Essays on Weak Identification

by
Zhaoguo Zhan

B.Eng., Renmin University of China, 2003
MSc, London School of Economics, 2005
M.A., Brown University, 2007

Submitted in partial fulfillment of the
requirements for the degree of Doctor of Philosophy
in Economics at Brown University

PROVIDENCE, RHODE ISLAND
May 2011

© Copyright 2011 by Zhaoguo Zhan

This dissertationby ZhaoguoZhan is acceptedin its presentform
by the Departrnentof Economicsas satisfying the
dissertationrequirementfor the degreeof Doctor of Philosophy.

nutu01(26(zou
FbankKleibergen,Advisor

Recommendedto the Graduate Council

oateO qf 2(rtr44
Blaise Mell

Dat"
:Qf2( hal/
Dhaerie,

Approved by the Graduate Council

DatePeter M. Weber.Dean of the GraduateSchool

llI

Vitae
The author was born on November 5th, 1980 in China. He received his Bachelor
in Engineering from Renmin University of China in 2003, and MSc in Econometrics
and Mathematical Economics from London School of Economics in 2005. He entered
Brown University in 2006 to pursue his degree in Economics. He received his M.A.
in 2007 and Ph.D. in 2011.

iv

Acknowledgements

I am deeply indebted to my advisor Frank Kleibergen for continuous support and
encouragement. I thank Blaise Melly and Sophocles Mavroeidis for inspiration and
insightful comments. Thanks to Geert Dhaene for reading the manuscript. I also
owe much to my colleagues, Toru Kitagawa, Yuya Sasaki, Alexei Abrahams and
Philipp Ketz, who spent time discussing econometrics with me throughout my graduate study. Lastly, I am most indebted to my wife Flora for her consideration and
support.

v

Abstract of “ Essays on Weak Identification ” by Zhaoguo Zhan, Ph.D., Brown
University, May 2011

Recent developments in econometrics have revealed weak identification in empirical
studies. The linear Instrumental Variable (IV) regression with weak instruments is an
example that has received sizable attention; when instruments are weak, conventional
asymptotics fail to function, and empirical results based on conventional asymptotics
are unreliable. The problem of weak identification, however, is not limited to the
linear IV regression or the Generalized Method of Moments (GMM) framework. In
a broad range of economic models, the quality of inference depends on the treatment
of weak identification.

In my dissertation, I consider three issues related to weak identification. The
first chapter proposes a method to detect whether weak identification exists. The
objective of this chapter is to develop an intuitive tool which can be universally used
in IV and GMM applications. The second chapter investigates the empirical studies
of the Capital Asset Pricing Model (CAPM) and Consumption CAPM, in which
various risk factors are suggested to explain the variation in asset returns. I find that
irrelevant risk factors may still appear useful in explaining the variation, because of
the weak identification problem induced by irrelevant factors. The third chapter
targets the ongoing debate on whether technology shocks increase the hours worked.
The debate derives from the highly persistent time series of hours, which makes
the impact of technology shocks on hours weakly identified. I construct confidence
intervals for this impact by adopting an approach robust to weak identification.

CONTENTS

Vitae

iv

Acknowledgments

v

1 Detecting Weak Identification by Bootstrap

1

1.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.2

Motivation and Strategy . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.2.1

Weak Identification, Rank Tests . . . . . . . . . . . . . . . . .

5

1.2.2

Test Strategy by Bootstrap . . . . . . . . . . . . . . . . . . .

7

1.2.3

An Alternative . . . . . . . . . . . . . . . . . . . . . . . . . .

13

1.3

Estimation and Test . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

1.4

IV and Bootstrap . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

1.4.1

Model Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

1.4.2

An Empirical Example: Card (1995) . . . . . . . . . . . . . .

28

1.5

Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

1.6

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

vi

2 Weak Identification in (C)CAPM

43

2.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

2.2

Model and Fama-MacBeth . . . . . . . . . . . . . . . . . . . . . . . .

46

2.2.1

Linear Factor Model . . . . . . . . . . . . . . . . . . . . . . .

46

2.2.2

Fama-MacBeth . . . . . . . . . . . . . . . . . . . . . . . . . .

48

2.3

A Rank Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

2.4

Distribution of R2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

2.4.1

Assumption . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

2.4.2

Four Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

2.5

Examples of (C)CAPM . . . . . . . . . . . . . . . . . . . . . . . . . .

57

2.6

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

3 Does a Technology Shock Increase or Decrease Hours

70

3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

3.2

Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

3.2.1

VAR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

3.2.2

Long Run Restriction . . . . . . . . . . . . . . . . . . . . . . .

74

3.2.3

Model Simplification . . . . . . . . . . . . . . . . . . . . . . .

76

Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

78

3.3.1

AR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

78

3.3.2

Wald . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

80

3.4

Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84

3.5

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87

3.3

vii

CHAPTER

One

Detecting Weak Identification by Bootstrap

1

2

1.1

Introduction

The Instrumental Variable (IV) regression and Generalized Method of Moments
(GMM) are becoming the standard toolkit for empirical economists, and it is now
well known that both IV and GMM applications may suffer from the problem of weak
identification. An example that has received sizable attention is the linear IV regression with weak instruments studied in Staiger and Stock (1997). When the strength
of identification is weak, the finite sample distribution of IV/GMM estimators is
poorly approximated by the normal distribution, which further induces the malfunction of conventional inference methods that rely on the property of asymptotic normality.1 Two approaches co-exist to handle weak identification: the first approach is
to use the robust methods in Anderson and Rubin (1949), Stock and Wright (2000),
Kleibergen (2002)(2005), and Moreira (2003), which can produce confidence intervals/sets with the correct coverage, regardless of the strength of identification; the
second approach that is popular in practice is to rule out weak identification by
pretesting. Although identification-robust methods are available, excluding weak
identification in IV/GMM applications has practical importance: if identification is
not weak, then the rich set of conventional methods is applicable, making statistical
inference and economic decisions much easier. For instance, other than its confidence
interval, the point estimator of a parameter is usually preferred by policy makers,
but it is not consistent or meaningful unless weak identification is excluded.

In this chapter, I propose a method based on bootstrap resampling to detect
whether weak identification exists in IV/GMM applications. This method has the
unique feature of providing a graphic view of the strength of identification. In the
econometric literature, there exists a group of tests on identification that this chap1

See the article by Stock et al. (2002) for a survey.

3
ter is in line with: for example, in the linear IV model, Hahn and Hausman (2002)
and Stock and Yogo (2005) provide tests for the null of strong instruments and weak
instruments respectively, and the first stage F test with the F > 10 rule of thumb proposed in Stock and Yogo (2005) is widely used; in Hansen (1982)’s GMM framework,
which nests the linear IV model, the suggested tests include Wright (2002)(2003),
Inoue and Rossi (2008), Bravo et al. (2009), etc.

The proposed method is illustrated by Figure 1.1. The exact finite sample distribution of IV/GMM estimators is generally unknown, but can be approximated
either by the limiting normal distribution or by the bootstrap distribution. When
the IV/GMM models are strongly identified, these two approximation methods are
both valid, i.e. both the normal distribution and the bootstrap distribution are close
to the exact distribution, and in practice, they can be used exchangeably. Consequently, strong identification implies that the bootstrap distribution is not far away
from the normal distribution. When the bootstrap distribution is substantially different from the normal distribution, inference based on these two distributions might
contradict; in this situation, it is inappropriate to consider the identification strength
as strong. As a result, whether or not weak identification exists can be inferred by
comparing the bootstrap distribution with the normal distribution.

Since its introduction by Efron (1979), the bootstrap has become a practical tool
for statistical inference. The properties of the bootstrap are explained using the
theory of the Edgeworth expansion in Hall (1992), and its econometric applications
are illustrated in Horowitz (2001). As an alternative to the limiting distribution, the
bootstrap approximates the distribution of a targeted statistic by resampling the
data, and there is considerable evidence that it performs better than the first-order
limiting distribution in finite samples.2 However, the bootstrap does not always
2

See, for example, Horowitz (1994).

4
work well. When the IV/GMM models are not well identified, for instance, the
bootstrap is known to be problematic when it is used to approximate the commonly
used t/Wald statistic, as explained in Hall and Horowitz (1996). Nevertheless, the
fact that the bootstrap fails still conveys useful information: as illustrated by Figure
1.1, the substantial difference between the bootstrap distribution and the normal
distribution indicates that it is problematic to approximate the exact finite sample
distribution of IV/GMM estimators by the normal distribution; in other words, the
identification strength is weak.

Given the above introduction, it is tempting to apply normality tests, e.g. the
Kolmogorov-Smirnov test, to examine whether the bootstrap distribution is normal
in order to investigate the strength of identification. However, this route is not productive, when normality tends to be rejected as the number of bootstrap replications
becomes large. This is due to the fact that, in general, the bootstrap distribution
coincides with the normal distribution only at the limit where there are an infinite number of data points. In an IV/GMM application with a finite sample, the
bootstrap distribution is not equivalent to the normal distribution, hence the null
hypothesis for normality tests that the bootstrap distribution is identical to the normal distribution does not hold in empirical applications. Although normality tests
do not work well, empirical researchers can still eyeball the graph of the bootstrap
distribution to evaluate the strength of identification, and the test proposed in this
chapter does not impose the equivalence of the bootstrap distribution and the normal
distribution; only a substantial difference of these two distributions would induce the
rejection of strong identification.

The rest of the chapter is organized as follows: weak identification and the bootstrap strategy are illustrated in Section 2; in Section 3, the bootstrap-based method
for determining whether weak identification exists is proposed; the linear IV regres-

5
sion model is used as an example in Section 4, with an application to Card (1995);
in Section 5, Monte Carlo results are presented; Section 6 concludes. Although the
linear IV model is employed in this chapter for expository purposes, the proposed
method can be extended to the more general GMM framework.

Throughout the chapter, the following notations are used: for an m by n matrix
A, Ai (sometimes ai is used when A is a vector) is its ith row, and PA = A(A′ A)−1 A′ ,
MA = Im − PA , Im is the m by m identity matrix; vec(A) is the column vector containing the column by column vectorization of elements in A; for an object O, O∗ is
p

its bootstrap counterpart; ⇒ stands for weak convergence; → stands for convergence
in probability; N (µ, σ 2 ) is the normal distribution with mean µ, variance σ 2 ; ⊗ is
the Kronecker product; ||.|| is the Euclidean norm.

1.2

1.2.1

Motivation and Strategy

Weak Identification, Rank Tests

Let θ denote the population parameter of interest in an IV/GMM application, and θ̂n
is the estimator for θ, e.g. the two stage least squares estimator. The subscript n for
θ̂n refers to that θ̂n is computed by finite sample of n observations. Assuming that the
rank condition and other regularity conditions are satisfied (cf., Wooldridge (2002)),
√
the conventional first-order asymptotic theory yields that θ̂n is n−consistent, and
asymptotically normally distributed:
√

p

n(θ̂n − θ) ⇒ N (0, σ 2 ), and there exists σ̂ → σ

(1.1)

6
As surveyed by Stock et al. (2002), the exact finite sample distribution of

√

n(θ̂n −

θ) can be substantially different from the normal distribution N (0, σ 2 ), especially
when the rank condition is weakly satisfied. The scenario that (1.1) does not provide
a good approximation in finite sample applications is known as weak identification.

It is helpful to distinguish weak identification from identification/noidentification:
identification/noidentification refers to whether θ can be identified at the population
level, hence it is unrelated to the sample size n; in contrast, weak identification
targets that the first-order asymptotic theory may provide a poor approximation in
finite samples. When θ is identified in the population, it is possible that the firstorder approximation of (1.1) under a given sample3 does not function well, hence
weak identification may exist even though there is identification; the case that θ is
unidentified is also nested by weak identification, since the first-order approximation
of (1.1) also breaks down under noidentification. Loosely speaking, if the first-order
approximation based on (1.1) is poor, then weak identification is concerned.

Since the conventional result of (1.1) breaks down under weak identification, it
is important to investigate the identification strength in IV/GMM applications. A
natural way is to examine the rank condition, as weak identification is more likely to
exist when the rank condition is only weakly satisfied. There are rank tests available
to serve this purpose. For example, in the linear IV model with a single endogenous
regressor, the rank condition corresponds to that the correlation of the endogenous
regressor and the instruments is non-zero, and Stock and Yogo (2005) propose the F
test to examine this correlation: if the F statistic is greater than the tabled critical
values, typically around 10 for small number of instruments, then the rank condition
is considered strongly satisfied, and weak identification is excluded.
3

Even when the sample size is very large, weak identification could exist, e.g. in some regressions
of Angrist and Krueger (1991), there are 329000 observations, but their instruments are weak.

7
The approach of examining the rank condition, however, has some limitations
when it is extended to the more general GMM framework: first of all, the rank
statistic in the non-linear GMM may depend on the weakly identified parameters,
hence could not be consistently estimated, see, e.g. Wright (2003); secondly, it is not
clear how large the rank statistic needs to be in order to decide that identification in
GMM is not weak, to the best of my knowledge. In the linear IV model, which can
be seen as a special case of GMM, although the tabled critical values of the F test are
available, they are derived under the i.i.d and homoscedasticity assumptions, hence
it is not appropriate to apply them if heteroscedasticity or non-i.i.d. takes place.

Given the importance of detecting weak identification in IV/GMM applications,
and the limitations of the rank test, a question naturally arises: is there a tool that
does not have these limitations, and is universally applicable to both IV and GMM
applications? This chapter suggests that the bootstrap could be the tool.

1.2.2

Test Strategy by Bootstrap

Let θ̂n∗ denote the bootstrap estimator of θ, and θ̂n∗ is the counterpart of θ̂n . θ̂n is
computed by the data sample of the IV/GMM application, while θ̂n∗ is computed by
resampling the data sample or the model estimated from the data. The bootstrap
√
√
counterpart of n(θ̂n − θ) is n(θ̂n∗ − θ̂n ).
When identification is not weak, together with mild regularity conditions, the
√
bootstrap distribution of n(θ̂n∗ − θ̂n ) asymptotically coincides with the distribution
√
of n(θ̂n − θ) (cf., Horowitz (2001)):
√

n(θ̂n∗ − θ̂n ) ⇒ N (0, σ 2 )

(1.2)

8
√
There are now three objects: the exact distribution of n(θ̂n − θ), the normal
√
distribution N (0, σ̂ 2 ), and the bootstrap distribution of n(θ̂n∗ − θ̂n ). By (1.1) and
(1.2), both of which hold under mild conditions, these three distributions are asymptotically equivalent.

The proposed bootstrap strategy for detecting weak identification is to compare
√
the bootstrap distribution of n(θ̂n∗ − θ̂n ) with the normal distribution N (0, σ̂ 2 ),
or equivalently, compare the standardized bootstrap distribution of

∗ −θ̂
θ̂n
√n
σ̂/ n

with the

standard normal distribution. Since the difference between these two distributions
is negligible when identification is strong, a substantial difference is the evidence
against strong identification.

I find that this bootstrap strategy is appealing: (i) it provides a simple and intuitive way to detect weak identification, i.e. empirical researchers can draw the
graph of the bootstrap distribution and compare it with the normal distribution to
evaluate the strength of identification; (ii) it is universally applicable to IV/GMM
applications, i.e. although the data of IV/GMM applications may not be i.i.d. and
homoscedastic, there exist various bootstrap methods to construct the bootstrap distribution, e.g. the block bootstrap in Hall and Horowitz (1996) for time series data in
GMM, the pair and wild bootstrap used in Davidson and MacKinnon (2010) for heteroscedasticity in IV. By the bootstrap strategy, as long as it is feasible to construct
the bootstrap distribution, detecting weak identification reduces to the comparison
of two distributions: the bootstrap distribution, and the normal distribution.
Definition 1. Conditional on the sample of the observed data, the standardized
bootstrap estimator X =

∗ −θ̂
θ̂n
√n
σ̂/ n

follows a distribution with c.d.f. F (x). From now on,

the bootstrap distribution refers to F (x) in this chapter.

Normality tests appear to be the intuitive choice, since comparing the bootstrap

9
distribution with the normal distribution is proposed to infer the strength of identification. However, applying normality tests here will almost always induce the
rejection of normality, and lead to the conclusion of weak identification.

Take the classic Kolmogorov-Smirnov (KS) test for example. With B bootstrap
replications, let Xi =

∗i −θ̂
θ̂n
√ n,i
σ̂/ n

= 1, ..., B denote the i.i.d. bootstrapped estimators

after standardization. To test the hypothesis that Xi , i = 1, ..., B are B points
drawn from the standard normal distribution Φ(x), the KS statistic is the supremum
distance between Φ(x) and the empirical c.d.f. F̂ (x), scaled by the square root of the
number of points. As B → ∞, the KS statistic goes to infinity, instead of converging
to the Kolmogorov distribution:

KS =

√

B sup |F̂ (x) − Φ(x)| → ∞
x

where F̂ (x) =

1
B

∑B
i=1

1(Xi ≤ x) =

1
B

∑B
i=1

∗i

θ̂
n −
√ n ≤ x).
1( θ̂σ̂/
n

This is because the hypothesis F (x) = Φ(x) does not hold when n < ∞. In
other words, in empirical applications where the sample size is finite, F (x) differs
from Φ(x), although the difference may not be substantial, e.g. by the Edgeworth
expansion in Horowitz (2001), F (x) − Φ(x) = O(n−1/2 ). Consequently, even when
the difference between F (x) and Φ(x) is minor, as B → ∞, KS → ∞, i.e. the KS
test tends to reject normality when the bootstrap replication gets large.

