On Variance Estimation for 2SLS

When Instruments Identify Different LATEs

Seojeong Lee∗†

June 30, 2014

Abstract

Under treatment effect heterogeneity, an instrument identifies the instrument-

specific local average treatment effect (LATE). If a regression model is esti-

mated by the two-stage least squares (2SLS) using multiple instruments, then

2SLS is consistent for a weighted average of different LATEs. In this case, the

conventional heteroskedasticity-robust variance estimator for 2SLS is incorrect

because the postulated moment condition evaluated at the 2SLS estimand does

not hold unless those LATEs are the same. I provide a correct formula for the

asymptotic variance of 2SLS by using the result of Hall and Inoue (2003).

Keywords: local average treatment effect, treatment heterogeneity, two-stage

least squares, variance estimator, model misspecification

JEL Classification: C13, C31, C36

1 Introduction

Since the series of seminal papers by Imbens and Angrist (1994), Angrist and Imbens

(1995), and Angrist, Imbens, and Rubin (1996), the local average treatment effect

(LATE) has played an important role in providing useful guidance to many policy

questions. The key underlying assumption is treatment effect heterogeneity. That is,

each individual has a different causal effect of treatment on outcome. If an instrument

∗School of Economics, Australian School of Business, UNSW, Sydney NSW 2052 Australia, email:jay.lee@unsw.edu.au, homepage: https://sites.google.com/site/misspecified/†I thank Tue Gorgens for stimulating my interest on this topic. I also thank Bruce Hansen, Han

Hong, and Toru Kitagawa as well as session participants at SETA 2014 conference for their helpfuldiscussions and suggestions.

1

is available, then the average treatment effect (ATE) of those whose treatment status

can be changed by the instrument, thus the local ATE, can be identified. Hence,

LATE is instrument-specific. When multiple instruments are available, the two-stage

least squares (2SLS) estimator is commonly used to estimate the causal effect. It is

well known that the 2SLS estimand is a weighted average of different LATEs and

a rejection of the overidentification test (the J test, hereinafter) is merely due to

treatment effect heterogeneity, rather than due to invalid instruments.

What is less well known and often overlooked in the literature is that the rejection

of the J test implies that the postulated moment condition model is likely to be mis-

specified. If so, the conventional standard errors are no longer consistent regardless of

how small the differences among individual LATEs are. This fact has been neglected

and the conventional standard errors have been routinely calculated even with small

p values of the J test, e.g. see Angrist and Evans (1998) and Angrist, Lavy, and

Schlosser (2010), among others.

In this paper, I propose a consistent estimator for the correct asymptotic variance

of 2SLS under treatment effect heterogeneity. Recently, Kolesar (2013) shows that

under treatment effect heterogeneity the 2SLS estimand is a convex combination of

LATEs while the limited information maximum likelihood (LIML) estimand may

not. Angrist and Fernandez-Val (2013) constructs an interesting estimand for new

subpopulations by reweighting covariate-specific LATEs. However, neither of the two

papers considers correct variance estimation.

2 2SLS with multiple instruments

2.1 Random coefficient model

The notations follow Imbens and Angrist (1994). Assume a binary treatment, Di and

an outcome variable Yi. Let Ydi denote the potential outcome of individual i given

treatment status Di = 1 or 0. Y1i and Y0i denote the response with and without

the treatment, respectively. The individual treatment effect is Y1i − Y0i which is

assumed to be heterogeneous, but we never observe both values at the same time.

