ECON 61001: Lecture 8
Alastair R. Hall
The University of Manchester
Alastair R. Hall ECON 61001: Lecture 8 1 / 27
Outline of today’s lecture
Instrumental Variables estimation
Instrument relevance
Construction of estimator
Large sample properties and inference Weak instruments
2SLS
Empirical application
Alastair R. Hall ECON 61001: Lecture 8 2 / 27
Relevance condition
Last time discussed how IV estimation is based on the information about the parameter vector in the population moment condition:
E[ziui(β0)] = 0 (♯)
As we will see, for estimation to be “successful” also need (♯) to
represent unique information about β0 that is, E[ziui(β)] ̸= 0 for all β ̸= β0.
This condition is equivalent to rankE[zixi′] = k
This is known as the identification condition for β0.
Alastair R. Hall ECON 61001: Lecture 8 3 / 27
Terminology
Three key conditions: E[ziui(β0)] = 0
known as the orthogonality condition rank{E[zixi′]} = k,
known as relevance condition ⇒ instruments are “sufficiently related” to regressors.
rank{E[zizi′]}=q,wherezi isq×1.
uniqueness condition ⇒ each moment condition provides some
unique information.
Alastair R. Hall ECON 61001: Lecture 8 4 / 27
Relationship between q and k
q < k: not enough information → β0 is under-identified.
q = k: same # of pieces of information as unknowns → β0 is
just-identified.
q > k: more pieces of information as unknowns → β0 is over-identified.
Alastair R. Hall ECON 61001: Lecture 8 5 / 27
Just-identified case: q = k
In this case can apply MoM principle and IV estimator βˆIV is
defined as solution to sample moment conditions that is,
N
N−1ziui(βˆIV) = N−1Z′u(βˆIV) = 0
i=1
where Z is N × q matrix with ith row zi′.
And so:
βˆIV = (Z′X)−1Z′y
Alastair R. Hall ECON 61001: Lecture 8 6 / 27
Over-identified case: q > k
In this case, MoM does not work as have more equations than
unknowns.
Instead, define βˆIV as value of β that is closest to solving sample
moment conditions.
How do we measure how far sample moment function is from zero? Answer via
QIV(β) = u(β)′Z(Z′Z)−1Z′u(β),
where (we have assumed) rank(Z) = q and so (Z′Z)−1 is p.d. ⇒
QIV (β) satisfies
QIV (β) ≥ 0 for all β
QIV (β) = 0 iff Z′u(β) = 0.
Alastair R. Hall ECON 61001: Lecture 8 7 / 27
Over-identified case: q > k
Define IV estimator to be
βˆIV = argminβ∈BQIV (β)
→ βˆIV = X′Z(Z′Z)−1Z′X−1 X′Z(Z′Z)−1Z′y (See Tutorial 8 Question 2)
Alastair R. Hall ECON 61001: Lecture 8 8 / 27
Large Sample analysis
As examples illustrate IV can be applied in cross-section or time series data.
Here we concentrate on IV in cross-section data. For times series case see Lecture Notes.
As with OLS need to impose certain assumptions. So start with those.
Alastair R. Hall ECON 61001: Lecture 8 9 / 27
Large Sample analysis – CS data
CS1-IV:yi =xi′β0+ui
CS2-IV: { (ui , xi′, zi′), i = 1, 2, . . .N} forms an independent
and identically distributed sequence.
CS3-IV: (i) E[zi zi′] = Qzz , finite, p.d.; (ii) E[zi xi′] = Qzx ,
rank{Qzx } = k. CS4-IV: E [ui |zi ] = 0.
CS5-IV: Var[ui|zi] = h(zi) > 0. Notice:
CS4-IV: ⇒ E [zi ui ] = 0 (via LIE), the orthogonality condition. CS3-IV(ii): is relevance condition; CS3-IV(i): is uniqueness
condition;.
it is now properties of ui conditional on zi that matter.