Said differently, the bootstrap distribution is not identical to the normal distribution, although it can be asymptotically equivalent to the normal distribution.
The KS test is for testing the hypothesis that the bootstrap distribution is identical
to the normal distribution. For the purpose of examining whether the bootstrap
distribution is close to the normal distribution, it is inappropriate to apply this test.

10
Instead of verifying the equivalence of the bootstrap distribution and the normal
distribution by normality tests, this chapter provides a quantitative measure of the
difference/distance between these two distributions: if the measure shows that the
difference is substantial, then identification is considered as weak ; on the contrary, if
the measure shows that the difference is negligible, then identification is considered
as strong.

The measure results from the comparison of two confidence intervals (C.I.): the
conventional C.I. derived by inverting the t/Wald test, and the bootstrap percentile
C.I.. Consider the practical task of constructing a confidence interval for θ, the
parameter of interest. The 100(1 − α)% C.I. of θ derived by inverting the t/Wald
test is written as:
σ̂
σ̂
Ct ≡ (θ̂n − z1−α/2 √ , θ̂n + z1−α/2 √ )
n
n

(1.3)

where z1−α/2 is the 1 − α/2 quantile of Φ(x). For example, when α = 5%, z1−α/2 ≈
1.96, and the 95% C.I. of θ is approximately (θ̂n − 1.96 √σ̂n , θ̂n + 1.96 √σ̂n ).
Alternatively, a C.I. can also be constructed by the bootstrap percentile method:
order the bootstrapped estimators {θ̂n∗i , i = 1, ..., B}, write the ordered sequence as
∗(i)

∗(i)

{θ̂n , i = 1, ..., B}, where θ̂n

∗
is the ith smallest of {θ̂n∗i , i = 1, ..., B}; define θ̂n,α/2

∗(⌈Bα/2⌉)

∗
∗
and θ̂n,1−α/2
: θ̂n,α/2
≡ θ̂n

∗(⌈B(1−α/2)⌉)

∗
, θ̂n,1−α/2
≡ θ̂n

, where ⌈x⌉ denotes the

integer ceiling of x. The bootstrapped 100(1 − α)% C.I. of θ is:
∗
∗
Cb ≡ (θ̂n,α/2
, θ̂n,1−α/2
)

(1.4)

The two intervals in (1.3) and (1.4) are asymptotically equivalent when identification is not weak. To see this, equalizing the boundaries of these two intervals

11
yields:
∗
∗
θ̂n,α/2
− θ̂n
θ̂n,1−α/2
− θ̂n
√
√
= −z1−α/2 ,
= z1−α/2
σ̂/ n
σ̂/ n

These two equalities approximately hold if the distribution of θ̂n∗ after standardization, i.e. subtracting the estimate and dividing by the standard error, is close to the
standard normal distribution.
In practice, the above two methods of constructing the 100(1 − α)% C.I. for θ are
both commonly used. From a practical point of view, no matter which method empirical researchers use, the correspondent intervals need not be substantially different
from each other. If these two intervals do substantially differ, then it is difficult to
make reliable economic inference: for instance, one interval may include zero while
the other one does not, hence decisions of whether θ is significantly different from
zero based on the two different C.I.’s could contradict.4 The difference between the
two intervals in (1.3) and (1.4) boils down to the difference between the bootstrap
distribution and the normal distribution. If these two intervals are substantially
different, it indicates the bootstrap distribution is substantially different from the
normal distribution, hence the identification strength is weak.

The idea of comparing alternative C.I.’s to investigate the identification status
comes from Wright (2002), where he compares the interval derived by inverting the
robust tests with the conventional interval derived by inverting the t/Wald test.
Different from Wright (2002), I use the bootstrap to construct a C.I. for comparison;
in addition, Wright (2002) provides an identification test, while in this chapter, I
target weak identification instead of identification.

Based on the comparison of the two intervals in (1.3) and (1.4), a measure of the
4

This problem is encountered in the empirical example of Card (1995).

12
difference between the bootstrap distribution and the normal distribution, as well as
a quantitative definition of weak identification, is provided below.
Definition 2. Define D as the measure of the difference between the bootstrap distribution F (x) and the standard normal distribution Φ(x):

D ≡

q1−α/2 − qα/2
−1
2z1−α/2

where z1−α/2 is the 1 − α/2 quantile of Φ(x), qα/2 , q1−α/2 are assumed to be the two
unique quantiles of the continuous c.d.f. F (x), i.e. F −1 (α/2) = qα/2 , F −1 (1−α/2) =
q1−α/2 .

D is the relative difference of the lengths between the α/2, 1 − α/2 quantiles of
the two distributions. D = 0 if F (x) = Φ(x). The deviation of D from 0 indicates
the deviation of F (x) from Φ(x), and hence suggests existence of weak identification.
Definition 3. Suppose there is a cutoff γ > 0. The identification strength is considered as weak in this chapter if |D| > γ, otherwise the identification strength is
strong.

It is important to set a non-zero γ: if γ = 0, then identification tends to be
considered as weak, because F (x) is not exactly normal except in the limit of n → ∞.
For the same reason, it is inappropriate to apply normality tests, which impose γ = 0
under their null hypothesis.

D can be estimated by the relative difference of (1.3) and (1.4):
q̂1−α/2 − q̂α/2
D̂ =
−1=
2z1−α/2

∗
θ̂n,1−α/2
−θ̂n
√
σ̂/ n

−

∗
θ̂n,α/2
−θ̂n
√
σ̂/ n

2z1−α/2

∗
∗
θ̂n,1−α/2
− θ̂n,α/2
√ −1
−1=
2z1−α/2 σ̂/ n

13
Empirical researchers may have a certain tolerance level for D, which is the
threshold5 γ. For example, a researcher may consider it acceptable if |D| ≤ γ = 0.25,
i.e. the relative difference of the two C.I.’s is less than a quarter. If |D| goes above
this tolerance level, then it is not unreasonable to determine that the identification
strength is weak.

To summarize, this chapter proposes to use the difference D between the bootstrap distribution F (x) and the standard normal distribution Φ(x) to evaluate the
strength of identification. If |D| is greater than a given threshold, identification is
weak.

1.2.3

An Alternative

To my knowledge, Stock and Yogo (2005) is the first to provide a quantitative definition of weak instruments/identification. Like Stock and Yogo (2005), the quantitative Definition 3 of weak identification is also practically motivated: from the
practical perspective, it is not appropriate to consider identification as strong if the
bootstrap and the limiting normal distribution provide substantially different confidence intervals. Unlike Stock and Yogo (2005), Definition 3 is not directly related
to the bias of the conventional IV/GMM estimator or the size of the t/Wald test,
while Stock and Yogo (2005) define identification as weak if the relative bias of the
conventional estimator or the size distortion of the t/Wald test exceeds a threshold. Although quantitatively different, the definition in this chapter is qualitatively
similar to the one in Stock and Yogo (2005): both definitions of weak identification
indicate that the distributions of the conventional IV/GMM estimators are poorly
approximated by the normal distribution, or said differently, the conventional ap5

A choice of the threshold γ is suggested in the later part of Monte Carlo studies.

14
proximation of (1.1) breaks down.

In this chapter, I follow the idea in Wright (2002) to use D, the relative difference
of the lengths of two alternative C.I.’s, to measure the strength of identification.
Alternatively, we could follow Stock and Yogo (2005) and define D as the difference
of coverage:
Da ≡ F (z1−α/2 ) − F (zα/2 ) − (1 − α)
Note that Da is zero if the bootstrap distribution F (x) coincides with the standard
normal distribution Φ(x). Following Stock and Yogo (2005), we can use γ a = 5%,
and define identification as weak if |Da | > γ a = 5%.
These definitions of (D, γ) and the alternative (Da , γ a ) are two sides of the
same coin: both the difference in length and the difference in coverage indicate the
bootstrap distribution and the normal distribution are different, and there is no clear
advantage if either definition is used. For the rest of this chapter, I use (D, γ)6 .

Note that weak identification refers to the severe disparity between the exact distribution and the normal distribution, while its quantitative definition stated above
rests on the disparity between the bootstrap distribution and the normal distribution. In essence, the bootstrap strategy for detecting weak identification is to use the
difference between the bootstrap distribution and the normal distribution as a proxy
for the the difference between the exact distribution and the normal distribution.

The so-called bootstrap principle (or the bootstrap analogy) in Hall (1992) can
help clarify the proposed strategy. The bootstrap principle states that the mapping
from the population to the sample (1st mapping) is similar to the mapping from
the sample, which is also the bootstrap population, to the bootstrap resample (2nd
6

The empirical and MC results are found to be similar, when (Da , γ a ) is used.

15
mapping). By this principle, the identification strength in the 2nd mapping is expected to be similar to the identification strength in the 1st mapping. Consequently,
the bootstrap strategy for detecting weak identification is to use the identification
strength in the 2nd mapping as the proxy for the identification strength in the 1st
mapping: the substantial disparity between the bootstrap distribution and the normal distribution corresponds to the weak identification strength in the 2nd mapping,
which further suggests the weak identification strength in the 1st mapping, i.e. the
disparity between the exact distribution and the normal distribution is also severe.

The advantage of the bootstrap strategy is clear: the bootstrap population is
the given sample, hence the identification strength in the 2nd mapping is known or
recoverable. The only randomness in this mapping comes from the randomness of
bootstrap resampling, and if the number of bootstrap replications is sufficiently large,
this randomness is negligible. Once the identification strength in the 2nd mapping
is recovered, it is used to infer the identification strength in the 1st mapping, since
they are expected to be similar by the bootstrap principle.

1.3

Estimation and Test

With the quantitative definition as well as the advantage of the bootstrap test strategy, detecting weak identification becomes straightforward. By definition, |D| > γ
implies weak identification. D, the distance between the bootstrap distribution and
the standard normal distribution, needs to be estimated.

Draw B i.i.d. observations from F (x) by bootstrap: Xi =

∗i −θ̂
θ̂n
√ n,
σ̂/ n

i = 1, ..., B.

The bootstrap distribution F (x) can be estimated by the empirical c.d.f. F̂ (x) almost

16
surely:

B
B
1 ∑ θ̂n∗i − θ̂n
1 ∑
a.s.
√ ≤ x) → F (x)
1(Xi ≤ x) =
1(
B i=1
B i=1
σ̂/ n

F̂ (x) =

Consequently, D can be estimated almost surely, by the continuous mapping
theorem:
D̂ =

q̂1−α/2 − q̂α/2
a.s.
−1 → D
2z1−α/2

It would be ideal if B could be made infinity. For the given B realizations from
F (x), though B can be arbitrarily large, whether |D| exceeds γ needs to be tested,
and a test serving this purpose is presented next.

Assume the following conditions hold for an IV/GMM model with the conventional estimator θ̂n , associated with standard error

√σ̂ :
n

Assumption 1. There exist {θ̂n∗i , i = 1, ..., B}, the i.i.d. draws of the bootstrapped
estimator θ̂n∗ ; conditional on the sample, the standardized random variable X =

∗ −θ̂
θ̂n
√n
σ̂/ n

has a continuous density function f (x) that is non-zero in a neighborhood of the two
quantiles qα/2 , q1−α/2 , and can be consistently estimated by the non-parametric kernel
p
estimation: fˆ(x) → f (x).

Comments:

1. Under Assumption 1, the joint distribution of the two quantile estimators,
namely, q̂α/2 =

∗
θ̂n,α/2
−θ̂n
√
,
σ̂/ n

q̂1−α/2 =

∗
θ̂n,1−α/2
−θ̂n
√
,
σ̂/ n

is asymptotically normal condi-

tional on the sample, as B → ∞ (see David and Nagaraja (2003)):

√ 
B

∗
θ̂n,α/2
−θ̂n
√
σ̂/ n
∗
θ̂n,1−α/2
−θ̂n
√
σ̂/ n


− qα/2
− q1−α/2







 0  

 ⇒ N   , Ω
0

17


where


Ω=


(1−α/2)α/2
f (qα/2 )2

(α/2)2
f (qα/2 )f (q1−α/2 )

(α/2)2
f (qα/2 )f (q1−α/2 )

(1−α/2)α/2
f (q1−α/2 )2




2. Silverman (1998) provides high level assumptions for the consistency of the
p
non-parametric kernel density estimator, while fˆ(x) → f (x) is directly as-

sumed here for simplicity. As

∗
−θ̂n
θ̂n,α/2
√
σ̂/ n

p

→ qα/2 , f (qα/2 ) is consistently estimated

∗

θ̂
−θ̂n
√
by fˆ( n,α/2
); similarly, f (q1−α/2 ) is consistently estimable. The covariance
σ̂/ n
p

matrix Ω is thus consistently estimable: there exists Ω̂ → Ω. The normal kernel and Silverman’s rule of thumb for choosing the bandwidth are used in the
empirical application and simulation studies of this chapter.
Theorem. Under Assumption 1, and conditional on the sample, the following result
holds as B → ∞:
√

B(D̂ − D) ⇒ N (0,

Ω11 + Ω22 − 2Ω12
)
2
4z1−α/2

(1.5)

where Ωi,j is the element of Ω at row i, column j.

The quantitative definition of strong and weak identification implies the following
decision rule: reject the null of strong identification when |D| > γ. There are
two cases that would induce the rejection, D > γ and D < −γ. Consequently,
strong identification is rejected when D̂ is significantly greater than γ (Case 1), or
significantly less than −γ (Case 2) in the test statistics below.
Case 1: strong identification is rejected at 5% if

b1 =

√

B

D̂ − γ
2
[(Ω̂11 + Ω̂22 − 2Ω̂12 )/(4z1−α/2
)]1/2

> z95%

18
Case 2: strong identification is rejected at 5% if

b2 =

√

B

D̂ + γ
2
[(Ω̂11 + Ω̂22 − 2Ω̂12 )/(4z1−α/2
)]1/2

< −z95%

Combining these two cases, reject strong identification at 5% if

√
|D̂| > γ + z95%

Ω̂11 + Ω̂22 − 2Ω̂12
2
4Bz1−α/2

As B → ∞, the test above ends up with a rule of thumb: reject strong identification if |D̂| > γ, hence this test can be substituted by the rule of thumb under
sufficiently large B. From now on, this test is referred to as the b test, since it is
based on the bootstrap.

1.4

IV and Bootstrap

In this section, a linear IV regression model is used as a specific example to further
illustrate the bootstrap approach for detecting weak identification, with an application to Card (1995). Most of the analytical results are well known, for example, the
convergence results in (1.1) and (1.2) under mild conditions, and proofs of the listed
results are attached in the appendix. The main objective of this section is to show
that the difference between the bootstrap distribution and the normal distribution
is a suitable proxy for the difference between the exact distribution and the normal
distribution, hence it is a reasonable indicator of the identification strength.

19

1.4.1

Model Setup


 Y = Xθ + U

 X = ZΠ + V

Y = (Y1 , ..., Yn )′ , X = (X1 , ..., Xn )′ are n × 1 vectors of endogenous observations, and
Z = (Z1 , ..., Zn )′ is the n × k matrix of instruments, k ≥ 1. U = (u1 , ..., un )′ , V =
(v1 , ..., vn )′ , where the error term (ui , vi )′ , i = 1, ..., n, is assumed
to have mean

 zero,
2
ρσu σv 
 σu
and to be i.i.d., homoscedastic with covariance matrix Σ = 
. The
2
ρσu σv
σv
parameter of interest is θ, and θ̂n is the IV estimator of θ:
θ̂n = (X ′ Pz X)−1 X ′ Pz Y

It is central to derive the distribution of θ̂n for statistical inference. The exact finite sample distribution of θ̂n , however, is unknown without making further
distributional assumptions. Instead, two alternative methods are often used to approximate the exact distribution in econometric applications: the limiting normal
distribution, and the bootstrap distribution.
Under the conventional asymptotic theory where the k × 1 vector Π is modeled
as non-zero and fixed, the IV estimator θ̂n is asymptotically normally distributed as
the sample size n gets large. In contrast, to explore the distribution of θ̂n when the
instruments are only weakly related to the endogenous variable, Staiger and Stock
(1997) develop weak instrument asymptotics, i.e. Π is modeled as local to zero.
Assumption 2. (a) Π = Π0 ̸= 0, and Π0 is fixed; (a’) Π = Πn =
fixed.

C
√
,
n

and C is

20
The asymptotics under Assumption 2(a) are called Strong Instrument Asymptotics, and the asymptotics under Assumption 2(a’) are called Weak Instrument
Asymptotics.

(a)(a’) are two alternative rank conditions, and in (a’) the rank

condition is only weakly satisfied. The following results and notations are used:
′

p

′

Z ′ Z/n → Qzz ≡ E(Zi′ Zi ), ( Z√nU , Z√nV ) ⇒ (Ψzu , Ψzv ), and (Ψzu , Ψzv )′ is distributed
N (0, Σ ⊗ Qzz ). The validity of these results follows from law of large numbers and a
central limit theorem, after assuming the existence of second moments. By the similar derivation as in Staiger and Stock (1997), the following two well-known results
hold.