Therefore, researchers focus on ATE, E[Y1i − Y0i]. However, unless the treatment

status is randomly assigned, a naive estimate of ATE is likely to be biased because

of selection into treatment. To overcome this endogeneity problem, researchers often

rely on instrumental variables. An instrument, Zi, is independent of Y1i and Y0i, and

2

correlated with the treatment, Di. Assume Zi is binary and define D1i and D0i be i’s

treatment status when Zi = 1 and Zi = 0, respectively. Since Di ≡ D0i+(D1i−D0i)Zi

and Yi ≡ Y0i + (Y1i − Y0i)Di, we can write a random coefficient model

Di = π0 + π1iZi + ξi, (2.1)

Yi = α0 + ρiDi + ηi,

where π0 ≡ E[D0i], π1i ≡ D1i−D0i, α0 ≡ E[Y0i] and ρi ≡ Y1i−Y0i. Note that π1i and

ρi are heterogeneous across individuals. Imbens and Angrist (1994) establish that

under regularity conditions,

Cov(Yi, Zi)

Cov(Di, Zi)=

E[Yi|Zi = 1]− E[Yi|Zi = 0]

E[Di|Zi = 1]− E[Di|Zi = 0]= E[ρi|D1i > D0i]. (2.2)

That is, the IV estimand (or the Wald estimand) is equal to ATE for a subpopulation

such that D1i > D0i. This is called LATE. The subpopulation whose treatment

status can be changed by the instrument is compliers. Those who take the treatment

regardless of the instrument status, D1i = D0i = 1, are always-takers, and those who

do not take the treatment anyway, D1i = D0i = 0, are never-takers. We cannot

identify ATE for always-takers and never-takers in general. By the monotonicity

assumption of Imbens and Angrist (1994), we exclude defiers who behave in the

opposite way with compliers, D1i = 0 and D0i = 1.

The above setting can be generalized to multiple instruments. Without loss of

generality, consider mutually exclusive binary instruments, Zji for j = 1, ..., q. Let

Djzi be i’s potential treatment status when Zj

i = z where z = 0, 1, and j = 1, ..., q.

Each instrument identifies a version of LATE because compliers may differ for each

Zji . It is well known that the 2SLS estimator using multiple instruments is consistent

for a weighted average of different LATEs. That is, the 2SLS estimand is a weighted

average of treatment effects for instrument-specific compliers (Angrist and Pischke,

2009; Kolesar, 2013):

ρa =

q∑j=1

ψj · E[ρi|Dj1i > Dj

0i],

where ψj is a nonnegative number and∑

j ψj = 1. For example, Angrist and Evans

(1998) use twin births and same-sex sibships as instruments to estimate the effect of

family size on mother’s labor supply. The twins instrument identifies ATE of those

3

who had more children than they otherwise would have had because of twinning,

while the same-sex instrument identifies ATE of those whose fertility was affected by

their children’s sex mix. These two compliers need not be the same. Their result

shows that the 2SLS estimate using both instruments is a weighted average of two

IV estimates using one instrument at a time.

Whether this weighted average of different LATEs is an interesting parameter

or not is context-specific. Complier groups may or may not overlap. If a complier

group is of researchers’ interest and other complier groups similarly overlap, then the

resulting 2SLS estimand would be the parameter of interest as it is similar to the

individual LATEs. In practice, this is often justified by a failure to reject the null of

the J test. But even a small difference in those LATEs would lead to a rejection of

the null hypothesis of homogenous LATEs asymptotically. If complier groups do not

overlap much and exhibit distinct characteristics, then combining them may make the

combined subpopulation more representative of the whole population. As a result,

the 2SLS estimate would be more informative than a single IV estimate. This case is

often followed by a significant J test statisitc.

2.2 Moment condition model

While the random coefficient model provides a theoretical ground for LATE, the

IV and 2SLS estimators can be derived from a slightly different but more general

framework, the moment condition model. Consider a linear model

Yi = α0 + ρ0Di + ηi ≡ X′iβ0 + ηi, (2.3)

where Xi = (1, Di)′ and β0 = (α0, ρ0)

′. If Di is endogeneous, i.e., Di is correlated

with ηi, β0 cannot be consistently estimated by OLS. If an instrument vector Zi =

(1, Z1i , Z

2i , ..., Z

qi )′ such that E[Ziηi] = 0 exists, then β0 can be consistently estimated

by the 2SLS estimator

β = (X′Z(Z′Z)−1Z′X)−1X′Z(Z′Z)−1Z′Y, (2.4)

where X ≡ (X′1, · · · ,X′n)′ is an n × 2 matrix, Z ≡ (Z′1, · · · ,Z′n)′ is an n × (q + 1)

matrix1, and Y ≡ (Y1, ..., Yn)′ is an n × 1 vector. The 2SLS estimator is a special

1Note that a constant is included in the instrument vector.