Alastair R. Hall ECON 61001: Lecture 8 10 / 27
Large Sample analysis – CS data
Consider just-identified case (q = k) – over-identified case (q > k) in Lecture Notes.
We have
ˆ N −1 N
βIV −β0 = N−1zixi′ N−1ziui, i=1 i=1
and using the WLLN, we have
N N−1zixi′ →p
E[zixi′] = Qzx, N−1ziui →p E[ziui]=0.
i=1 N
i=1
So using Slutsky’s Theorem: βˆ →p β + Q−1 ×0 = β .
IV 0 zx 0
Alastair R. Hall ECON 61001: Lecture 8 11 / 27
Large Sample analysis – CS data
ˆ N −1 N N1/2(βIV − β0) = N−1 zixi′ N−1/2 ziui.
i=1 i=1
As in Lecture 6: CLT →
N
N−1/2ziui →d N(0,Ω), whereΩ=limN→∞ΩN
i=1
and (with slight abuse of notation)
−1/2N ΩN=Var N ziui
i=1
Alastair R. Hall ECON 61001: Lecture 8 12 / 27
Large Sample analysis – CS data
What is Ω here?
Assumption CS2-IV ⇒ {ziui; i = 1,2,…N} are i.i.d. and so
Cov[ziui,zjuj] = 0 (i ̸= j). ⇒ΩN =Var[ziui].
Using E [zi ui ] = 0,
Var[ziui] = E[ui2zizi′] = E E[ui2|zi]zizi′ = E[h(zi)zizi′] = Ωh, say.
Alastair R. Hall ECON 61001: Lecture 8 13 / 27
Large Sample analysis – CS data
Therefore, under our assumptions, we have:
N
N−1/2ziui →d N(0,Ωh).
i=1
Under Assumptions CS1-IV-CS4-IV and CS5-IV: N1/2(βˆIV−β0)→d N(0,VIV).
where V = Q−1Ω (Q−1)′. IV zx h zx
Alastair R. Hall ECON 61001: Lecture 8 14 / 27
Large Sample analysis – CS data
To use this result as basis for inference, need a consistent estimator of VIV and so Ωh.
Can adapt ideas from discussion of OLS, and show that:
N
Ωˆ h = N − 1 e i 2 z i z i ′ →p Ω h ,
i=1
where have (re-)defined ei = yi − xi′βˆIV .
Alastair R. Hall ECON 61001: Lecture 8 15 / 27
Large Sample analysis – CS data
S e t Qˆ z x = N − 1 Z ′ X t h e n
Vˆ I V = Qˆ − 1 Ωˆ h ( Qˆ − 1 ) ′ →p V I V .
zx zx
Can then perform inference using same techniques as in Lecture 6 provided we use modified variance estimator.
For example, an approximate 100(1 − α)% confidence interval for β0,l is given by,
βˆIV ,l ± z1−α/2VˆIV ,l,l/N .
Alastair R. Hall ECON 61001: Lecture 8 16 / 27
Large Sample analysis
Recall that large sample is used as an approximation to the finite sampling distribution of test statistics.
It has been realized that if relevance condition holds but almost fails then large sample distribution theory derived above can be a very poor approximation even in very large samples.
In this scenario, instruments are said to be weak.
Example: Angrist & Krueger (1991) study of returns to education
(example 2 above).
used Qi , quarter of birth, as instrument. but edi is only very weakly related to Qi .
Alastair R. Hall ECON 61001: Lecture 8 17 / 27
IV estimation
Often IV viewed through the lens of simultaneous equations model. We explore this in case where only one endogenous regressor.
where
y =z′γ+αy+u 1,i 1,i 0 0 2,i 1,i
y2,i = z′ δ1,0 +z′ δ2,0 +u2,i 1,i 2,i
u1,i ∼iid 0 , σ12 σ1,2 , σ1,2̸=0 u2,i 0 σ1,2 σ2
E[zl,iuj,i] = 0forl,j=1,2.