Under Strong Instrument Asymptotics:
√

n(θ̂n − θ) ⇒ (Π′0 Qzz Π0 )−1 Π′0 Ψzu ∼ N (0, (Π′0 Qzz Π0 )−1 σu2 )

(1.6)

Under Weak Instrument Asymptotics:
−1
′ −1
θ̂n − θ ⇒ [(Qzz C + Ψzv )′ Q−1
zz (Qzz C + Ψzv )] (Qzz C + Ψzv ) Qzz Ψzu

(1.7)

If k = 1, i.e. the model is exactly identified, then (1.7) reduces to:
θ̂n − θ ⇒ (Qzz C + Ψzv )−1 Ψzu

The conventional result of (1.6) indicates that when instruments are strong, the
IV estimator θ̂n is both consistent and asymptotically normally distributed. In contrast, the result of (1.7) indicates that the IV estimator θ̂n is neither consistent nor
asymptotically normally distributed, if the rank condition is weak. As the magni-

21
tude of Π increases, however, the distribution of θ̂n in (1.7) gets closer to the normal
√
distribution in (1.6). In the extreme case that C = nΠ0 , (1.6) and (1.7) coincide.
Π, the vector of nuisance parameters, is thus the driving force of the linear IV regression model: it determines whether θ can be consistently estimated, and whether
the estimator θ̂n can be well approximated by the normal distribution.
As a function of Π, the concentration parameter µ2 is a unit-less measure of the
identification strength in the studies of weak instruments:

µ2 =

Π′ Z ′ ZΠ
σv2

The greater µ2 , the stronger the identification of the parameter θ, and the distribution of θ̂n gets closer to the normal distribution, as shown in Rothenberg (1984).
In addition, Stock and Yogo (2005) suggest that there is a threshold of the concentration parameter for the set of weak instruments, i.e. instruments as well as
identification are considered weak if µ2 /k is under the threshold. The first stage
F test is suggested in Stock and Yogo (2005) to check whether the threshold is exceeded: if the F statistic is greater than the tabled critical values, typically around
10 for small k, then instruments as well as identification are not weak:

F =

Π̂′n Z ′ Z Π̂n /k
σ̂v2

where Π̂n = (Z ′ Z)−1 Z ′ X, σ̂v2 = (X − Z Π̂n )′ (X − Z Π̂n )/(n − k).
As an alternative to the limiting normal distribution, the bootstrap provides
another way of approximating the distribution of θ̂n . For the linear IV regression
model under homoscedasticity, the residual bootstrap is a commonly used bootstrap
method. See, for example, Moreira et al. (2009). This bootstrap procedure or algo-

22
rithm is described as follows.

1. Û , V̂ are the residuals induced by θ̂n , Π̂n in the linear IV model:
Û = Y − X θ̂n , V̂ = X − Z Π̂n

2. Re-center Û , V̂ to yield Ũ , Ṽ , by pre-multiplying a constant matrix Me , where
Me = In − Pe , and e is the n by 1 vector of ones:

Ũ = Me Û , Ṽ = Me V̂

3. Sampling the rows of (Ũ , Ṽ ) and Z independently n times with replacement,
and let (U ∗ , V ∗ ) and Z ∗ denote the outcome. The dependent variables (X ∗ , Y ∗ )
are generated by:


 Y ∗ = X ∗ θ̂n + U ∗

 X ∗ = Z ∗ Π̂n + V ∗
4. As the counterpart of the IV estimator θ̂n , the bootstrapped IV estimator θ̂n∗
is computed by the bootstrap resample (X ∗ , Y ∗ , Z ∗ ):
′

′

θ̂n∗ = (X ∗ Pz∗ X ∗ )−1 X ∗ Pz∗ Y ∗

5. Re-do Steps 2-4 B times, and {θ̂n∗i , i = 1, ..., B} are B i.i.d. estimators.

The bootstrap data generation process (D.G.P.) above aims to mimic the D.G.P.
of the linear IV regression model: when instruments are strong, the equation Π̂n =
Π+Op (n−1/2 ) indicates that Π̂n is not substantially different from Π; in addition, the

23
variance of the bootstrap error term (u∗i , vi∗ ) converges to Σ, the variance of (ui , vi ).
Consequently, it is natural to expect that the mimicking process works well under
strong instruments, and the distributions of θ̂n∗ and θ̂n are alike. This conjecture on
the bootstrapped estimator θ̂n∗ is confirmed by the result below.
Under Strong Instrument Asymptotics:
√

′

′

n(θ̂n∗ − θ̂n ) ⇒ (Π0 Qzz Π0 )−1 Π0 Ψzu

(1.8)

The result of (1.8) motivates the usage of the bootstrap as a tool to detect the
√
identification strength: under strong identification, the distribution of n(θ̂n∗ − θ̂n )
√
is asymptotically identical to the distribution of n(θ̂n − θ), and the asymptotic
√
distribution is normal; if the distribution of n(θ̂n∗ − θ̂n ) is found to be substantially
different from normal, then it indicates that identification is weak.

The bootstrap mimicking process also helps explain why the bootstrap becomes
problematic when instruments are weak: first of all, if Π is local to zero, then the
connection Π̂n = Π + Op (n−1/2 ) implies that the difference between Π̂n and Π becomes substantial, hence the identification strength in the bootstrap resample is
substantially different from the identification strength in the sample; secondly, when
θ̂n does not consistently estimate θ under weak instruments, the residual Û does not
converge to the error term U , since Û = U − X(θ̂n − θ). Both of these two facts
contribute to that the bootstrap D.G.P. does not mimic the D.G.P. of the linear IV
model well under weak instruments, and consequently, the distributions of θ̂n∗ and θ̂n
are different.

24
To compare the identification strength in the bootstrap resample (X ∗ , Y ∗ , Z ∗ )
with the identification strength in the sample (X, Y, Z), consider the concentration
parameter µ2∗ , the bootstrap counterpart of µ2 :

′

2∗

µ

′

′

Π̂n Z ∗ Z ∗ Π̂n
V∗V∗
2∗
=
, where σv =
σv2∗
n

Under Strong Instrument Asymptotics:

µ2 → ∞, and µ2∗ → ∞

(1.9)

Under Weak Instrument Asymptotics:
′

C ′ Qzz C
C ′ Qzz C + 2C ′ Ψzv + Ψzv Q−1
zz Ψzv
2∗
µ →
,
and
µ
⇒
2
2
σv
σv
2 p

(1.10)

The result of (1.9) states that when identification is strong, µ2 and µ2∗ go to
infinity, implying that the distributions of θ̂n and θ̂n∗ are both asymptotically normal, as stated in (1.6) and (1.8). On the contrary, the result of (1.10) states that
when identification is weak, µ2∗ and µ2 do not go to infinity, and are asymptotically
different, which implies that the distributions of θ̂n and θ̂n∗ are both asymptotically
non-normal, and their asymptotical distributions are not identical. The asymptotic
difference in µ2 and µ2∗ is

′

2C ′ Ψzv +Ψzv Q−1
zz Ψzv
:
σv2

under homoscedasticity, it has mean k.

Table 1.1 reports the difference between µ2 and µ2∗ when k = 1 by Monte Carlo
studies: the relative difference is found to be substantial when µ2 is small, and
negligible when µ2 is large; overall, µ2∗ is greater than µ2 . Another interpretation of
(1.10) is that, loosely speaking, the F statistic of the bootstrap resample is above the

25
F statistic of the sample by 1, because of the relation E(F ) ≈ µ2 /k +1 in Stock et al.
(2002).
The comparison of µ2∗ and µ2 indicates a useful result in the linear IV model
where µ2 /k and µ2∗ /k are measures of the identification strength: on average, the
identification strength in the bootstrap resample is similar to, and slightly stronger
(plus 1) than the identification strength in the sample; consequently, if the identification strength in the bootstrap resample is weak, then the identification strength
in the sample must also be weak. In this sense, the proposed bootstrap strategy for
detecting weak identification is conservative.

The Edgeworth expansion provides another look at the bootstrap strategy for detecting weak identification. For the purpose of this chapter, it suffices to consider the
two-term expansion of the standardized estimator

θ̂n√
−θ
,
σ/ n

where σ 2 = (Π′0 Qzz Π0 )−1 σu2 .

Define Ri = (Zi Xi , Zi Yi , vec(Zi′ Zi )′ )′ , µ = E(Vi ) = (Qzz Π, Qzz Πβ, vec(Qzz )′ )′ .
∑
√
θ̂n√
−θ
Rewrite σ/
in the form of nA(R̄), where R̄ = n1 ni=1 Ri , and A(µ) = 0. The
n
following result is the application of the smooth function model and Theorem 2.2 in
Hall (1992). A similar result is available in Moreira et al. (2009).
Theorem. Under Assumption 2(a), and assume two conditions: (i) E(||Ri ||3 ) < ∞,
(ii) lim sup||t||→∞ |Eexp(it′ Ri )| < 1,

θ̂n√
−θ
σ/ n

and its bootstrap counterpart admit two-

term Edgeworth expansions uniformly in x:
θ̂n − θ
√ ≤ x) = Φ(x) + n−1/2 p(x)ϕ(x) + o(n−1/2 )
σ/ n

(1.11)

θ̂n∗ − θ̂n
√ ≤ x) = Φ(x) + n−1/2 p∗ (x)ϕ(x) + o(n−1/2 )
σ̂/ n

(1.12)

P(
P(

where p(x) is a polynomial of degree 2, with coefficients depending on θ, Π, and moments of Ri up to order 3, p∗ (x) is the bootstrap counterpart of p(x) with coefficients

26
depending on θ̂n , Π̂n , and moments of Ri∗ .

Figure 1.2 is drawn based on the results of the Edgeworth expansion. The distance between the exact distribution of

θ̂n√
−θ
σ/ n

and the standard normal distribution

has order O(n−1/2 ), which is the same as the order of the distance between the bootstrap distribution of

∗ −θ̂
θ̂n
√n
σ̂/ n

and the standard normal distribution. Compared with

the normal distribution, the bootstrap distribution has the well known property of
asymptotic refinement: it is closer to the the exact distribution, as the distance has
order O(n−1 ), which results from p∗ (x) − p(x) = O(n−1/2 ): θ̂n , Π̂n , and moments of
Ri∗ approach θ, Π, and moments of Ri at rate n−1/2 .
As discussed above, the bootstrap strategy for detecting weak identification is to
use the distance between the bootstrap distribution and the normal as a proxy of the
distance between the exact distribution and the normal distribution. This strategy is
supported by the Edgeworth expansion in two ways: firstly, these two distances have
the same order O(n−1/2 ); secondly, the price of using the proxy is low, i.e. the proxy
∗

θ̂
θ̂n√
−θ
n−
√ n ≤ x) − Φ(x)] = O(n−1 ).
≤ x) − Φ(x)] − [P ( θ̂σ̂/
error is O(n−1 ), because [P ( σ/
n
n

To summarize, the bootstrap distribution is a good proxy for the exact distribution for the purpose of this chapter. Under the null hypothesis of strong identification, the disparity between the exact distribution and the normal distribution
is reflected by the disparity between the bootstrap distribution and the normal distribution. If the bootstrap distribution is found to be substantially different from
normal, then it is appropriate to conclude that the exact distribution is substantially
different from normal as well, hence the identification strength is weak.

So far, the discussion is restricted to the linear IV model under the homoscedasticity assumption. However, it is well understood that this assumption is unlikely

27
to hold in practice: for example, E(u2i |Zi ) may not be constant, but depend on Zi .
As a result, it is common that empirical researchers need to take the existence of
heteroscedasticity into consideration.

Once the homoscedasticity assumption is loosened, the validity of the popular F
test for detecting weak instruments/identification is under doubt: the way in which
Stock and Yogo (2005) derive the critical values of the F test crucially depends
on the homoscedasticity assumption. Consequently, it is not clear whether these
critical values can still be used when heteroscedasticity instead of homoscedasticity
takes place.

The validity of the bootstrap test approach, on the contrary, stays unaffected,
when the pair bootstrap replaces the residual bootstrap. Freedman (1984) shows
that the pair bootstrap of the IV estimator remains valid under heteroscedasticity:
the IV estimator and its bootstrap counterpart asymptotically have the same normal
distribution under strong instruments. The idea of the pair bootstrap is to directly
resample the data. In the special case of k = 1, the pair bootstrap is to draw
the bootstrap resample (X ∗ , Y ∗ , Z ∗ ) from the empirical distribution of (X, Y, Z).
Compared with the residual bootstrap above, the pair bootstrap is non-parametric,
and it preserves the possible heteroscedastic relations in the IV model, as proved in
Freedman (1984). Consequently, if heteroscedasticity is concerned, the bootstrap test
procedure involves two minor modifications: (i) use the pair bootstrap to compute
θ̂n∗ ; (ii) use a heteroscedasticity-robust estimator for σ̂.

28

1.4.2

An Empirical Example: Card (1995)

In this section, an empirical application is investigated to illustrate the bootstrap
approach for detecting weak identification. The same data as in Card (1995) is used.
By employing the IV approach, Card (1995) answers the following question: what is
the return to education? Or specifically, how much more can an individual earn if
he/she completes an extra year of schooling?

The dataset is ultimately taken from the National Longitudinal Survey of Young
Men between 1966-1981 with 3010 observations, and there are two variables in the
dataset that measure college proximity: nearc2 and nearc4, both are dummy variables, and are 1 if there is a 2-year, 4-year college in the local area respectively.
See Card (1995) for the detailed description of the data. To identify the return to
education, Card (1995) considers a structural wage equation as follows:
lwage = α + θedu + W ′ β + u

where lwage is the log of wage, edu is the years of schooling, the covariate vector W
contains the control variables, and u is the error term. Among the set of parameters
(α, θ, β ′ ), θ measuring the return to education is of interest.

In the basic specification, Card (1995) uses five control variables: experience,
the square of experience, the dummy for race, the dummy for living in the south,
and the dummy for living in the standard metropolitan statistical area (SMSA).
To bypass the issue that experience is also endogenous, Davidson and MacKinnon
(2010) replace experience, the square of experience with age, the square of age.
Following Davidson and MacKinnon (2010), I used age, square of age, and the three
dummy variables as control variables, hence edu is the only endogenous regressor. In

29
addition, an extra instrument nearc2 ∧ nearc4 is constructed: nearc2 ∧ nearc4 is a
dummy variable, and is 1 if there are both a 2-year and a 4-year college in the local
area. Unlike Davidson and MacKinnon (2010), I use the three instruments, nearc2,
nearc2 ∧ nearc4, nearc4, one by one as the single instrument for edu to better
illustrate the approach discussed in this chapter, while Davidson and MacKinnon
(2010) simultaneously use more than one instrument.

The identification strength under the three potential IV’s is examined by the
first stage F test in Stock and Yogo (2005): if nearc2 is used as the IV, F ≈ 0.54; if
nearc2 ∧ nearc4 is used as the IV, F ≈ 6.98; if nearc4 is used as the IV, F ≈ 10.22.
According to the rule of thumb F > 10 suggested in Stock and Yogo (2005), these
F statistics suggest that nearc4 is a strong IV, nearc2, nearc2 ∧ nearc4 are not.
Table 1.2 reports the point estimate and 95% confidence interval of θ under each
of these instruments. If the point estimate of the return to education is of interest,
0.0936 derived by nearc4 is more reliable, compared with the other point estimates.
Based on these empirical results, an additional year of education increases the wage
by about 9.36%; however, the possibility that this effect is zero can not be rejected
at 95%.

As proposed in this chapter, the bootstrap can help evaluate the identification
strength. To allow for heteroscedasticity, the pair bootstrap is employed, and the
IV estimator of θ is computed B = 9999 times, using the three instruments one
by one: specifically, the bootstrap resample is directly drawn from the sample with
replacement, and the size of the resample is equal to the number of observations; the
bootstrapped IV estimator is computed by each bootstrap resample, and this process
is replicated B times. To make it comparable to the standard normal variate, the
bootstrapped estimator is standardized by subtracting the IV estimate and dividing
by the standard error of the IV estimator, where the standard error is computed in

30
the way of White (1980). The p.d.f and quantiles of the bootstrapped IV estimator
after standardization are plotted against the standard normal variate in Figure 1.3.
Figure 1.3 shows that, the bootstrap distribution is closer to the normal distribution,
when the instrument is stronger. The figure of the bootstrap distribution and the
Q-Q plot7 hence are useful tools to help detect weak identification, since they provide
empirical researchers a graphic evaluation of the identification strength.

Table 1.2 reports the 95% C.I. of the return to education by the bootstrap
percentile method, in addition, the identification-robust conditional likelihood ratio (CLR)8 test by Moreira (2003) is also applied to construct a C.I. for comparison.
Instead of the F test, the proposed b test is applied to determine whether the strength
of identification is strong or weak, with the tentative threshold γ = 0.25.

nearc2: the bootstrap C.I. of the return to education is (−4.3279, 4.8868), while
the C.I. by t/Wald is (−0.8188, 1.8346); the relative difference is significantly larger
than the threshold with D̂ ≈ 2.46 > γ, b1 ≈ 12.09 > z95% , hence the null of strong
identification under nearc2 is rejected.
nearc2 ∧ nearc4: the bootstrap C.I. of the return to education is found to be
(0.0019, 0.4664), while the C.I. by t/Wald is (−0.0099, 0.2692); the relative difference
is significantly larger than the threshold with D̂ ≈ 0.66 > γ, b1 ≈ 6.36 > z95% , hence
the null of strong identification under nearc2 ∧ nearc4 is rejected.
nearc4: the bootstrap C.I. of the return to education is found to be (0.0034, 0.2579),
while the C.I. by t/Wald is (−0.0027, 0.1899); the relative difference is significantly
larger than the threshold with D̂ ≈ 0.32 > γ, b1 ≈ 2.71 > z95% , hence the null of
7

Compared to the p.d.f., the Q-Q plot is known to be a better way of comparing distributions.
Note: The IV model under consideration is just identified, hence CLR is equivalent to the AR
test in Anderson and Rubin (1949) and K test in Kleibergen (2002).
8

31
strong identification under nearc4 is rejected.