4

case of a GMM estimator using (n−1Z′Z)−1

as a weight matrix based on the moment

condition

0 = E[Ziηi] = E[Zi(Yi − α0 − ρ0Di)]. (2.5)

When (2.5) holds at a unique parameter vector β0 = (α0, ρ0), the moment condition is

correctly specified. The model is overidentified if the dimension of moment condition

is greater than that of the parameter vector and just-identified when they are equal.

For example, if Z1i is the only instrument available, the model is just-identified and

the solution is given by

α0 = E[Yi]− ρ0 · E[Di], (2.6)

ρ0 =Cov(Yi, Z

1i )

Cov(Di, Z1i )

= E[ρi|D11i > D1

0i].

Thus, LATE for Z1i is identified.

However, when multiple instruments are used so that the model is overidentified,

the postulated moment condition becomes problematic because it restricts all the

instrument-specific LATEs to be identical by construction. If those LATEs’ are dif-

ferent, then there may be no parameter that satisfies (2.5) simultaneously even if the

instrument vector is uncorrelated with the error term ηi. To see this, suppose that

there are two instruments, Z1i and Z2

i . The moment condition is

0 = E[Yi − α0 − ρ0Di], (2.7)

0 = E[Z1i (Yi − α0 − ρ0Di)],

0 = E[Z2i (Yi − α0 − ρ0Di)].

Solving the first equation for α0 and substituting it into the second and third equa-

tions, we have

ρ0 =Cov(Yi, Z

1i )

Cov(Di, Z1i )

=Cov(Yi, Z

2i )

Cov(Di, Z2i ). (2.8)

But this implies that the two LATEs are the same, which is not true in general. Thus,

(2.8) does not hold and the moment condition does not have a solution that satisfies

the three equations simultaneously.

In general, if the moment condition does not hold for all possible values of pa-

rameter in the parameter space, the model is misspecified. In our case, the model is

misspecified due to heterogeneous treatment effect across different complier groups

5

although all the instruments are valid. The J test is commonly used in practice to

test whether the moment condition is correctly specified or not, which implies homo-

geneous treatment effect in this framework. It is not surprising that researchers often

face a significant J test statistic when multiple instruments are used because even a

small difference in LATEs across complier groups will result in a rejection of the J

test asymptotically.

Although it is well known that 2SLS with multiple instruments estimates a weighted

average of different LATEs and a rejection of the J test can be merely due to het-

erogeneity of treatment effects, the consequence of misspecification on the variance

estimation for 2SLS has been overlooked in the literature. This is surprising because

the conventional heteroskedasticity-robust variance estimator would be inconsistent

for the true asymptotic variance of 2SLS if the underlying moment condition is mis-

specified. Since 2SLS is a special case of GMM, the argument can be easily shown by

using that of Hall and Inoue (2003).

Proposition 1. Let (Yi,Xi,Zi)ni=1 be an iid sample, where Xi = (1, Di)

′ and Zi =

(1, Z1i , Z

2i , · · · , Z

qi )′. Suppose Theorem 2 (the LATE theorem) of Imbens and Angrist

(1994) holds for each Zji and the model is given by (2.3). Then β is consistent for

βa = (αa, ρa)′ where αa = E[Yi]−ρaE[Di] and ρa is a linear combination of individual

LATE parameters. In addition, the asymptotic distribution is

√n(β − βa)

d→ N(0, H−1ΩH−1),

where H = E[XiZ′i] (E[ZiZ

′i])−1E[ZiX

′i] and Ω is a (k + 2q + 2) × (k + 2q + 2)

variance-covariance matrix of the following process

√n

1

n

n∑i=1

Zi(Yi −X′iβa)− E[Zi(Yi −X′iβa)]

(XiZ′i − E[XiZ

′i])(E[ZiZ

′i])−1E[Zi(Yi −X′iβa)]

(E[ZiZ′i])−1 (E[ZiZ

′i]− ZiZ

′i)(1n

∑ni=1 ZiZ

′i

)−1 .