Alastair R. Hall ECON 61001: Lecture 8 18 / 27
IV estimation
Equation of interest:
y =z′γ+αy +u =x′β+u
1,i 1,i 0 0 2,i 1,i i 0 1,i Estimate β0 via IV based on:
where
zi=[z′ ,z′ ]′
1,i 2,i u1,i(β)=y1,i −xi′β.
E[ziu1,i(β0)] = 0
Alastair R. Hall ECON 61001: Lecture 8
19 / 27
IV estimation
Note:
z1,i are instruments for themselves and z2,i are instruments for y2,i
z2,i does not appear on rhs of equation of interest.
Alastair R. Hall ECON 61001: Lecture 8 20 / 27
IV and 2SLS estimation
IV can be implemented via a two-step procedure: regress y2,i on zi (via OLS) and obtain yˆ2,i .
regress y1,i on (z ′ , yˆ2,i ) (via OLS) → β ̃N . 1,i
β ̃N is known as Two Stage Least squares (2SLS) estimator of β0. It can be shown that β ̃N = βˆIV .
Alastair R. Hall ECON 61001: Lecture 8 21 / 27
IV estimation
In this context, relevance relates to relationship between y2,i and z2,i controlling for z1,i. Can be assessed by looking at the first stage regression
y =z′δ +z′δ +u 2,i 1,i 1,0 2,i 2,0 2,i
Instruments are relevant if δ2,0 ̸= 0.
This can be tested using F test described in Lectures 3/4:
H0 :δ2,0 =0vsHA :δ2,0 ̸=0.
H0 ⇒ instruments do not satisfy relevance condition.
HA ⇒ relevance condition satisfied.
Alastair R. Hall ECON 61001: Lecture 8 22 / 27
Example: Economic development and institutions
Acemoglu et al (2001):
ln[yi] = μ0 +riα0 +wiγ0+ui
where
yi is income per capita in developing country i
ri is quality of institutions in developing country i wi is life expectancy
ri is likely correlated with ui due to reverse causality.
Alastair R. Hall ECON 61001: Lecture 8 23 / 27
Example: Economic development and institutions
Base estimation on the population moment condition: E[ziui] = 0
where zi = [1,wi,z1,i,z2,i,z3,i,z4,i]′ and
z1,i is log settler mortality of country i, z2,i is the absolute latitude of country i, z3,i is the mean temperature of country i,
z4,i is the proportion of land area within 100km of the seacoast.
Alastair R. Hall ECON 61001: Lecture 8 24 / 27
Example: Economic development and institutions
αˆ 95%CIforα0 γˆ 95%CIforγ0 OLS 0.287 (0.186,0.387) 0.0496 (0.036,0.063) 2SLS 0.744 (0.335,1.153) 0.016 (-0.018,0.051)
point estimate of α0 is higher with 2SLS s.e.’s are larger for 2SLS than OLS
Alastair R. Hall ECON 61001: Lecture 8 25 / 27
Example: Economic development and institutions
Are instruments relevant? Assess this using first stage regression: r = δ0 + δ1 ∗ w + δ2 ∗ z1 + δ3 ∗ z2 + δ4 ∗ z3 + δ5 ∗ z4 + “error”
Test:
H0 : δ2 = 0,δ3 = 0,δ4 = 0,δ5 = 0 ⇒ instruments not relevant
HA : δi ̸= 0 for at least one i = 2,3,4,5 ⇒ instruments relevant
F = 2.27 with p-value of 0.0740 ⇒ only marginal evidence in support of instrument relevance (may be in weak instrument territory)
Alastair R. Hall ECON 61001: Lecture 8 26 / 27
Further reading
Notes: Chap 5.
Greene:
8.1 (models with endogenous regressors)
8.2 (assumptions) 8.3.1 (OLS)
8.3.2 (IV)
Alastair R. Hall ECON 61001: Lecture 8 27 / 27