To conclude, both the F test and the bootstrap based b test can detect the weak
instrument nearc2, nearc2∧nearc4 and support the view that nearc4 is the strongest
among the three potential instruments. The difference is, the b test considers nearc4
as weak, while the F test treats nearc4 as strong (but just across the threshold).
It is surprising that, although F > 10 holds under nearc4, the relative difference of
the C.I. by t/Wald and the bootstrap C.I. is as large as 0.32. It thus indicates that
the F > 10 rule is not strict enough, i.e. the disparity between the two C.I.’s is still
severe, although F > 10 holds in this example.

The b and F tests are proposed on different grounds, i.e. the b test is based on
the comparison of the length of confidence intervals, while the F test is based on the
threshold of the rank condition, hence it is not surprising that these two tests can
imply different outcomes. In contrast with the F test, the b test has its advantages:
firstly of all, it provides a graphic view of the identification strength, i.e. the Q-Q
plot in Figure 1.3 shows that the bootstrap distribution is not very close to the
normal distribution, hence the identification strength appears weak; secondly, unlike
the F test, the b test does not rely on the restrictive homoscedasticity assumption,
and has the potential of being extended to the generalized GMM framework.

1.5

Simulation

This section presents Monte Carlo results to evaluate the power of the proposed
b test. The disparity between the two intervals, (1.3) and (1.4), is reported. The
threshold γ is calibrated to the F > 10 rule.

32
The linear IV model described inSection
employed
in the D.G.P.,
 with
 1.4.1 is 
  
 ui 
 0   1 ρ 
the following choice of parameters: 
 ∼ N ID   , 
, where
vi
0
ρ 1
ρ ∈ {0.99, 0.50, 0.01} to introduce high, moderate and low degrees of endogeneity,
respectively; θ = 0; zi ∼ N ID(0, 1), i = 1, ..., n, and n = 1000.
A sequence of µ2 , µ2 ∈ {2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 40, 60, 80}, is chosen by
assigning different values to Π. For each µ2 , the data of (X, Y, Z) is generated by
the linear IV model with the parameters specified above, and the bootstrapped IV
estimator θ̂n∗ is computed B = 9999 times by the residual bootstrap. The number of
replications equals 1000.

The results of the Monte Carlo studies are reported in Table 1.3.

C0 vs. Ct : C0 denotes the interval derived by taking the 2.5%, 97.5% quantiles of
θ̂n . Ct is the 95% C.I. derived by inverting the t test. Table 1.3 reports the relative
difference in lengths of these two intervals: the median absolute difference in the
lengths of C0 , Ct weighted by the length of Ct is reported. The departure of Ct from
C0 is more severe for larger ρ, and as expected, the departure of Ct from C0 shrinks
as µ2 increases. It is found that the relative difference can be as high as 0.62 when
µ2 = 10. This is a bit surprising since the identification strength under µ2 = 10 is
generally not considered as very weak: as reported in the table, there is about half
chance that the F test will consider the identification under µ2 = 10 as strong. In
other words, although the F test and its decision rule may treat the identification
strength under µ2 = 10 as strong, the departure of Ct from C0 can still be severe.

Cb vs. Ct : the 95% C.I. Cb by the bootstrap is compared with Ct . Table 1.3
reports the median absolute value of the relative difference in the lengths of Cb , Ct .

33
As expected, the difference shrinks as µ2 increases. In particular, when µ2 > 10, the
relative difference does not exceed 0.50 in absolute value; when µ2 ≥ 20, the relative
difference does not exceed 0.25 in absolute value; when µ2 ≥ 80, the difference is
negligible.

b test: the proposed b test is applied to test the null of strong identification, and
the percentage of rejecting strong identification is reported. Two tentative cutoffs,
γ = 0.25, 0.50 are considered. Under the stricter rule of γ = 0.25, the b test rejects
the null more often.

F test: the F test and its decision rule in Stock and Yogo (2005) are applied to
provide a benchmark for the b test. Table 1.3 reports the percentage of concluding
weak identification by the F test (the frequency of the F statistic is less than its
critical value) for different ρ’s. As discussed above, the F test examines whether
µ2 /k exceeds the cutoff, hence it does not depend on ρ, the degree of endogeneity.

By comparing the performance of the b test with the F test, γ = 0.25 appears
to be a reasonable choice. It corresponds to µ2 around 10 with endogeneity close
to zero. With this threshold, the frequency of concluding weak identification by the
b test is comparable to the F test, although b rejects strong identification slightly
more often. This chapter hence suggests γ = 0.25 as the quantitative threshold for
distinguishing strong and weak identification, and this simple rule is not off the mark
for practical reasons.

34

1.6

Conclusion

This chapter suggests that the bootstrap is a useful tool for detecting weak identification in IV/GMM applications. The distinguishing feature of the bootstrap based
approach is that it provides a graphic view of the identification strength. By eyeballing the graph of the bootstrap distribution, and comparing it with the normal
distribution through the Q-Q plot, empirical researchers can evaluate whether or
not weak identification exists. The underlying reason is simple: strong identification
implies the bootstrap distribution is close to, and asymptotically identical to the
normal distribution.

A quantitative threshold for distinguishing strong and weak identification is suggested based on the comparison of two C.I.’s for the parameter of interest: the C.I.
by inverting the t/Wald test and the bootstrap percentile C.I.. The difference of
these two C.I.’s boils down to the difference between the bootstrap distribution and
the normal distribution, and exceeding the threshold implies that the relative difference of the two C.I.’s is at least as large as a quarter. For practical purposes,
the identification strength is considered as weak in this chapter once this threshold is exceeded. Monte Carlo experiments show that this threshold is comparable
to and slightly stricter than the F > 10 rule of thumb in Stock and Yogo (2005).
Even in the i.i.d. and homoscedasticity setting, F > 10 is found to be not strict
enough, when it comes to the comparison of C.I.’s: the relative difference of the two
commonly used C.I.’s named above can be very large, even when F > 10 holds; an
application to Card (1995) also makes the same point.

35
Figure 1.1: Three Related Distributions

Normal Distribution

Exact Distribution

Bootstrap Distribution

Figure 1.2: Expressions and Distances

Notes: The normal distribution and the bootstrap distribution are two approximations to
θ̂n√
−θ
the exact distribution of IV/GMM estimators. If the exact distribution of σ/
is well
n
approximated by the normal distribution N (0, 1), then the bootstrap distribution of
∗ −θ̂
θ̂n
√n
σ̂/ n

is well approximated by N (0, 1) as well, because of the same magnitude O(n−1/2 )

of the approximation error. A substantial difference between
the difference between
of weak identification.

θ̂n√
−θ
σ/ n

∗ −θ̂
θ̂n
√n
σ̂/ n

and N (0, 1) indicates

and N (0, 1) is also substantial, hence provides the evidence

36
Table 1.1: A Monte Carlo study of µ2 and µ2∗

µ2∗

12

µ2
14

2

4

6

8

10

16

18

20

40

60

80

3.1

5.1

7.1

9.1

11.2 13.2 15.2 17.2 19.2 21.2 41.4 61.5 81.6

Notes: This table compares the concentration parameter µ2 with µ2∗ , the bootstrap
counterpart of µ2 , by a Monte Carlo study. For each µ2 , the data of X, Z, V are
generated by xi = zi Π + vi , where: (i) xi , zi , vi are the ith elements of X, Z, V ; (ii)
zi ∼ N ID(0, 1), vi ∼ N ID(0, 1), i = 1, ..., 1000; (iii) Π is determined by the value of µ2 .
The reported µ2∗ is the sample average of 1000 replications; in each replication, the
residual bootstrap is conducted 1000 times.

Table 1.2: Return to Education

IV
F statistic
θ̂n
95% C.I. by t/Wald
by bootstrap
by AR/CLR/K

nearc2

nearc2 ∧ nearc4

nearc4

0.54
0.5079
(-0.8188, 1.8346)
(-4.3279, 4.8868)
(−∞, −0.1750] ∪ [0.0867, +∞)

6.98
0.1297
(-0.0099, 0.2692)
(0.0019, 0.4664)
[0.0133, 0.5253]

10.22
0.0936
(-0.0027, 0.1899)
(0.0034, 0.2579)
[0.0009, 0.2550]

Notes: This table presents the estimate θ̂n and confidence interval for return to education
using the data of Card (1995). The first stage F statistic is reported for the three
instrumental variables, nearc2, nearc2 ∧ nearc4, nearc4, which are used one by one for
the endogenous years of schooling. The included control variables are age, square of age,
and three dummy variables for race, living in the south, living in the SMSA.

37

Figure 1.3: The bootstrap distribution, p.d.f and Q-Q plot
1

4

0.9

3

Quantiles of Bootstrap Sample

0.8
0.7
0.6
0.5
0.4
0.3

2

1

0

−1

−2

0.2
−3

0.1
0
−4

−3

−2

−1

0

1

2

3

−4
−4

4

1

−3

−2

−1
0
1
Standard Normal Quantiles

2

3

4

−3

−2

−1
0
1
Standard Normal Quantiles

2

3

4

−3

−2

−1
0
1
Standard Normal Quantiles

2

3

4

4

0.9

3

Quantiles of Bootstrap Sample

0.8
0.7
0.6
0.5
0.4
0.3

2

1

0

−1

−2

0.2
−3

0.1
0
−4

−3

−2

−1

0

1

2

3

−4
−4

4

1

4

0.9

3

Quantiles of Bootstrap Sample

0.8
0.7
0.6
0.5
0.4
0.3

2
1
0
−1
−2

0.2
−3

0.1
0
−4

−3

−2

−1

0

1

2

3

4

−4
−4

Notes: The p.d.f. and Q-Q plot of the bootstrap distribution are presented, under three
instrumental variables in the application of Card (1995). 1st row: nearc2 as IV; 2nd
row: nearc2 ∧ nearc4 as IV; 3rd row: nearc4 as IV; Left: p.d.f of the bootstrapped IV
estimator after standardization (dotted) against the standard normal (solid); Right: the
Q-Q plot. 9999 bootstrap replications are conducted.

38

Table 1.3: The performance of b, F for detecting weak identification

2

4

6

8

10

12

ρ = 0.99

5.24

1.64

1.37

0.80

0.62

0.60

0.50

2.62

1.10

0.83

0.62

0.41

0.01

1.90

0.95

0.53

0.39

ρ = 0.99

1.87

1.61

1.10

0.50

1.56

1.06

0.01

1.45

ρ = 0.99

µ2
14

16

18

20

40

60

80

0.50

0.41

0.38

0.37

0.22

0.18

0.14

0.38

0.33

0.29

0.29

0.26

0.15

0.13

0.11

0.32

0.27

0.23

0.21

0.21

0.19

0.12

0.10

0.08

0.86

0.61

0.45

0.37

0.33

0.27

0.23

0.11

0.07

0.05

0.72

0.47

0.37

0.29

0.22

0.19

0.16

0.14

0.07

0.05

0.03

0.97

0.61

0.40

0.29

0.22

0.18

0.15

0.13

0.12

0.05

0.04

0.03

90.1

84.3

75.9

65.4

53.9

40.3

30.4

22.9

14.4

10.1

0.1

0.0

0.0

0.50

83.3

69.8

57.7

43.5

34.0

24.3

16.9

13.6

9.0

5.7

0.0

0.0

0.0

0.01

81.4

68.2

52.3

38.3

27.4

18.2

12.3

7.7

4.7

3.0

0.0

0.0

0.0

ρ = 0.99

96.4

96.0

94.4

89.7

81.7

73.3

63.1

56.6

44.7

35.5

1.6

0.0

0.0

0.50

93.6

87.5

82.6

67.6

60.3

49.5

37.9

29.0

23.8

17.9

0.6

0.0

0.0

0.01

92.2

87.1

77.2

62.2

50.2

39.2

29.5

22.6

15.7

11.5

0.2

0.0

0.0

ρ = 0.99

94.1

82.7

70.5

53.0

43.3

32.1

22.6

16.1

12.5

7.9

0.0

0.0

0.0

0.50

93.9

82.1

66.3

56.9

42.9

32.7

22.4

15.7

12.5

8.2

0.0

0.0

0.0

0.01

93.4

83.7

69.4

54.9

45.7

33.3

22.3

15.2

11.6

7.4

0.0

0.0

0.0

Comparison of C.I.
C0 vs. Ct :

Cb vs. Ct :

Rejection freq. of b
γ = 0.50:

γ = 0.25:

Rejection freq. of F

Notes: This table presents the Monte Carlo results of comparing the bootstrap based b
test proposed in this chapter with the F test of Stock and Yogo (2005). The frequencies
of concluding weak identification by the b test is reported for each µ2 , and the frequencies
of concluding weak identification by the F test is also reported. µ2 is the concentration
parameter, and the greater µ2 is, the stronger the strength of identification; ρ is the
correlation coefficient, and the greater ρ is, the stronger the endogeneity; C0 , Ct and Cb
are the C.I. derived by taking the 2.5%, 97.5% quantiles of θ̂n , the 95% C.I. by inverting
t, the 95% C.I. by the bootstrap respectively, and their relative difference in length (the
median absolute value) is reported; γ is the threshold, identification is considered as weak
by the b test if the relative difference between Ct and Cb exceeds γ.

39

Appendix

Proof. (1.5)

By the joint normal distribution of quantile estimators, the difference of two quantile
estimators is also normally distributed:
√

[(
B

∗
∗
− θ̂n
θ̂n,1−α/2
− θ̂n θ̂n,α/2
√
√
−
σ̂/ n
σ̂/ n

)

]
− (q1−α/2 − qα/2 ) ⇒ N (0, Ω11 + Ω22 − 2Ω12 )

Rewrite the LHS:
√

[(
B

∗
∗
θ̂n,1−α/2
− θ̂n,α/2
√
−1
2z1−α/2 σ̂/ n

)

(
−

)]
q1−α/2 − qα/2
− 1 · 2z1−α/2 ⇒ N (0, Ω11 + Ω22 − 2Ω12 )
2z1−α/2

The result follows after the substitution of D̂ =

∗
∗
θ̂n,1−α/2
−θ̂n,α/2
√
2z1−α/2 σ̂/ n

− 1, D =

Proof. (1.6)
X ′ Pz X
n

=
=

X ′ Z(Z ′ Z)−1 Z ′ X
n
Z ′ Z ′ −1 ′
X ′ Z(Z ′ Z)−1
(Z Z) Z X
n

p

→ Π′0 Qzz Π0
X ′ Pz U
√
n

=

Z ′U
X ′ Z(Z ′ Z)−1 √
n

⇒ Π′0 Ψzu
√

n(θ̂n − θ)

=

(

X ′ Pz X −1 X ′ Pz U
√
)
n
n

⇒ (Π′0 Qzz Π0 )−1 Π′0 Ψzu

q1−α/2 −qα/2
2z1−α/2

− 1.

40

Proof. (1.7)
X ′ Pz X

=
=

X ′ Z Z ′ Z −1 Z ′ X
√ (
) √
n n
n
′
′
ZZ
Z V Z ′ Z −1 Z ′ Z
Z ′V
(
C + √ )′ (
) (
C+ √ )
n
n
n
n
n

⇒ (Qzz C + Ψzv )′ Q−1
zz (Qzz C + Ψzv )

X ′ Pz U

=

X ′ Z Z ′ Z −1 Z ′ U
√ (
) √
n n
n

⇒ (Qzz C + Ψzv )′ Q−1
zz Ψzu
θ̂n − θ

=

(X ′ Pz X)−1 X ′ Pz U

−1
′ −1
⇒ [(Qzz C + Ψzv )′ Q−1
zz (Qzz C + Ψzv )] (Qzz C + Ψzv ) Qzz Ψzu

If exactly identified :
θ̂n − θ

=

(Z ′ X)−1 Z ′ U

⇒ (Qzz C + Ψzv )−1 Ψzu

Proof. (1.8)

Π̂∗n

≡

′

′

(Z ∗ Z ∗ )−1 Z ∗ X ∗
′

=
=

′

Z ∗ Z ∗ −1 Z ∗ V ∗
)
n
n
′
′
′
′
Z ∗ Z ∗ −1 Z ∗ V ∗
Z Z −1 Z V
)
+(
)
Π0 + (
n
n
n
n
Π̂n + (

p

→ Π0
′

Z∗ U ∗
√
n

⇒ Ψzu by Lyapunov’s Central Limit Theorem:

41
′

Let mi = Zi∗ u∗i , rewrite

′

∗ U∗
Z√
n

∑n

=

′
Z ∗ u∗
i=1
√ i i

n

pendent with mean µi = 0, variance σi2 =

=

∑n
mi
i=1
√
.
n

Z ′ Z U˜′ Ũ
n
n ,

By construction, m′i s are indep

and σi2 → Qzz σu2 under Strong Instru-

ment Asymptotics. To verify the Lyapunov’s condition: for 1 > δ > 0, the expected values
∑n
Op (n)
2+δ ] = lim
E[|mi |2+δ ] < ∞, and limn→∞ ∑n 1 2+δ
n→∞ O (n+ nδ ) = 0.
i=1 E[|mi − µi |
(

i=1

σi2 )

p

2

2

Combining the two results above:
√

[
n(θ̂n∗

− θ̂n )

∗′

=

∗ −1 Z

X Z (Z Z )
[

=

∗′

∗

(
′
Π̂∗n
′

′

Z∗ Z∗
n

n
]−1

)
Π̂∗n
′

∗′ Z ∗

⇒ (Π0 Qzz Π0 )−1 Π0 Ψzu

]−1
∗′

∗ −1

(Z Z )

∗′ ∗
∗Z U
Π̂n √

n

∗′

Z X

∗

′

Z∗ U ∗
′
′
X ∗ Z ∗ (Z ∗ Z ∗ )−1 √
n

42
Proof. (1.9)

µ2

=

Π′ Z ′ ZΠ
σv2

=

Op (n)

→ ∞
′

p

p

Π̂n → Π, σv2∗ → σv2 ,

′

2∗

µ

Z∗ Z∗ p
→ Qzz
n
′

=

Π̂n Z ∗ Z ∗ Π̂n
σv2∗

=

Op (n)

→ ∞

Proof. (1.10)

′

µ2

=
p

→

C ′ ZnZ C
σv2
′
C Qzz C
σv2
′

2∗

µ

=

′

′

[C + ( ZnZ )−1 Z√nV ]

′

Z∗ Z∗
n [C
2∗
σv
′

⇒

′

′

+ ( ZnZ )−1 Z√nV ]

C ′ Qzz C + 2C ′ Ψzv + Ψzv Q−1
zz Ψzv
2
σv

CHAPTER

Two

Weak Identification in (C)CAPM

43

44

2.1

Introduction

The Fama-MacBeth (FM) procedure analyzed by Shanken (1992) is widely used in
empirical studies of the (C)CAPM. This two-pass procedure involves a first stage
time series regression to derive β, the correlation matrix of financial assets with risk
factors, and a second stage cross-sectional regression using β estimated from the first
stage as regressors. The R2 at the second stage is typically reported by empirical
researchers to illustrate how much cross-sectional variation of asset returns can be explained by their proposed factors, and it often increases dramatically when a new factor is added: see, e.g., Jagannathan and Wang (1996), Lettau and Ludvigson (2001),
Acharya and Pedersen (2005), Lustig and Van Nieuwerburgh (2005), Li et al. (2006),
Santos and Veronesi (2006), Hansen et al. (2008). Although it is common to evaluate how well models fit real data by looking at the R2 , what is surprising is that
the empirical (C)CAPM literature heavily depends on the usage of the R2 for model
comparison. The various versions of the (C)CAPM with new risk factors are favored,
at least partially due to their large R2 .
This chapter shows that the R2 of the FM two-pass procedure can be large even
when risk factors are irrelevant, hence a large R2 does not imply risk factors are
relevant. The underlying reason is, when factors are irrelevant, the β matrix at the
first stage does not have full rank, which further induces a spurious distribution of the
R2 at the second stage. I call this problem weak identification, because it displays the
similar feature as the weak instrument problem surveyed by Stock et al. (2002): the
validity of the FM procedure crucially depends on the quality of regressors generated
from its first stage; if irrelevant factors are included, the rank condition of the second
stage is violated, hence the outcome of the FM procedure is no longer reliable.