Proof. By the Weak Law of Large Numbers (WLLN), the continuous mapping the-

orem (CMT) and the LATE theorem, consistency of β for βa immediately follows.2

Let ε ≡ (ε1, ...εn)′ be an n× 1 vector where εi ≡ Yi −X′iβa. Note that E[Ziεi] 6= 0.

2This is the 2SLS estimand and also called the pseudo-true value in a general context.

6

From the first-order condition, substitute Xβa +ε for Y and rearrange terms to have

β − βa = (X′Z(Z′Z)−1Z′X)−1X′Z(Z′Z)−1Z′ε, (2.9)

=

(1

nX′Z

(1

nZ′Z

)−11

nZ′X

)−11

nX′Z

(1

nZ′Z

)−1(1

nZ′ε− E[Ziεi]

)+

(1

nX′Z− E[XiZ

′i]

)(1

nZ′Z

)−1E[Ziεi]

+ E[XiZ′i]

((1

nZ′Z

)−1− E[ZiZ

′i]−1

)E[Ziεi]

.

The second equality holds because the population first-order condition of GMM holds

regardless of misspecification, 0 = E[XiZ′i]E[ZiZi]

−1E[Ziεi]. The expression (2.9) is

different from the usual one because E[Ziεi] 6= 0. As a result, the asymptotic variance

matrix of√n(β − β) includes additional terms, which are assumed to be zero in the

standard asymptotic variance matrix of 2SLS. By taking the limit of the right-hand-

side of (2.9), the Proposition follows. Q.E.D.

Remark 1. Since the proof of Proposition 1 deriving the asymptotic distribution is

essentially the same as that of Hall and Inoue (2003), there is little technical contri-

bution. The marginal contribution of this paper is to show that 2SLS using multiple

instruments under treatment effect heterogeneity is a special case of misspecified

GMM. Thus, the analysis of its asymptotic behavior can be significantly simplified.

The next Proposition proposes a consistent estimator for the asymptotic variance

matrix of 2SLS when 0 = E[Ziεi] may not hold. The proof is straightforward by

using WLLN and CMT, and thus omitted.

Proposition 2. A treatment-heterogeneity-robust asymptotic variance estimator for

2SLS is given by

ΣMR = n ·(X′Z (Z′Z)

−1Z′X

)−1(∑i

ψiψ′i

)(X′Z (Z′Z)

−1Z′X

)−1(2.10)

7

where εi = Yi −X′iβ, ε = (ε1, ε2, ..., εn)′, and

ψi =1

nX′Z

(1

nZ′Z

)−1(Ziεi −

1

nZ′ε

)(2.11)

+

(XiZ

′i −

1

nX′Z

)(1

nZ′Z

)−11

nZ′ε

− 1

nX′Z

(1

nZ′Z

)−1(ZiZ

′i −

1

nZ′Z

)(1

nZ′Z

)−11

nZ′ε.

The formula of ΣMR is different from that of the conventional heteroskedasticity-

robust variance estimator for 2SLS:

ΣC = n ·(X′Z (Z′Z)

−1Z′X

)−1(∑i

ZiZ′iε

2i

)(X′Z (Z′Z)

−1Z′X

)−1. (2.12)

Although both ΣMR and ΣC converge in probability to the same limit under treat-

ment effect homogeneity, but they are generally different in finite sample. ΣMR is

consistent for the true asymptotic variance matrix regardless of whether the postu-

lated moment condition is misspecified or not, while ΣC is consistent only if the

underlying LATEs are the same. This is also true for the standard errors based on

ΣMR and ΣC .

