECA 5304 MATH 1064 ECSE444 DATA 604 COMP7105 MIE1615 ECON4004 ENV200H1S EC481 MKT 306 EC1B1 MATH08051 ALY6140 CS260C CSE30 MSIN0226 CS4103 CSC3064 AM16 COMP11212 CSE4101 PSTAT 174 Math 380W CNT 5106C STAT 310 FIN2028 CS5014 STAT 425 COMP 4102A ELEC2140 MANF9544 QRM 9 QRM 8 ECON 457 COMPSCI 4091 MATH 523 DNSC-4211 PEAS 6000 MSIN0180 COMP2119 A2A HT 2022 IB93F0 MAS316/414 ACCT2522 MSIN0181 ATMS 201 COMP3425 MATH7861 SOLA2060 MA 325 7CCEMCTH ECON 432 INFT2051 SOLA 2060 ECOS2002 Stats141XP BIOA02H3 S2022 MA4601 K5 COMP5329 EG 284 IB3A70 WECO1020 COMP3027 Energy ACS61011 MAST20026 STAT 533 MATA31 ENVS257 BEM2031 EC 486 COMP0172 CS3214 EECE5644 IB2B20 COSC2536 COSC1147 COMP6214 5CCM242A COMP0130 AMME2000 MTMG50: EG238 ACF6006 QBUS2810 COSC2391 ECON7560 MATH377 STA305 CS 486 COMP1117B EL648 ELEC3909 ECON 241 COMP 4901W BUSI4428 ENEN20002 ENGN4524 ELEC3115 CSC 413 PSYC 100B ECON1002 MATH256 ANSC20005 MTHS2008 MATH40082 COSC 2406 FIT3174 STA 238 CMT202 Differential Equations MTHM017 GARCH 101 MATH2069 CSC336S ACSC12 ETF2121 CEME 2001 DB B474F ARHT1001 EECS 3311 3220 STATS 786 CS 484 COMP20008 MATH4091 ECE 455 CSCI 204 BUS B200F BA 875 BU5562 COSC 1147 X11 GEOG 101 IMM250F CSE 584A MATH38172 LAB 4 ELEC5882M FINS5523 EE 430 COMP 465 6CCS3PRE 7CCSMNSE FINANCE ACMB02H3 ENGG2112 ELEC5616 SCM B371 CHEM3120 CS22 ECMT2130 MAST30011 ADSPBF706 CENG0013 ACCOUNTING BUSS1040 CHEM 181 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