45
I start by taking the influential work by Lettau and Ludvigson (2001) for example to illustrate the main point of this chapter. Lettau and Ludvigson (2001)
report larger values of R2 after adding a conditioning variable, the log consumptionwealth ratio cay, as a new factor. I consider four specifications of the (C)CAPM in
Lettau and Ludvigson (2001), and replace cay with caysim , where caysim is randomly
drawn from a normal distribution; the other settings remain unchanged. Table 2.1
presents the R2 under cay and caysim , and it shows that the R2 under caysim is comparable to the R2 under cay. In other words, the irrelevant risk factor caysim can
also dramatically improve the R2 . In this chapter, I explain why the result in Table
2.1 could happen, by deriving the distribution of the R2 under irrelevant factors; in
addition, I use the rank test of Kleibergen and Paap (2006) as a tool to investigate
whether irrelevant risk factors commonly exist in empirical studies of the (C)CAPM.
Although the formal analysis of the R2 is rare, the inadequacy of the FM twopass procedure has been examined in several aspects. Kan and Zhang (1999) study a
single irrelevant factor model, and provide theoretical results and simulation evidence
to show the existence of bias in the second stage statistical tests and overflation in
the cross-sectional R2 . Employing techniques from the weak identification literature,
Kleibergen (2009) shows that the statistical inference based on the FM two-pass
estimator is misleading when β is equal or close to zero. Lewellen et al. (2010) find
that the R2 of the FM two-pass procedure is not informative and offer suggestions
to improve its performance, but their main point is not on the magnitude of β.
This article extends this discussion by deriving the distribution of the R2 of a linear
multifactor pricing model, and the findings cast doubt on the success of the recent
conditional versions of the (C)CAPM: an increase in the R2 could be a byproduct
of introducing irrelevant risk factors, hence are not strong evidence to support the
conditional (C)CAPM.

46
I evaluate the performances of some recent versions of the (C)CAPM, including
the following: Lettau and Ludvigson (2001), Lustig and Van Nieuwerburgh (2005),
Yogo (2006), Santos and Veronesi (2006), Li et al. (2006). Previous studies indicate
that risk factors proposed in these papers are successful in explaining the crosssectional variation in asset returns. However, I can not rule out these factors are in
fact irrelevant.

The rest of the chapter is organized as follows: the linear factor model is set
up in Section 2; a rank test is suggested in Section 3; the distribution of the crosssectional R2 based on the FM two-pass procedure is derived in Section 4; in section
5, the performances of several versions of the (C)CAPM are evaluated; Section 6
d

concludes. The following notations are used throughout this chapter: “ →” indicates
p

convergence in distribution, and “ →” indicates convergence in probability. For a T ×
n matrix A = (a1 , ..., an ), PA = A(A′ A)−1 A′ , MA = IT − PA , IT is the T × T identity
matrix, ιT is the T ×1 vector of ones, vec(A) = (a′1 , ..., a′n )′ , vecinvn ((a′1 , ..., a′n )′ ) = A,
and ⊗ is the Kronecker operator.

2.2

2.2.1

Model and Fama-MacBeth

Linear Factor Model

The linear factor model is constructed by three equations:

E(Rt ) = ιn λ1 + βλF
cov(Rt , Ft ) = βvar(Ft )
E(Ft ) = µF

(2.1)
(2.2)
(2.3)

47
where excess asset returns Rt , market risk factors Ft , are n by 1, k by 1 vectors
respectively.

Equation (2.1) reflects Sharpe (1964) and Lintner (1965)’s view: in equilibrium
average returns should be priced by the risk measure β, plus the risk free return λ1 .
Equation (2.2) defines β as the n by k correlation matrix of asset returns and risk
factors. See Cochrane (2001) for a book-length discussion.

The unconditional version of CAPM considers the market return as the single factor, hence k = 1; in the unconditional CCAPM, the single factor is nondurable consumption growth. Fama and French (1992)(1993) show that a single
factor model could only explain a small fraction of the total cross-sectional variation. The failings of the unconditional (C)CAPM have induced researchers to
introduce new risk factors. For instance, a conditional version of the (C)CAPM
adds at least a conditioning variable to Ft , hence k ≥ 2. There exists a sizeable empirical literature on the (C)CAPM, suggesting various factors can successfully explain the cross-sectional variation of asset returns, and these factors include: the consumption-to-wealth ratio in Lettau and Ludvigson (2001), the labor
income-to-consumption ratio in Santos and Veronesi (2006), the housing-collateral
ratio in Lustig and Van Nieuwerburgh (2005), the investment growth rate in Li et al.
(2006), the growth in durable consumption in Yogo (2006), the consumption risk in
Parker and Julliard (2005), the housing expenditure in Piazzesi et al. (2007), etc.

Equations (2.1)(2.2)(2.3) imply the following model:

Rt = ιn λ1 + βλF + βvt + ut
= ιn λ1 + β(F̄t + λF ) + ϵt
Ft = µF + vt

48
where ut , vt are n×1, k×1 vectors of errors, F̄t = Ft − T1

∑T
t=1

Ft , ϵt = ut +β T1

∑T
t=1

vt ,

and ut , vt are uncorrelated because of Equation (2.2).

The data observed by empirical researchers are Rt , Ft , t = 1, ..., T . To make
expressions neat and facilitate the econometric analysis of the R2 , define matrices
R, F , and a vector R̄:

Rn×T = (R1 , R2 , ..., RT )
Fk×T = (F1 , F2 , ..., FT )
T
1∑
Rt
R̄n×1 =
T t=1

2.2.2

Fama-MacBeth

The commonly used FM two-pass procedure in empirical studies of the (C)CAPM is
as follows: (i) at the first stage, estimate β in a time series regression, i.e. regressing
Rt on Ft with intercepts; (ii) estimate the risk premium λF using β̂ in a second stage
cross-sectional regression, i.e. regressing R̄ on β̂ with intercepts. Shanken (1992)
provides a detailed econometric analysis of this approach. The expressions of β̂, λ̂F ,
and the R2 of the FM two-pass procedure are given below, where MιT , Mιn are two
projection matrices of constants (see Appendix A):
β̂ = RMιT F ′ (F MιT F ′ )−1
λ̂F = (β̂ ′ Mιn β̂)−1 β̂ ′ Mιn R̄
R2 =

R̄′ Mιn β̂(β̂ ′ Mιn β̂)−1 β̂ ′ Mιn R̄
R̄′ Mιn R̄

(2.4)
(2.5)
(2.6)

Given the setup of the model, a large value of the cross-sectional R2 is expected

49
in empirical studies of the (C)CAPM. This is because the R2 converges to 1 in large
samples (see Appendix B), when the risk factors chosen by empirical researchers
coincide with the factors in the above model. Consequently, a small value of the R2
suggests that the chosen risk factors do not coincide with the factors in the model,
hence are not useful in explaining the variation of asset returns.

2.3

A Rank Test

Note that the validity of the FM two-pass procedure relies on the full rank of the
matrix β: in the second stage regression, the n×k matrix β is the matrix of regressors.
If the rank of β is less than k, the rank condition of the ordinary least squares
estimator is violated, and the validity of the FM two-pass procedure demonstrated
by Shanken (1992) collapses.

When only a single factor is considered, k = 1, then the rank condition fails if this
single factor is irrelevant, which is unlikely to happen. However, as the empirical
literature on the (C)CAPM starts to propose various multifactor models, k ≥ 2,
there is a larger chance that β does not have full rank. For example, if one of the
k factors is irrelevant, then a column of β is zero, hence its rank is reduced; more
generally, if one column of β can be written as a linear combination of the other
columns, then the rank condition fails to hold.

There exist several rank tests to examine the rank condition, given a consistent estimator β̂ of the matrix β is available by the first-pass of the FM procedure: see Anderson (1951), Cragg and Donald (1996), etc.

The rank test of

Kleibergen and Paap (2006) is used in this chapter as this novel test overcomes some

50
deficiencies of other tests: it is robust to heteroscedasticity, while homoscedasticity
is assumed in Anderson (1951); in addition, it is easier for implementation, while the
rank test of Cragg and Donald (1996) involves numerical optimization.

If the rank test of Kleibergen and Paap (2006) suggests that β does not have
full rank, it indicates the violation of the rank condition, and further casts doubt
on the validity of the risk premium estimator and the cross-sectional R2 from the
second-pass of the FM procedure. Specifically, if an irrelevant factor is introduced
by empirical researchers, then the rank of β in the empirical model is not full, hence
the FM two-pass procedure is no longer valid; the rank test of Kleibergen and Paap
(2006) helps determine whether this is the case.

Despite of its importance, the necessity of a rank test has not been recognized in
empirical studies of the (C)CAPM, to the best of my knowledge. I now use the rank
test of Kleibergen and Paap (2006) to show it is helpful for excluding irrelevant risk
factors.

Take a specification of the conditional CAPM from Lettau and Ludvigson (2001)
for example: the log consumption-wealth ratio cay, the value weighted return Rvw ,
and the interaction term cay · Rvw are the three factors. The estimate of β, the
associated t statistic and p value using 25 size and book-to-market sorted portfolios
are presented in Table 2.2. A feature observed from Table 2.2 that has motivated
this chapter is that, Rvw is significantly related to all of the 25 portfolio returns,
while cay is insignificant at 5% except for 2 portfolios only. The small magnitude of
the estimate of β for cay induces the suspicion that the column of β corresponding to
cay is zero, hence β does not have full rank. The result of the Kleibergen and Paap
(2006) rank test is in line with Table 2.2: it tests the null that the rank of the β
matrix (its dimension is 25 by 3 in this example) is 2, and reports a p value around

51
0.77. The large p value implies the failure of rejecting the null at 5% significance
level, indicating that the rank condition is violated.

Table 2.3 presents the rank test results for three specifications of the (C)CAPM
suggested in Lettau and Ludvigson (2001), together with the unconditional (C)CAPM,
and the Fama-French three factor model. I can not reject that the three versions of
the conditional (C)CAPM violate the rank condition, as their p values are large. In
contrast, p values for the unconditional (C)CAPM and the Fama-French three factor
model are all close to zero, indicating that the rank condition is satisfied for these
models.

2.4

Distribution of R2

As discussed above, if at least one risk factor is irrelevant, the rank condition is
not satisfied. In this situation, the FM estimator of risk premium is inconsistent,
as shown in Kleibergen (2009); however, it is not clear how the commonly used
cross-sectional R2 behaves. The presumption may be that irrelevant risk factors
could not dramatically increase the R2 at the second stage. This section shows this
presumption is incorrect: the R2 converges to a random variable if β does not have
full rank, hence it could be large, even when factors are irrelevant.
Kan and Zhang (1999) have analytical results for the distribution of the R2 under
k = 1, which is generalized here to consider k ≥ 1, because multifactor models are
favored in recent empirical studies of the (C)CAPM. I am interested in answering
the following question: if an irrelevant risk factor is used together with some relevant
factors, how may the R2 be affected?

52
To visualize the asymptotic distribution of R2 , Monte Carlo simulations are conducted to draw the densities of the risk premium estimator λ̂F and the R2 . In the
underlying data generating process (D.G.P.), n = 25 as the 25 Fama-French size and
book-to-market sorted portfolios are the focus of many empirical studies. The risk
premium λF is fixed to 1 if it exists, and the variance of errors are obtained from
Lettau and Ludvigson (2001). 10000 replications are conducted. For simplicity, a
three factor model is used in D.G.P. of the Monte Carlo study, unless otherwise
stated.

2.4.1

Assumption

The same assumption as in Kleibergen (2009) is made. Assumption 1 is a statement
of a central limit theorem, which implies that R̄ and β̂ follow two independent normal
distributions as stated in Lemma 1. Proof of this lemma is contained in the appendix
of Kleibergen (2009).

Assumption 1:

√1
T


Lemma 1:









∑T  1 
 d  φR 
⊗
(R
−
ι
λ
−
β(
F̄
+
λ
))



→

t
n
1
t
F
t=1
Ft
φβ






√  R̄ − ιn λ1 − βλF  d  ψR 
T

→
ψβ
vec(β̂ − β)

where ψR (n × 1), ψβ (nk × 1) are two independent normally distributed random variables with mean 0, and covariance matrices Ω, VF−1
⊗Ω. Ω = var(ϵt ), VF F = var(Ft ),
F
′ 


 1   1  
(φ′R , φ′β )′ ∼ N (0, V ), V = Q ⊗ Ω, Q = E 
 .

Ft
Ft

53
Lemma 1 describes the distributions of R̄ and β̂, the main components of the R2
in Equation (2.6). The asymptotic distribution of the R2 is derived in terms of the
distributions of R̄ and β̂.

2.4.2

Four Cases

I consider four cases, depending on the values of β.
.
.
Case 1: If (ιn ..β)′ (ιn ..β) has full rank, and risk factors used in the empirical study
p

coincide with those in the model, then R2 → 1.

In this ideal setting, as the sample size T increases to ∞, λ̂F converges to λF ,
and the R2 goes to 1. See Figure 2.1(a)(b). Figure 2.1(a)(b) present the densities of
λ̂F and R2 as the sample size T increases. Figure 2.1(a) shows that λ̂F shrinks to
the true value λF = 1, and Figure 2.1(b) shows that the cross-sectional R2 converges
to 1.
Case 1 illustrates the foundation of reporting the R2 in practice: if the same risk
factors Ft as in the underlying model are chosen, then the sample R2 tends to be
large. As a result, a large value of the cross-sectional R2 is a positive indicator, since
it suggests that the correct risk factors have been discovered. The R2 thus serves as
a criterion for model comparison in the empirical (C)CAPM literature: risk factors
that yield larger R2 are generally favored.
Not surprisingly, the way of using the R2 to compare models has its pitfalls,
many of which are documented in Lewellen et al. (2010). In this chapter, I focus on
the problem induced by the magnitude of β: the second stage regression fails if β is

54
not large enough, and the R2 consequently becomes a misleading measure. Case 2
describes an extreme case: useless factors can yield large values of the R2 .

d

Case 2: If β = 0 and E(R̄) = c, then R2 →
d

R2 →

′ P
ψR
Mιn Ψβ ψR
′ M ψ
ψR
ιn R

c′ PMιn Ψβ c
c′ Mιn c

. Specially, if c = ιn λ1 , then

, where Ψβ = vecinvk (ψβ ), PMιn Ψβ = Mιn Ψβ (Ψ′β Mιn Ψβ )−1 Ψ′β Mιn .

This is the extreme setting that all k factors included in the model are irrelevant,
in the sense that these factors are uncorrelated with asset returns. Although it is
tempting to think that in this setting the R2 is zero as all factors have no power of
pricing average asset returns, in fact the sample R2 can be close to 1 even in large
samples. See Figure 2.1(c)(d). Figure 2.1(c) shows that λ̂F does not converge to a
point when the sample size increases, instead it stays randomly centered around 0.
Similarly, Figure 2.1(d) shows that the R2 does not converge to 0 or 1, but stays as
a random variable, as Case 2 states.
The behaviors of the R2 in Figure 2.1(b) and 2.1(d) are different: in Figure 2.1(d),
the R2 stays as a random variable, instead of converging to a fixed point. The
randomness of the R2 in the limiting distribution makes it a misleading criterion
for model comparison, i.e. irrelevant factors have a positive probability of yielding
large R2 even in large samples, hence a model made of irrelevant factors always has
a chance of being favored by empirical researchers, if the R2 is the sole criterion of
model comparison.