Remark 2 (Generalization of LATE) The conclusions of Propositions 1 and 2 do

not change with covariates. Let Wi be the vector of covariates including a constant.

Then Xi = (W′i, Di)

′ and Zi = (W′i, Z

1i , · · · , Z

qi )′ in Propositions 1 and 2. In addition,

it is possible that Zji or Di can take multiple values, or even continuous. With

covatiates or with variable treatment/instrument intensity, only the interpretation of

LATE parameters changes.

Remark 3 (Invalid Instruments) The treatment-heterogeneity-robust variance es-

timator ΣMR is also robust to invalid instruments, i.e. instruments correlated with

the error term. Consider a generic linear model

Yi = X′iβ0 + ηi, (2.13)

where Xi is a (k + p) × 1 vector of regressors. Among k + p regressors, p are endo-

geneous, i.e. E[Xiηi] 6= 0. If a k + q vector of instruments Zi is available such that

8

E[Ziηi] = 0 and q ≥ p, then β0 can be consistently estimated by 2SLS or GMM. If

any of the instruments is invalid, then E[Ziηi] 6= 0 and β0 may not be consistently

estimated. Since the moment condition does not hold, the model is misspecified.

There are two types of misspecification: (i) fixed or global misspecification such that

E[Ziηi] = δ where δ is a constant vector containing at least one non-zero component,

and (ii) local misspecification such that E[Ziηi] = n−rδ for some r > 0. With a par-

ticular choice of r = 1/2, this framework has beeen used to analyse the asymptotic be-

havior of the 2SLS estimator with invalid instruments by Hahn and Hausman (2005),

Bravo (2010), Berkowitz, Caner, and Fang (2008, 2012), Otsu (2011), Guggenberger

(2012), DiTraglia (2013), among others. Either under fixed misspecification or under

local misspecification, ΣMR in Proposition 2 is consistent for the true asymptotic

variance matrix. However, the conventional variance estimator ΣC is inconsistent

under fixed misspecification. Under local misspecification, ΣC is consistent for the

true asymptotic variance matrix but its finite sample performance can be poor.

Remark 4 (Bootstrap) Bootstrapping can be used to get more accurate t tests

and confidence intervals (CI’s) based on β, in terms of having smaller errors in the

rejection probabilities or coverage probabilities. This is called asymptotic refinements

of the bootstrap. Since the model is overidentified and possibly misspecified, the

misspecification-robust bootstrap of Lee (2014) should be used. In contrast, the

conventional bootstrap methods for overidentified moment condition models of Hall

and Horowitz (1996), Brown and Newey (2002), and Andrews (2002) assume correctly

specified model, in this case, treatment effect homogeneity. Therefore, they achieve

neither asymptotic refinements nor consistency. Suppose one wants to test H0 :

βm = βa,m or to construct a CI for βa,m where βa,m is the mth element of βa. The

misspecification-robust bootstrap critical values for t tests and CI’s are calculated

from the simulated distribution of the bootstrap t statistic

T ∗n =β∗m − βm√Σ∗MR,m/n

where β∗m and βm are the mth elements of β∗ and β, respectively, Σ∗MR,m is the mth

diagonal element of Σ∗MR, and β∗ and Σ∗

MR are the bootstrap versions of β and

ΣMR based on the same formula using the bootstrap sample rather than the original

sample.

9

3 Simulation

The random coefficient model with two mutually exclusive instruments is considered

for simulation. First, a single binary instrument Z0i is randomly generated. Next,

uini=1 are generated from the uniform distribution U [0, 1] and individual compliance

types with respect to Z0i : An individual i is a never-taker if 0 ≤ ui < 0.2, a complier

if 0.2 ≤ ui < 0.8, and an always-taker if 0.8 ≤ ui ≤ 1. Treatment status is determined

accordingly. Then, Z0i is decomposed into two mutually exclusive instruments ran-

domly, Z1i and Z2

i , such that Z1i + Z2

i = Z0i . Thus, in this setting, always-takers and

never-takers are common to both instruments, but compliers for Z0i is decomposed

into three subgroups: Common compliers for both Z1i and Z2

i , compliers for Z1i only,

and compliers for Z2i only.