In practice, it is unlikely that all of the factors used by empirical researchers are
unrelated with asset returns, hence the settings in Case 2 that β is zero for all k
factors are very restrictive. What is possible, however, is that β is sizeable for most
factors, but close to zero for one factor (or more). Case 3 corresponds to the setting

55
that a factor uncorrelated with asset returns is included in the model, together with
several other relevant factors. The question under consideration is, if an irrelevant
factor is introduced into the model as the conditioning variable, what will happen
to the cross-sectional R2 in the conditional (C)CAPM?

.
Case 3: Suppose k1 + 1 factors with β̃ = (βk1 ..0) are chosen in the empirical
√
.
d
model, where βk1 is n × k1 , 0 is n × 1, k1 < k, T (β̃ˆ − (βk1 ..0)) → ψ̃β , Ψβ =
.
vecinvk1 +1 (ψ̃β ) = (Ψk1 ..Ψ1 ), and E(R̄) = c, then:
d

R2 →

c′ Mιn (V1 + V2 + V3 + V4 )Mιn c
c′ Mιn c

where random variables V1 , V2 , V3 , V4 are defined as follows:
V0 ≡ (βk′ 1 Mιn βk1 − βk′ 1 Mιn Ψ1 (Ψ′1 Mιn Ψ1 )−1 Ψ′1 Mιn βk1 )−1
V1 ≡ βk1 V0 βk′ 1
V2 ≡ −βk1 V0 βk′ 1 Mιn Ψ1 (Ψ′1 Mιn Ψ1 )−1 Ψ′1
V3 ≡ −Ψ1 (Ψ′1 Mιn Ψ1 )−1 Ψ′1 Mιn βk1 V0 βk′ 1
V4 ≡ Ψ1 ((Ψ′1 Mιn Ψ1 )−1 + (Ψ′1 Mιn Ψ1 )−1 Ψ′1 Mιn βk1 V0 βk′ 1 Mιn Ψ1 (Ψ′1 Mιn Ψ1 )−1 )Ψ′1

This setting corresponds to the scenario that an irrelevant factor is added to
an empirical model with other k1 relevant factors. Case 3 states that adding an
irrelevant factor causes the limiting distribution of the R2 to be spurious. As shown
by simulation, the value of the R2 could be increased even if the added factor is
irrelevant. See Figure 2.1(e)(f ). The Fama and French (1993) three factor model is
used in the D.G.P. of the simulation for Figure 2.1(e)(f ). A single factor from the
D.G.P., and an extra useless factor are chosen to compute the R2 in Figure 2.1(f ).

56
The useless factor is constructed by a normally distributed variate independent of
other variables. Figure 2.1(e) plots the density of λ̂F corresponding to the useless
factor, and Figure 2.1(f ) shows that the R2 converges to a random variable after a
useless factor is added.

The restriction in Case 3 can be loosened to allow the added factor to be weakly
related to assets, i.e. β corresponding to this added factor is close, but not strictly
equal to zero. This is considered in Case 4. The result is similar: the R2 after
including a nearly irrelevant factor is still random in its limiting distribution.

.
Case 4: Suppose k1 + 1 factors with β̃ = (βk1 .. √bT ) are chosen in the empirical
√
.
d
model, where βk1 is n × k1 , b is n × 1, k1 < k, T (β̃ˆ − (βk1 .. √bT )) → ψ̃β , Ψβ =
.
vecinvk1 +1 (ψ̃β ) = (Ψk1 ..Ψ1 ), and E(R̄) = c, then:
d

R2 →

c′ Mιn (Ṽ1 + Ṽ2 + Ṽ3 + Ṽ4 )Mιn c
c′ Mιn c

where random variables Ṽ1 , Ṽ2 , Ṽ3 , Ṽ4 are defined as follows:
Ψ̃1 ≡ b + Ψ1
Ṽ0 ≡ (βk′ 1 Mιn βk1 − βk′ 1 Mιn Ψ̃1 (Ψ̃′1 Mιn Ψ̃1 )−1 Ψ̃′1 Mιn βk1 )−1
Ṽ1 ≡ βk1 Ṽ0 βk′ 1
Ṽ2 ≡ −βk1 Ṽ0 βk′ 1 Mιn Ψ̃1 (Ψ̃′1 Mιn Ψ̃1 )−1 Ψ̃′1
Ṽ3 ≡ −Ψ̃1 (Ψ′1 Mιn Ψ̃1 )−1 Ψ̃′1 Mιn βk1 Ṽ0 βk′ 1
Ṽ4 ≡ Ψ̃1 ((Ψ′1 Mιn Ψ̃1 )−1 + (Ψ̃′1 Mιn Ψ̃1 )−1 Ψ̃′1 Mιn βk1 Ṽ0 βk′ 1 Mιn Ψ̃1 (Ψ̃′1 Mιn Ψ̃1 )−1 )Ψ̃′1

Case 3 and 4 help explain the success of the conditional versions of the (C)CAPM.
If the conditioning variable has β close to zero, then adding a conditioning variable

57
to the model is similar to adding an irrelevant factor, which causes the R2 to fail
to converge, hence it is not surprising to get large R2 ’s in empirical studies, even
though risk factors are irrelevant. The motivating example presented in Table 2.1 in
the introduction section makes the same point: irrelevant factors can appear useful
in explaining the cross-sectional variation of asset returns.

To summarize, the analytical results in this section formalize the view that the
R2 is a misleading measure for the empirical studies of the (C)CAPM, if the rank
condition on β is not satisfied. With irrelevant or nearly irrelevant factors, it is not
surprising to find a sizeable value of the R2 , because the limiting distribution of the
R2 is spurious.

2.5

Examples of (C)CAPM

In the early part of this chapter, I show that for the specifications of the (C)CAPM
in Lettau and Ludvigson (2001), the possibility of violating the rank condition can
not be ruled out. Does this concern commonly exist in the empirical (C)CAPM
literature, or is it just unique in Lettau and Ludvigson (2001)? To answer this
question, I evaluate several versions of the (C)CAPM in this section by checking
whether the rank condition is satisfied. As discussed above, a large p value of the
Kleibergen and Paap (2006) rank test suggests the violation of the rank condition.

The 25 Fama-French size and book-to-market sorted portfolios are used as assets. Quarterly returns are compounded by monthly returns obtained from Kenneth
French’s web site, together with the Fama-French three factors, Rm-Rf, SMB, and
HML. Details of constructing the portfolios and benchmark factors are available on

58
the web site.
I use the following factors: the nondurable consumption growth △cN dur and
the durable consumption growth △cDur in Yogo (2006), the housing-collateral ratio myf a in Lustig and Van Nieuwerburgh (2005), the labor income-to-consumption
ratio sw in Santos and Veronesi (2006), the investment growth rate in the household sector HHOLDS, the nonfinancial corporate business NFINCO, and the financial cooperations FINAN in Li et al. (2006). The data of these factors are either
directly offered by the authors, or constructed following the descriptions in their
paper. 1952Q1-2001Q4 is the time period within which all of the factors have data
available.

One specification of the (C)CAPM from each paper listed above is used as an
example in Table 4. The p value of the suggested rank test by Kleibergen and Paap
(2006) is reported, together with estimation results based on the ordinary least
squares (OLS), generalized least squares (GLS) (see Lewellen et al. (2010) for the
GLS method).
The OLS R2 in Table 4 is encouraging: all of the models explain at least 41%
of the cross-sectional variation in average portfolio returns, when the FM two-pass
procedure is applied. However, the results of the rank test are discouraging: except
the benchmark Fama-French three factor model, all the other models have large p
values.
The evaluation based on the GLS R2 supports the outcome of the rank test: only
the Fama-French three factor model has a large GLS R2 , while all the other models
have the GLS R2 under 20%. Compared to the OLS R2 , the GLS R2 is much smaller
for most models, except for the Fama-French model. These results hence support

59
the view in Lewellen et al. (2010): the GLS R2 appears to be a better criterion than
the OLS R2 .1

Overall, the violation of the rank condition can not be ruled out. Since introducing an irrelevant or nearly irrelevant factor into the linear multifactor model would
make the R2 spurious, as demonstrated in the last section, the large values of the
OLS R2 based on the FM two-pass procedure are not reliable.

2.6

Conclusion

This chapter cautions that if irrelevant risk factors are included in the empirical
studies of the (C)CAPM, they may appear useful in explaining the cross-sectional
variation of asset returns. With one or more irrelevant risk factors in a linear multifactor model, the full rank condition of the β matrix no longer holds, which further
induces the spurious limiting behavior of the cross-sectional R2 . Consequently, the
value of the sample R2 can be large, even though factors are irrelevant.

From the perspective of the empirical (C)CAPM literature, this chapter highlights the necessity of applying a rank test on the β matrix, which helps make empirical results more reliable. The empirical findings are easy to summarize: I can not
rule out the possibility that some recent versions of the (C)CAPM violate the rank
condition, while the only model that remains trustworthy is the Fama and French
(1993) three factor model.

1

Lewellen, Nagel, and Shanken (2008) also use these factors as examples, and they use a different
time period, 1963-2004, with 30 extra industry portfolios to expand the test assets.

60

Appendix

A. The cross-sectional R2 based on the FM two-pass procedure
Equations (2.1)(2.2)(2.3) imply the following model:

Rt = ιn λ1 + βλF + βvt + ut
Ft = µF + vt
where ut , vt are n × 1, k × 1 vectors of errors, and ut , vt are uncorrelated. The model
implies the following:

Rt = ιn λ1 + β(λF + Ft − µF ) + ut
= (ιn λ1 + βλF − βµF ) + βFt + ut

Use R = (R1 , R2 , ..., RT ), F = (F1 , F2 , ..., FT ), U = (u1 , u2 , ..., uT ):
R = ι′T ⊗ (ιn λ1 + βλF − βµF ) + βF + U
MιT R′ = MιT F ′ β ′ + MιT U ′

Hence the OLS estimator β̂ is:
β̂ = [(F MιT F ′ )−1 F MιT R′ ]′ = RMιT F ′ (F MιT F ′ )−1

61
Similary, λ̂F = (β̂ ′ Mιn β̂)−1 β̂ ′ Mιn R̄, where R̄ =
R2 =

(Mιn β̂ λ̂F )′ (Mιn β̂ λ̂F )
(Mιn R̄)′ (Mιn R̄)
λ̂′F β̂ ′ Mιn β̂ λ̂F
R̄′ Mιn R̄

=

R̄′ Mιn β̂(β̂ ′ Mιn β̂)−1 β̂ ′ Mιn β̂(β̂ ′ Mιn β̂)−1 β̂ ′ Mιn R̄
R̄′ Mιn R̄

=

n β̂

By definition:

=

=

where PMι

1 T
T Σt=1 Rt .

R̄′ Mιn β̂(β̂ ′ Mιn β̂)−1 β̂ ′ Mιn R̄
R̄′ Mιn R̄
′
R̄ PMι β̂ R̄
n

R̄′ Mιn R̄

= Mιn β̂(β̂ ′ Mιn β̂)−1 β̂ ′ Mιn .

B. Proof of Case 1
Start from Equation (2.1):

E(Rt ) = ιn λ1 + βλF
Mιn E(Rt ) = Mιn ιn λ1 + Mιn βλF
Mιn E(Rt ) = Mιn βλF
(Mιn βλF )′ (Mιn βλF )
(Mιn E(Rt ))′ (Mιn E(Rt ))

= 1

p

p

p

In the ideal setting, as T → ∞, β̂ → β, λ̂F → λF and R̄ → E(Rt ), by Slutsky’s
theorem:

R2

=
p

→

(Mιn β̂ λ̂F )′ (Mιn β̂ λ̂F )
(Mιn R̄)′ Mιn R̄
(Mιn βλF )′ (Mιn βλF )
(Mιn E(Rt ))′ (Mιn E(Rt ))

p

→ 1

C. Proof of Case 2
If β = 0 and E(R̄) = c, Lemma 1 reduces to:

62








√  R̄ − c  d  ψR 
T
→

ψβ
vec(β̂)
Hence, the following are true:
p

Mιn R̄ → Mιn c
√
d
T β̂ → vecinvk (ψβ ) = Ψβ

Apply the continuous mapping theorem:

R2

=
d

→
d

→

R̄′ Mιn β̂(β̂ ′ Mιn β̂)−1 β̂ ′ Mιn R̄
R̄′ Mιn R̄
c′ Mιn Ψβ (Ψ′β Mιn Ψβ )−1 Ψ′β Mιn c
c′ Mιn c

c′ PMιn Ψβ c
c′ Mιn c

As a special case, if c = ιn λ1 , then:
√

T Mιn R̄

=

√

T Mιn (R̄ − ιn λ1 )

d

→ M ιn ψ R

Apply the continuous mapping theorem:

R2

=
d

→
d

→

R̄′ Mιn β̂(β̂ ′ Mιn β̂)−1 β̂ ′ Mιn R̄
R̄′ Mιn R̄
′ M Ψ (Ψ′ M Ψ )−1 Ψ′ M ψ
ψR
ιn β
β ιn R
β ιn β
′ M ψ
ψR
ιn R

′ P
ψR
Mιn Ψβ ψR
′
ψ R M ιn ψ R

D. Proof of Case 3
If β̃ = (βk1 : 0) and E(R̄) = c, Lemma 1 reduces to:

63

√ 
T


R̄ − c
ˆ
vec(β̃ − (βk1





 d  ψR 
→

: 0))
ψ̃β

Hence, the following are true:
p

R̄ → c
√

d
ˆ
T (β̃ − (βk1 : 0)) → Ψβ = (Ψk1 : Ψ1 )
√
√
1
1
ˆ
β̃ = (βk1 + Ψk1 / T + o(T − 2 ) : Ψ1 / T + o(T − 2 ))

This gives the expression of R2 as:

R

2

=

=

ˆ
ˆ
ˆ ˆ
R̄′ Mιn β̃(β̃ ′ Mιn β̃)−1 β̃ ′ Mιn R̄
R̄′ Mιn R̄

−1
√
− 12
− 12
′
′
βk1 Mιn Ψ1 / T + o(T )  ˆ
ˆ  βk1 Mιn βk1 + O(T )
R̄′ Mιn β̃ 
 β̃ ′ Mιn R̄
√
1
−2
′
−1
′
Ψ1 Mιn Ψ1 /T + o(T )
Ψ1 Mιn βk1 / T + o(T )
R̄′ Mιn R̄

The formula of inverse of a block matrix states, with SD = A − BD−1 C:


−1

 A B 


C D



= 

−1
SD

−1
−SD
BD−1

−1
−1
−D−1 CSD
D−1 + D−1 CSD
BD−1





Introduce new notations to make expressions neat:


−1

 A B 


C D



= 


A∗

B∗

C ∗ D∗




Apply the formula and rearrange terms:
R

2

=

√
√
1
R̄′ Mιn (βk1 A∗ βk′ 1 + βk1 B ∗ Ψ′1 / T + Ψ1 C ∗ βk′ 1 / T + Ψ1 D∗ Ψ′1 /T + O(T − 2 ))Mιn R̄
R̄′ Mιn R̄

64
Derive asymptotic distributions of the four leading terms above:

A∗ → (βk′ 1 Mιn βk1 − βk′ 1 Mιn Ψ1 (Ψ′1 Mιn Ψ1 )−1 Ψ′1 Mιn βk1 )−1 ≡ V0
d

β1 A∗ βk′ 1
√
β1 B ∗ Ψ′1 / T
√
Ψ1 C ∗ β1′ / T
Ψ1 D∗ Ψ′1 /T

→ βk1 V0 βk′ 1 ≡ V1
d

→ −βk1 V0 βk′ 1 Mιn Ψ1 (Ψ′1 Mιn Ψ1 )−1 Ψ′1 ≡ V2
d

→ −Ψ1 (Ψ′1 Mιn Ψ1 )−1 Ψ′1 Mιn βk1 V0 βk′ 1 ≡ V3
d

→ Ψ1 (Ψ′1 Mιn Ψ1 )−1 (1 + Ψ′1 Mιn βk1 V0 βk′ 1 Mιn Ψ1 (Ψ′1 Mιn Ψ1 )−1 )Ψ′1 ≡ V4
d

Apply the continuous mapping theorem:
d

R2 →

c′ Mιn (V1 + V2 + V3 + V4 )Mιn c
c′ Mιn c

E. Proof of Case 4
If β̃ = (βk1 :

√b )
T

and E(R̄) = c, Lemma 1 reduces to:

√ 
T


R̄ − c
ˆ
vec(β̃ − (βk1





 d  ψR 
→

: √bT ))
ψ̃β

Hence, the following are true:
p

R̄ → c
√

b
d
ˆ
T (β̃ − (βk1 : √ )) → Ψβ = (Ψk1 : Ψ1 )
T
√
√
1
1
ˆ
β̃ = (βk1 + Ψk1 / T + o(T − 2 ) : (b + Ψ1 )/ T + o(T − 2 ))

Define Ψ̃1 ≡ b + Ψ1 , and follow the steps in the proof of Case 3.

65

Table 2.1: A motivating example, R2 under cay and caysim

Specification (4)
(5)
(6)
(7)

cay
0.31
0.31
0.77
0.75

caysim
0.44
0.29
0.72
0.69

Notes: This table presents the values of the cross-section R2 of the Fama-MacBeth
two-pass procedure using the versions of the (C)CAPM in Lettau and Ludvigson (2001).
R2 ’s under cay are identical to those reported in Table 1 of Lettau and Ludvigson (2001),
and the results are replicated using the same data as in Lettau and Ludvigson (2001).
R2 ’s under caysim are the average of 10000 simulations: caysim is randomly drawn from a
normal distribution, with the same mean and variance as cay. In each simulation, we
compute the R2 replacing cay with caysim , and take the average of 10000 simulations.
Each row in this table corresponds to a specification of the (C)CAPM in Table 1 of
Lettau and Ludvigson (2001). For instance, Specification (6) corresponds to a five-factor
version of the (C)CAPM: the log consumption-wealth ratio cay, the value weighted return
Rvw , the labor income growth △y, and the interaction terms cay · Rvw , cay · △y; in the
simulation, we use caysim , Rvw , △y, caysim · Rvw , caysim · △y to generate the R2 under
caysim . This table shows that the irrelevant caysim can increase the R2 , just like cay does.

66

Table 2.2: An example of the β matrix in Lettau and Ludvigson (2001)

25 portfolios
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25

βcay
0.02
0.03
0.02
0.02
-0.01
0.03
0.02
0.01
0.02
-0.04
0.04
0.07
0.03
0.03
-0.02
0.06
0.03
-0.01
-0.01
-0.04
-0.05
0.02
0.04
-0.01
-0.04

cay
t
0.30
0.47
0.30
0.31
-0.09
0.56
0.35
0.30
0.63
-0.83
1.04
2.21
0.87
0.82
-0.47
1.95
1.05
-0.30
-0.48
-0.95
-2.09
0.89
1.44
-0.36
-0.96

p
0.76
0.64
0.76
0.76
0.93
0.57
0.73
0.76
0.53
0.41
0.30
0.03
0.39
0.41
0.64
0.05
0.29
0.77
0.63
0.34
0.04
0.37
0.15
0.72
0.34

βRvw
1.60
1.43
1.32
1.25
1.28
1.54
1.35
1.24
1.14
1.18
1.43
1.21
1.10
1.03
1.09
1.25
1.16
1.05
1.02
1.11
1.04
0.95
0.77
0.83
0.80

Rvw
t
17.17
17.48
17.64
16.93
14.74
21.81
21.97
22.99
21.44
17.49
26.61
27.24
25.72
23.73
17.49
30.45
35.03
31.86
25.52
19.35
30.46
33.89
22.93
24.80
14.81

p
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

cay · Rvw
βcay·Rvw
t
0.21
0.31
0.27
0.45
0.31
0.56
0.08
0.14
0.60
0.92
0.07
0.14
-0.20
-0.44
0.00
0.01
0.29
0.73
0.65
1.29
-0.32
-0.79
0.02
0.05
0.02
0.06
0.47
1.46
-0.01
-0.02
-0.27
-0.89
-0.33
-1.35
0.27
1.10
0.75
2.50
1.05
2.44
0.30
1.17
0.19
0.91
-0.25
-1.01
0.10
0.40
0.16
0.39

p
0.76
0.66
0.58
0.89
0.36
0.89
0.66
1.00
0.47
0.20
0.43
0.96
0.95
0.15
0.99
0.37
0.18
0.27
0.01
0.01
0.24
0.36
0.31
0.69
0.70

Notes: This table presents the estimate of the correlation matrix
β = (βcay , βRvw , βcay·Rvw ), and its associated t statistic and p value, from the first stage
time series regression of the Fama-MacBeth two-pass procedure. The three risks factors
are the log consumption-wealth ratio cay, the value weighted return Rvw , and the
interaction term cay · Rvw in Lettau and Ludvigson (2001). The quarterly returns of 25
Fama-French size and book-to-market sorted portfolios are from 1963Q3 to 1998Q3.

67

Table 2.3: p value of the rank test for 6 specifications in Lettau and Ludvigson (2001)

cay
1

Rvw
-0.32

△y

SMB

HML

cay·Rvw

cay·△y

△c

cay·△c

R2
0.01

p value
0.00

0.16

0.00

0.80

0.00

0.31

0.77

0.77

0.55

0.70

0.33

(0.96)

2

0.22
(0.18)

3

4

5

6

1.33

0.47

1.46

(1.59)

(0.10)

(0.12)

-0.52

-0.06

1.14

(3.28)

(1.47)

(0.46)

-0.44

-1.99

0.56

0.34

-0.17

(0.45)

(1.58)

(0.44)

(0.32)

(0.12)

-0.13

0.02

0.06

(0.40)

(0.16)

(0.02)

Notes: This table presents p values of the rank test in Kleibergen and Paap (2006), for 6
specifications of the (C)CAPM. A large p value indicates that the rank condition is likely
violated. These 6 specifications are: the unconditional CAPM is Specification 1, the
unconditional CCAPM is Specification 2, the Fama-French three factor model is
Specification 3, and the other three specifications are from Lettau and Ludvigson (2001).
We use the same data as Lettau and Ludvigson (2001), hence the OLS estimates of the
risk premium and the R2 are identical to Lettau and Ludvigson (2001). Standard errors
with Shanken (1992) correction are in brackets. The quarterly returns of 25 Fama-French
size and book-to-market sorted portfolios are from 1963Q3 to 1998Q3.

68

Table 2.4: p value of the rank test for 5 specifications of (C)CAPM

Model
Fama-French (1993)
OLS
GLS

Lustig-Nieuwerburgh (2005)
OLS
GLS

Li-Vassalou-Xing (2006)
OLS
GLS

RM
-1.06

Factors
SMB
0.46

HML
1.34

(1.30)

(0.11)

(0.15)

-1.50

0.59

1.06

(0.98)

(0.06)

(0.11)

myf a
0.02

△c
0.14

myf a△c
0.05

(0.03)

(0.33)

(0.03)

-0.01

-0.05

0.02

(0.01)

(0.12)

(0.01)

HHOLDS
0.03

NFINCO
0.01

FINAN
0.09

(0.02)

(0.01)

(0.04)

0.01

-0.01

0.01

(0.01)

(0.01)

(0.02)

Santos-Veronesi (2006)
OLS

RM
0.53

sw
-2.38

(1.73)

(1.14)

GLS

-0.99

-0.15

(0.68)

(0.29)

Yogo (2006)
OLS

RM
0.06

△cN dur
0.68

(1.70)

(0.38)

(0.30)

GLS

-1.03

-0.01

-0.03

(0.69)

(0.10)

(0.12)

R2

p value

0.73

0.00

0.79

0.74

0.54

0.18

0.58

0.57

0.19

0.41

0.96

0.04

△cDur
-0.12

0.54

0.78

0.03

Notes: The table presents the OLS, GLS estimates of risk premium, the R2 , and p value
from the rank test of Kleibergen and Paap (2006). 5 specifications of the (C)CAPM are
considered, including the benchmark Fama-French three factor model. For each
specification, the OLS, GLS estimates of the risk premium with the corresponding R2
separated by rows are reported: OLS in the first row, GLS in the second row. Standard
errors with Shanken (1992) correction are in brackets. The quarterly returns of 25
Fama-French size and book-to-market sorted portfolios used for this table are from
1952Q1 to 2001Q4.

69
Figure 2.1: Densities of λ̂F and R2

7

25

6
20
5
15

4

3

10

2
5
1

0
−3

−2

−1

0

1
OLS λF

2

3

4

5

0

0

0.2

25

0.2

20

0.15

15

0.1

10

0.05

5

−1

0
OLS λF

0.8

1

0.6

0.8

1

(b)

0.25

−2

0.6
2

OLS R

(a)

0
−3

0.4

1

2

3

0

0

0.2

0.4
OLS R2

(c)

(d)

0.8

5
4.5

0.7

4
0.6
3.5
0.5

3

0.4

2.5
2

0.3

1.5
0.2
1
0.1
0
−3

0.5
−2

−1

0
OLS λF

(e)

1

2

3

0

0

0.2

0.4

0.6

0.8

1

OLS R2

(f)

Notes: This figure presents the densities of λ̂F and R2 as the sample size T
increases. T = 1400(dotted), 14000(dash-dotted), 70000(dashed), 140000(solid).
Case 1 is in Row 1, where factors are correctly chosen; Case 2 is in Row 2, where
all factors are irrelevant; Case 3 is in Row 3, where one irrelevant factor is chosen
with one relevant factor.

CHAPTER

Three

Does a Technology Shock Increase or Decrease Hours

70

71

3.1
1

Introduction

Structural Vector Autoregressions (SVAR) have recently been employed to investi-

gate the impact of technology shocks on production inputs, e.g. the hours worked.
The empirical findings are conflicting, and there is currently a debate on whether
a positive technology shock would increase or decrease hours: for example, Gali
(1999), Shea (1999), Francis and Ramey (2005) find that a positive technology shock
decreases hours at short horizons, while Christiano et al. (2003) report the increase
in hours after a positive technology shock. The sign of the impact of technology
shocks on hours is of interest, because real business cycle models have the typical
implication that the hours worked will increase after a positive shock to technology,
while the implication of models with sticky prices is often the opposite: see, e.g.
Rebelo (2005).

The widely used identification strategy of the aforementioned empirical studies
in SVAR is to impose the same restriction that only technology shocks have the long
run effect on labor productivity, see e.g., Gali (1999). In addition, it is found that
empirical outcome crucially depends on whether the time series of hours are specified
in levels or in first differences: as stated in Pesavento and Rossi (2005), the decrease
in hours is typically found in the first difference specification of hours, while the
increase is often found in the level specification. Since the series of hours is highly
persistent, and unit root tests are often not powerful enough to determine which
specification should be chosen, Pesavento and Rossi (2005), Gospodinov (2010) etc.
model hours as local-to-unity, an instrument to nest both the level and difference
specifications. However, it has been shown that identification through the longrun restriction becomes weak when the variables that enter the model are highly
1

This chapter is based on a joint project with Sophocles Mavroeidis.

72
persistent, and under the local-to-unity asymptotics, Gospodinov (2010) proves that
the structural parameters as well as the impulse response functions (IRF) of interest
can not be consistently estimated: this failure in estimation is interpreted as a weak
identification problem in Gospodinov (2010), by using the framework of the linear
instrumental variable (IV) regression in Staiger and Stock (1997).

From the weak identification perspective, although a consistent point estimator
may not exist, it is often feasible to construct a confidence interval for the parameter
of interest. In this chapter, we propose two tests to construct such confidence intervals, and we consider these intervals as robust, because no matter whether the true
specification of hours is in levels or in first differences, these intervals would cover the
structural parameter of interest with the probability at least as high as the nominal
coverage rate. Given the robust intervals for structural parameters are derivable,
and the IRF is a function of structural parameters, we consequently construct the
bounds for the IRF to investigate whether a positive technology shock would increase
or decrease the hours worked. The empirical findings are as follows: the impact of a
positive technology shock on hours is likely positive at short horizons, but the 95%
confidence interval of the contemporaneous effect also includes a negative region.

The rest of the chapter contains the following parts: the VAR(p + 1) model is
described in Section 2; our robust approach of deriving the confidence intervals are
presented in Section 3; Section 4 contains the empirical findings; Section 5 concludes.

73

3.2

Model

Following the convention, we use the following notations throughout this chapter: let
lt , ht denote labor productivity and the hours worked respectively; structural shocks
ϵt = (ϵzt , ϵdt )′ , where ϵzt is the technology shock and ϵdt is the non-technology shock; the
object of interest is the IRF

∂ht+j
,
∂ϵzt

i.e. the response of hours to positive technology

shocks, and we would like to explore whether its sign is positive or negative for small
j’s; in the special case that j = 0,

∂ht
∂ϵzt

is the contemporaneous effect of the technology

shock on the hours worked.

3.2.1

VAR

We consider the same setup as in Gospodinov
 etal. (2009), which is an extension
 lt 
of the model in Gospodinov (2010). Ỹt = 
, t = 1, ..., T , are assumed to be
ht
generated by a bivariate VAR(p + 1) model of (3.1) with assumptions i − iv:
Ψ(L)(I − ΦL)Ỹt = ut

where Ψ(L) = I −





(3.1)




 ψ11 (L) ψ12 (L) 
 1 β(ρ − 1) 
i
Ψ
L
=
,
Φ
=



.
i
i=1
ψ21 (L) ψ22 (L)
0
ρ

∑p







2
 σ1 σ12 
 u1,t 
is
i.i.d.
with
covariance
Σ
=
i. ut = 
 and finite fourth


u
2
σ12 σ2
u2,t
moments;

ii. the largest roots of the system are contained in Φ with the conditions that
lt has a unit-root, while ht follows a local-to-unity process, i.e. ρ = 1 + Tc ,

74
and c is a fixed constant: this device nests both the first difference and level
specifications of ht , depending on whether c = 0 or c < 0;
iii. |Ψ(z)| = 0 has roots outside the unit circle and Ψ(z)−1 is one-summable;
iv. there is an off diagonal element β(ρ − 1) in Φ.

Gospodinov et al. (2009) and Gospodinov (2010) both emphasize that it is appropriate to use the model of (3.1) to study the impact of technology shocks on
hours, and the off diagonal element β(ρ − 1) in Φ is crucial: β ̸= 0 allows the low
frequency co-movement between lt and ht in the level specification of ht , and this
co-movement is assumed to be removed by the first difference filter, i.e. when c = 0,
Φ reduces to an identity matrix. Gospodinov et al. (2009) offer arguments for why
β(ρ − 1) needs to be in place. For example, if technology shocks have lasting effects
on the labor market, then the low frequency co-movement between lt and ht may be
plausible. Following Gospodinov et al. (2009), the lower-left element in Φ is set to
zero, to avoid that ht is I(2) when c = 0 or I(1) when c ̸= 0. If β = 0, (3.1) reduces
to the model considered in Gospodinov (2010), hence (3.1) is an extension of the
setup in Gospodinov (2010).

3.2.2

Long Run Restriction








 △lt 
 1 β(1 − ρ)L 
Rewrite (3.1) as A(L)Yt = ut , with A(L) = Ψ(L) 
.
, Yt = 
ht
0
1 − ρL

75
A(L)Yt = ut is thus equivalent to:












 ψ11 (L) ψ12 (L)   1 β(1 − ρ)L   △lt 
 u1,t 



 = 

ψ21 (L) ψ22 (L)
0
1 − ρL
ht
u2,t



Premultiplying the equation above by B0 = 


1
−b21

−b12 
 yields the SVAR
1

B(L)Yt = ϵt , with B(L) = B0 A(L), ϵt = B0 ut :







1
−b21











−b12   ψ11 (L) ψ12 (L)   1 β(1 − ρ)L   △lt  




=

1
ψ21 (L) ψ22 (L)
0
1 − ρL
ht
ϵdt
ϵzt

The identification restriction that ϵdt has no long run effect on lt implies two
different expressions for the structural parameter b12 , depending on whether c = 0
or not:

i. if c ̸= 0:




Let M1 (L) = 


1
−b21





−b12   ψ11 (L) ψ12 (L)   1 β(1 − ρ)L 


, then
1
ψ21 (L) ψ22 (L)
0
1 − ρL

the long run restriction corresponds to that M1 (1) is lower-triangular, i.e.
the upper-right element of the 2 by 2 matrix M1 (1) is zero, implying b12 =
ψ12 (1)+βψ11 (1)
;
ψ22 (1)+βψ21 (1)

ii. if c = 0:






Let M2 (L) = 

1
−b21



−b12   ψ11 (L) ψ12 (L) 
, then the long run restric
ψ21 (L) ψ22 (L)
1

76
tion corresponds to that the upper-right element in M2 (1) is zero, implying
b12 =

ψ12 (1)
.
ψ22 (1)

Gospodinov et al. (2009) use the above discontinuity in the solution of b12 to
explain why the level specification and the first difference specification of ht produce substantially different IRF’s in empirical studies: the IRF’s are functions of the
structural parameters; using a different specification of ht implies a different identification condition for b12 , as described above, hence it is not surprising that IRFs
substantially differ when the specifications of ht differ, if β ̸= 0.
The condition β ̸= 0 is thus crucial to help reconcile the conflicting empirical
results: the discontinuity in the solution of b12 disappears once β = 0 is imposed, i.e.
b12 =

ψ12 (1)
,
ψ22 (1)

b12 by

no matter whether c = 0 or c ̸= 0. Gospodinov (2010) proposes to infer

ψ12 (1)
,
ψ22 (1)

after imposing β = 0; without imposing β = 0, Gospodinov (2010)’s

approach of deriving b12 through

ψ12 (1)
ψ22 (1)

is no longer applicable in the current setup.

In the later part of this chapter, we will propose methods to construct C.I.’s for b12 ,
without assuming β = 0.