The potential outomes with and without treatment are generated as

Y0i ∼ N(0, 32), Y1i = Y0i + ρi. (3.1)

Without loss of generality, ρi is assumed to be identical within each subgroup. Let

ρi = 4 if i is an always-taker, ρi = 1 if i is a never-taker, ρi = 2 if i is a common

complier, ρi = 3 if i is a complier for Z1i only, and ρi = 1 if i is a complier for Z2

i only.

LATEs for Z1i and Z2

i are weighted averages of 2 and 3, and 1 and 2, respectively.

Note that ρi of always-takers and never-takers do not matter in calculating LATE.

The 2SLS estimand ρa is a weighted average of the two LATEs, which are different.

Thus, the potulated moment condition is misspecified.

Table 1 shows the mean and the standard deviation of the 2SLS estimator, the

mean of conventional/misspecification-robust standard errors, and the rejection prob-

n 100 1,000 10,000

mean(ρ) 1.8926 1.9213 1.9235s.d.(ρ) 1.1770 0.3618 0.1140

mean(s.e.ΣMR) 1.1699 0.3607 0.1137

mean(s.e.ΣC) 1.1188 0.3485 0.1100

reject J test at 5% 32.6% 99.8% 100%

Table 1: The standard deviation and standard errors of ρ when the 2SLS estimandis a weighted average of different LATEs

10

ability of the J test at 5% for different sample sizes. The number of Monte Carlo

repetition is 10,000. The result clearly shows that the conventional standard error

based on ΣC is inconsistent for the standard deviation of the estimator, and under-

estimate it for any sample size n. In contrast, the misspecification-robust standard

error estimates the standard deviation more accurately. The mean of ρ is slightly less

than two, because the number of compliers for Z2i is larger than that for Z1

i .

Figure 1 shows a negative relationship between the p-values of the J test and

the percentage difference of the two standard errors s.e.ΣMRand s.e.ΣC

for different

sample sizes. When n = 100, it is quite possible that the J test does not reject

the false null hypothesis at a usual significance level. However, as the p-value gets

smaller, it becomes more likely that the two s.e.’s are different. Since only ΣMR is

consistent for the true asymptotic variance under treatment effect heterogeneity, it

is recommended to report s.e.ΣMRespecially when the p-value is small. Since the J

test is consistent, the p-values become more concentrated around 0 as n increases.

Around zero p-values, the difference of the two s.e.’s can be substantial.

4 Conclusion

Estimating a weighted average of LATEs with multiple instruments using 2SLS is

a common practice for applied researchers. The resulting inferences and confidence

intervals are often justified when estimated LATEs are similar and the overidentifying

restrictions test does not reject the assumed model. However, when researchers face

a rejection of the overidentifying restrictions test, there has been no guidance on

how to proceed. Routinely reported standard errors by a statistical program are

likely to be inconsistent because they do not take into account the consequence of

possible misspecification of the postulated moment condition model. This paper

provided a solution to such dilemmas. The proposed formula of the variance estimator

for 2SLS is consistent for the true asymptotic variance regardless of whether the

underlying LATEs are heterogenous or not. In addition, this variance estimator is

robust to invalid instruments, and can be used for boostrapping to achieve asymptotic

refinements.

11

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−10

0

10

20

30

40

50

60

p value of the J test

perc

enta

ge d

iffer

ence

bet

wee

n th

e tw

o s.

e.’s

(a) n = 100

0 0.05 0.1 0.15 0.2 0.250

2

4

6

8

10

12

14

p value of the J test

perc

enta

ge d

iffer

ence

bet

wee

n th

e tw

o s.

e.’s

(b) n = 1, 000

Figure 1: Relationship between p-values of the J test and percentage difference be-tween the two standard errors, s.e.ΣMR

and s.e.ΣC

14