3.2.3

Model Simplification

Define Ã(L) = Ψ(L)(I − ΦL), A∗ (L) = L−1 (I − Ã(L)) = A∗0 + A∗1 L + ... + A∗p Lp ,
∑p
∗
A∗∗
i = −
j=i+1 Aj .
The Vector Error Correction (VEC) form of (3.1) is as follows:

∗

△Ỹt = (A (1) − I)Ỹt−1 +

p
∑
i=1

A∗∗
i−1 △Ỹt−i + ut

77
More explicitly:


 △lt = (ψ12 (1) + βψ11 (1)) c ht−1 + ∑p a∗∗ △lt−i + ∑p a∗∗ △ht−i + u1,t
i=1 i−1,11
i=1 i−1,12
T
∑
∑

p
∗∗
 △ht = (ψ22 (1) + βψ21 (1)) c ht−1 + pi=1 a∗∗
i−1,21 △lt−i +
i=1 ai−1,22 △ht−i + u2,t
T
To simplify the VEC form above, project △lt , △ht and ht−1 on the predetermined
variables △lt−1 , ..., △lt−p , △ht−1 , ..., △ht−p , and save the residuals from projection
as △˜lt , △h̃t and h̃t−1 , i.e. regress △lt , △ht and ht−1 on △lt−1 , ..., △lt−p , △ht−1 , ...,
△ht−p , and the residuals are denoted by △˜lt , △h̃t and h̃t−1 . By projection, the VEC
form is simplified to (up to op (1) terms):



△˜lt = (ψ12 (1) + βψ11 (1)) c h̃t−1 + u1,t
T

(3.2)



△h̃t = (ψ22 (1) + βψ21 (1)) c h̃t−1 + u2,t
T

Let c∗ ≡ (ψ22 (1) + βψ21 (1))c, and use the condition b12 =

ψ12 (1)+βψ11 (1)
ψ22 (1)+βψ21 (1)

implied

by the long run restriction, we get:



△˜lt = b12 c∗ h̃t−1 + u1,t
T


△h̃t =

Note that the condition b12 =

c∗
h̃
T t−1

(3.3)

+ u2,t

ψ12 (1)+βψ11 (1)
ψ22 (1)+βψ21 (1)

holds only when c ̸= 0, hence (3.3)

is derived under c ̸= 0. However, this simplified model remains correct when c = 0:
c = 0 implies c∗ = 0, hence b12 c∗ = 0, no matter which expression of b12 is used.
In other words, no matter whether or not c equals 0, it is valid to simplify (3.1) to
(3.3).

78
∗∗
Premultiplying the VEC form by B0 yields the structural form below, with Bi−1
=

B0 A∗∗
i−1 :

∗

B0 △Ỹt = B0 (A (1) − I)Ỹt−1 +

p
∑

∗∗
Bi−1
△Ỹt−i + ϵt

i=1

Impose the long run restriction and write the structural form more explicitly:


∑
∑p ∗∗

z
△lt = b12 △ht + pi=1 b∗∗
i−1,11 △lt−i +
i=1 bi−1,12 △ht−i + ϵt

∑
∑p ∗∗

d
△ht = b21 △lt + b∗22 ht−1 + pi=1 b∗∗
i−1,21 △lt−i +
i=1 bi−1,22 △ht−i + ϵt
where b∗22 = [(ψ22 (1) + βψ21 (1)) − b21 (ψ12 (1) + βψ11 (1))]c/T . After projecting out
the lags, the structural form reduces to:



△˜lt = b12 △h̃t + ϵzt


△h̃t = b21 △˜lt + b∗22 h̃t−1 + ϵdt

3.3

3.3.1

(3.4)

Tests

AR

For a given c, a confidence interval for b12 can be constructed by inverting AR test.
Consider the auxiliary regression below for testing H0 : b12,0 = b12 . Under H0 , θ = 0:

△˜lt − b12,0 △h̃t = θh̃t−1 + ϵzt

79
The tAR statistic is the t statistic for testing θ = 0 in the auxiliary regression
above:

∑
∑
( Tt=2 h̃2t−1 )−1 Tt=2 h̃t−1 (△˜lt − b12,0 △h̃t )
tAR (b12,0 ) =
∑
( Tt=2 h̃2t−1 )−1/2 σ̂ϵ
where σ̂ϵ 2 is the sample variance of the OLS residual.
Theorem. Under assumptions i − iv, and H0 : b12 = b12,0 , tAR has the asymptotic
distribution that depends on c:

tAR =⇒ ρvϵ τc +

√

1 − ρ2vϵ z

∑p
∗∗
where ρvϵ is the long run correlation of vt and ϵzt , vt ≡
i=1 ai−1,21 △lt−i + u2,t ,
(∫
)−1/2 ∫
1
1
τc ≡ 0 Jc (s)2 ds
J (s)dW (s), z is a standard normal variate.
0 c

Proof.
∑T

∑
h̃2t−1 )−1 Tt=2 h̃t−1 ϵzt
tAR (b12,0 ) =
∑
( Tt=2 h̃2t−1 )−1/2 σ̂ϵ
∑
∑
( Tt=2 h̃2t−1 )−1/2 Tt=2 h̃t−1 ϵzt
=
σ̂ϵ
(

t=2

Apply two results in Hansen (1995), with a(1) ≡ ψ22 (1) + βψ21 (1):
∫ 1
T
1 ∑ 2
−2 2
J c (s)2 ds
h̃ ⇒ a(1) σv
T 2 t=2 t−1
0
)
( ∫ 1
∫ 1
T
1∑
c
†
−1
c
2 1/2
z
J (s)dW (s)
J (s)dW (s) + (1 − ρvϵ )
h̃t−1 ϵt ⇒ a(1) σv σϵ ρvϵ
T t=2
0
0

80
(∫

)−1/2 (

1
2

tAR (b12,0 ) =⇒

Jc (s) ds
√
=⇒ ρvϵ τc + 1 − ρ2vϵ z

∫
ρvϵ

0

3.3.2

1

Jc (s)dW (s) +
0

√

∫
1 − ρ2vϵ

)

1

†

Jc (s)dW (s)
0

Wald

A joint confidence set for (b12 , c∗ ) can be constructed by inverting Wald test.

c∗0
˜
t=2 h̃t−1 △lt − b12,0 T 
 (
Let ϕT (b12,0 , c∗0 ) = 
. The Wald statis∑
∑
c∗
( Tt=2 h̃2t−1 )−1 Tt=2 h̃t−1 △h̃t − T0
tic is:
[
]
T
∑
W (b12,0 , c∗0 ) = ϕT (b12,0 , c∗0 )′ Σ̂−1
h̃2t−1 ϕT (b12,0 , c∗0 )
u


∑T

2
−1
t=2 h̃t−1 )

∑T

t=2

Theorem. Under assumptions i − iv, and H0 : b12 = b12,0 , c∗ = c∗0 , W (b12,0 , c∗0 ) has
the asymptotic distribution dependent on c:
W (b12,0 , c∗0 ) =⇒ (ρv2 τc + (1 − ρ2v2 )1/2 z2 )2 + (ρv1−2 τc + (1 − ρ2v1−2 )1/2 z1 )2

where ρv2 is the long run correlation of vt and u2,t , ρv1−2 as the long run correlation
of vt with u1−2 , z1 , z2 are independent standard normal variates, and u1−2 results
from the decomposition of u1,t = σ1 ( ρσu2 u2,t + (1 − ρ2u )1/2 u1−2 ), ρu is the correlation of
u1,t , u2,t .

Proof. Decompose u1,t :

u1,t = σ1 (

ρu
u2,t + (1 − ρ2u )1/2 u1−2 )
σ2

81
where ρu is the correlation of u1,t and u2,t , and u1−2 is uncorrelated with u2,t : the subscript indicates that it comes from u1,t after excluding u2,t ; in addition, the variance
of u1−2 is 1.
T
T
∑
∑
2
−1/2
Let D1 denote the convergence outcome of (
h̃t−1 )
h̃t−1 u1−2 :
t=2

t=2

T
T
∑
∑
−1/2
2
h̃t−1 u1−2 ⇒ D1
(
h̃t−1 )
t=2

t=2

where D1 = ρv1−2 τc + (1 − ρ2v1−2 )1/2 z1 , and z1 is a standard normal variate. Similarly:
T
T
∑
∑
(
h̃2t−1 )−1/2
h̃t−1 u2,t ⇒ σ2 D2
t=2

t=2

where D2 = ρv2 τc + (1 − ρ2v2 )1/2 z2 , and z2 is a standard normal variate independent
of z1 .
T
T
∑
∑
2
−1/2
With the above notations, rewrite (
h̃t−1 )
h̃t−1 u1,t as:
t=2

[
σ1
[

t=2

T
T
∑
∑
ρu
2
−1/2
(
h̃t−1 )
h̃t−1 ( u2,t + (1 − ρ2u )1/2 u1−2 )
σ2
t=2
t=2

]

T
T
T
T
∑
∑
ρu ∑ 2 −1/2 ∑
2 1/2
2
−1/2
= σ1
(
h̃ )
h̃t−1 u2,t + (1 − ρu ) (
h̃t−1 )
h̃t−1 u1−2
σ2 t=2 t−1
t=2
t=2
t=2
[
]
⇒ σ1 ρu D2 + (1 − ρ2u )1/2 D1

]

82
Now the Wald statistic becomes:


′

T
T
∑
∑
h̃2 )−1/2
h̃t−1 u1,t 
 (

 t=2 t−1


−1
t=2

 Σu 
T
T
 ∑


∑



h̃2t−1 )−1/2
h̃t−1 u2,t
(


t=2

t=2

(
(

T
∑
t=2
T
∑

h̃2t−1 )−1/2
h̃2t−1 )−1/2

t=2

T
∑
t=2
T
∑


h̃t−1 u1,t 




h̃t−1 u2,t

t=2

′


[
]
[
]
2 1/2
σ ρu D2 + (1 − ρ2u )1/2 D1 
 σ1 ρu D2 + (1 − ρu ) D1 
−1  1
⇒ 
Σ
 u 

σ2 D2
σ2 D2

 


[
] ′
[
]
2 1/2
2
2 1/2
 σ1 ρu D2 + (1 − ρu ) D1   σ2 −σ12   σ1 ρu D2 + (1 − ρu ) D1 

 


2
σ 2 D2
−σ12 σ1
σ 2 D2
⇒
2
σ12 σ22 − σ12
⇒ D12 + D22

We are ultimately most interested in the IRF

∂ht+j
:
∂ϵzt

the effect of a positive

technology shock on the hours worked. Once we get b12 by inverting AR or Wald,
we derive

∂ht+j
∂ϵzt

in the following manner.

Given b12 , there is a correspondent b21 : b21 =

b12 σ22 −σ12
.
b12 σ12 −σ11

This relation holds

because the covariance matrix E(ϵt ϵ′t ) of the
is diagonal, and
 structural shocks

−b12 
 1
B0 E(ut u′t )B0′ = E(ϵt ϵ′t ). Consequently, B0 = 
 can be recovered.
−b21
1
Combining B0 with A1 , ..., Ap+1 , which are estimated by VAR(p + 1) of Yt =

83




 △lt 

, the IRF
ht

∂ht+j
∂ϵzt

can be derived through the companion matrix F :





 A1 A2 ... Ap+1

 I

F = 

...


I










Take the j th power of F , and denote the up left 2 by 2 matrix of F j by F̃ j , then
∂Yt+j
∂ϵt ′

= F̃ j B0−1 , and

∂ht+j
∂ϵzt

is the (2, 1) element of

∂Yt+j
.
∂ϵt ′

As an example, when j = 0, the contemporaneous effect of

∂ht
∂ϵzt

is:

∂ht
b21
=
z
∂ϵt
1 − b12 b21
b12 σ22 − σ12
=
2b12 σ12 − σ11 − b212 σ22

Once a robust confidence interval of b12 is available, we derive the confidence interval for

∂ht+j
∂ϵzt

by using the point estimates of A1 , ..., Ap+1 , Σu , and all possible values

of b12 in its confidence interval: the maximum/minimum value of
upper/lower bound, respectively.

∂ht+j
∂ϵzt

becomes its

84

3.4

Application

The same data as in Christiano et al. (2003) and Pesavento and Rossi (2005) is used
in our empirical application: the quarterly data of labor productivity and the hours
worked (denoted by lt , ht respectively in this chapter) between 1948Q1 − 2001Q4 are
from the DRI Economics Database; labor productivity is measured by the output
per hour in the business sector, while hours is measured by the hours worked in the
business sector divided by the population; both labor productivity and hours are in
natural logarithm. See Christiano et al. (2003) for further details.

Although Christiano et al. (2003) and Pesavento and Rossi (2005) use the same
dataset, their conclusions contradict each other: Christiano et al. (2003) argue that
the level specification of hours should be chosen over the difference specification, and
report that a positive technology shock will drive hours up, while Pesavento and Rossi
(2005) employ an agnostic procedure which works for both the level and difference
specifications of hours, and report that a positive productivity shock drives hours
down.

The recent work by Gospodinov et al. (2009), Gospodinov (2010) suggest that it
is appropriate to use the model of (3.1) to investigate the impact of technology shocks
on hours, and we adopt the robust approach for (3.1) to construct the confidence
interval for the response of hours to technology shocks. Since we are using the same
dataset, it will be interesting to compare our results with Christiano et al. (2003),
Pesavento and Rossi (2005), etc.

As a starting point, we consider the two specifications of hours, in levels and
in first differences. The conventional identification procedure (see, e.g., Gali (1999))
under the restriction that only technology shocks have a long run effect on labor pro-

85
ductivity is applied for the two specifications, respectively: following Christiano et al.
(2003), a VAR model with a drift and four lags is employed, where lt is specified in
first differences, ht is specified either in levels or in first differences.

The response of the hours worked to the positive technology shocks is plotted in
Figure 3.1, and it is consistent with the current debate: at short horizons, the effect
of a positive technology shock on the hours worked is found positive under the level
specification of hours, and this effect is negative under the difference specification.
The 95% confidence bounds are constructed by the bootstrap.

For future use, we estimate the following
matrices: (i) the coefficient matrix


 △lt 
A1 , A2 , A3 , A4 in the VAR(4) model of 
; (ii) the covariance matrix Σu of
ht
the reduced error ut ; (iii) the long run matrix Ψ(1).




 △lt 
i. Â1 , Â2 , Â3 , Â4 : estimate a VAR(4) model with a drift for 
, Â1 , Â2 , Â3 , Â4
ht
are the OLS estimates of the coefficients before 
the first, second, third,
 fourth
 A1 A2 A3 A4 


 I
0 0 0 


lag, respectively; the companion matrix F = 
 is thus
 0 I
0 0 




0 0 I
0

86
estimated by:



−0.1008
0.0698
0.0317
−0.2719
−0.0616
0.0858
−0.0286
0.1344




 0.1611 1.4193 0.2083 −0.4265 0.1303 0.0131 0.0503 −0.0329 






 1.0000

0
0
0
0
0
0
0






0
1.0000
0
0
0
0
0
0


F̂ = 



0
0
1.0000
0
0
0
0
0








0
0
0
1.0000
0
0
0
0






0
0
0
0
1.0000
0
0
0




0
0
0
0
0
1.0000
0
0





 0.8707 0.0015 
ii. Σ̂u : save the residuals of the VAR(4) model above, and Σ̂u = 

0.0015 0.5390
is the sample covariance matrix of the residuals.


 △lt 
iii. Ψ̂(1): estimate a VAR(3) model with a drift for 
, Ψ̂1 , Ψ̂2 , Ψ̂3 are the
△ht
OLS estimates of the coefficients before the first, second, third lag, respectively,


and:



 1.0895 0.2986 
Ψ̂(1) = 1 − Ψ̂1 − Ψ̂2 − Ψ̂3 = 

−0.4372 0.4970

Using the same dataset, Pesavento and Rossi (2005) report that the CADF (Covariate Augmented Dickey-Fuller) statistic is around −3.072, and the 95% confidence
interval of ρ is (0.897, 0.997). When T = 216 in the application, it implies that the
95% C.I. of c is: (cl , cu ) = (−22.2480, −0.6480).
Take the following steps to invert AR and construct C.I. for the IRF:

1. For each c ∈ (cl , cu ), invert tAR for a C.I. of b12 ;

87
2. Bonferroni: take the union of C.I.’s of b12 from above;
3. For each b12 in its C.I., compute

∂ht+j
;
∂ϵzt

4. Upper/lower bounds of C.I. come from max/min of

∂ht+j
.
∂ϵzt

Similarly, take the following steps to invert Wald test and construct C.I. for the
IRF:

1. For each (b12,0 , c∗0 ), compute the Wald statistic, and save (b12,0 , c∗0 ) if W (b12,0 , c∗0 )
does not exceed the critical value;
2. Construct the C.I. of b12,0 by projection;
3. For each b12 in its C.I., compute

∂ht+j
;
∂ϵzt

4. Upper/lower bounds of C.I. come from max/min of

∂ht+j
.
∂ϵzt

The response of the hours worked to the positive technology shocks by AR and
Wald is plotted in Figure 3.2. It is found that the IRF lies mostly in the positive region at short horizons, although at 95%, we can not reject that the contemporaneous
effect is negative.

3.5

Conclusion

We contribute to the current debate on whether a positive technology shock will
bring the hours worked up or down by providing new robust results: this effect on
hours is found most positive at short horizons, although we do not rule out the
possibility that the contemporaneous effect is negative at 95%.

88

Figure 3.1: IRF under level and first-difference

hours: in levels
1.5

1

0.5

0

−0.5

−1

0

2

4

6

8
10
12
Periods After Shock

14

16

18

20

14

16

18

20

hours: in first differences
1.5

1

0.5

0

−0.5

−1

0

2

4

6

8
10
12
Periods After Shock

∂h

Notes: this figure reports the IRF ∂ϵt+j
z , i.e. the effect of positive technology shocks
t
on the hours worked, and the IRF is constructed under the long run restriction by
the same approach as in Christiano et al. (2003). The solid line is the point
estimate, with ht in levels in the upper panel, and ht in first differences in the low
panel; the dashed line is the 95% upper and lower bounds by the bootstrap.

89

Figure 3.2: IRF bounds by AR and Wald
Robust: AR
1.5

1

0.5

0

−0.5

−1

0

5

10
Periods After Shock

15

20

15

20

Robust: Wald
1.5

1

0.5

0

−0.5

−1

0

5

10
Periods After Shock

∂h

Notes: this figure reports the 95% bounds of the IRF ∂ϵt+j
z , i.e. the effect of positive
t
technology shocks on the hours worked, and the bounds are constructed by the AR and
Wald approach proposed in this chapter.

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