Queen’s Economics Department Working Paper No. 1135 (Revised)
Wild Bootstrap Tests for IV Regression
Russell Davidson
McGill University
James G. MacKinnon
Queen’s University
Department of Economics
Queen’s University
94 University Avenue
Kingston, Ontario, Canada
K7L 3N6
Revised 3-2008
Wild Bootstrap Tests for IV Regression
Russell Davidson
GREQAM
Centre de la Vieille Charit´e
2 rue de la Charit´e
13236 Marseille cedex 02, France
Department of Economics
McGill University
Montreal, Quebec, Canada
H3A 2T7
email: Russell.Davi dson@mcgill .ca
and
James G. MacKinnon
Department of Economics
Queen’s University
Kingston, Ontario, Canada
K7L 3N6
email: jgm@econ.queensu.ca
Abstract
We propose a wild bootstrap procedure for linear regression models estimated by
instrumental variables. Like other bootstrap procedures that we have proposed else-
where, it uses efficient estimates of the reduced-form equation(s). Unlike them, it takes
account of possible heteroskedasticity of unknown form. We apply this procedure t o
t tests, including heteroskedasticity-robust t tests, and to the Anderson-Rubin test. We
provide simulation evidence that it works far better than older methods, such as the
pairs bootstrap. We also show how to obtain reliable confidence intervals by inverting
bootstrap tests. An empirical example illustrates the utility of these procedures.
Keywords: Instrumental variables estimation, two-stage least squares, weak instru-
ments, wild bootstrap, pairs bootstrap, residual bootstrap, confidence intervals,
Anderson-Rubin test
JEL codes: C12, C15, C30
This research was supported, in part, by grants from the Soc ial Sciences and Hum anities
Research Council of Canada, the Canada Research Chairs program (Chair in Econom ics,
McGill University), and the Fonds Qu´eb´ecois de Recherche sur la Soci´et´e et la Cult ure. We
are grateful to Arthur Sweetman for a valuable suggestion and to two referees and an associate
editor for very helpful comments.
Revised, March 2008
Minor corr ections, May 2009, November 2011, and November 2013
1. Introduction
It is often difficult to make reliable inferences from regressions estimated using instru-
mental va riables. This is especially true when the instruments are weak. There is
an enormous l iterature on thi s subject, much of it quite recent. Most of the papers
focus on the case in which there is just one endogenous vari able on the right-hand
side of the regression, and the problem is to test a hypothesis about the coefficient of
that variabl e. In this paper, we al so focus on t hi s case, but, in addition, we discuss
confidence intervals, and we al low the number of endogenous variables to exceed two.
One way to obtain reliable inferences is to use statistics with better properties than
those of the usual IV t statistic. These include the famous Anderson-Rubin, or AR,
statistic proposed i n Anderson and Rubin (1949) and extended in Dufour and Taamouti
(2005, 2007), t he Lagrange Multiplier, or K, statistic proposed in Kleibergen ( 2002),
and the conditional likelihood ratio, or CLR, test proposed in Moreira (2003). A
detailed analysis of several tests is found in Andrews, Morei r a, and Stock ( 2006).
A second way to obta in reliable inferences is to use the bootstrap. This approach
has been much less popular, probably because the simplest bootstrap methods for thi s
problem do not work very well. See, for ex ample, Flores-Lagunes (2007). H owever, the
more sophisti cated bootstrap methods recently proposed in Davidson and MacKinnon
(2008) work very much better than traditional bootstrap procedures, even when they
are combined with the usual t statistic.
One advantage of the t statistic over the AR, K, and CLR statistics is that it can
easily be modified to be asympt otically valid in the presence of heteroskedasticity of
unknown form. But existing procedures for bootstrapping IV t statistics either are
not valid in this case or work badly in general. The main contribution of this paper
is to propose a new bootstrap data generating process (DGP) which is valid under
heteroskedasticity of unknown form and works well in finite samples even when the
instruments are quite weak. This is a wild bootstrap version of one of the methods
proposed in Davidson and MacKinnon (2008). Using this bootstra p method together
with a heteroskedasticity-robust t statistic generally seems to work remarka bly well,
even though it is not asymptotically valid under weak instrument asymptot ics. The
method can also be used with other test sta tistics that are not heteroskedasticity-
robust. It seems to work particularly well when used with the AR stati st ic, probably
because the resulting test is asymptotically valid under weak instrument a symptotics.
In the next section, we discuss six bootstrap methods that can be applied to test
statistics for the coefficient of the single right-hand si de endogenous variable in a linear
regression model estimated by IV. Three of these have been available for some time,
two were proposed in Davidson and MacKinnon (2008), and one is a new procedure
based on the wild boo tstrap. In Section 3, we discuss the asymptotic validity of
several tests based on this new wild bootstrap method. In Section 4, we investigate
the finite-sample performance of the new bootstrap method and some existing ones by
simulation. Our simulation results are quite ex tensive and a re presented graphically.
– 1 –
In Section 5, we briefly discuss the more general case in which there are two or more
endogenous variables on the right-hand side. In Section 6, we discuss how to obtain
confidence intervals by inverting bootst rap tests. Finally, in Section 7, we present an
empirical application that involves estimation of the return to schooling.
2. Bootstrap Methods for IV Regression
In most of this paper, we deal with the two-equation model
y
1
= βy
2
+ Zγ + u
1
(1)
y
2
= Wπ + u
2
. (2)
Here y
1
and y
2
are n--vectors of observations on endogenous varia bl es, Z is an n × k
matrix of observations on exogenous varia bl es, and W is a n n ×l mat rix of exogenous
instruments with the property t hat S(Z), the subspace spanned by the columns of Z,
lies in S(W ), the subspace spanned by the columns of W. E quation (1) is a structural
equation, and equatio n (2) is a reduced-form equation. Observations are indexed by i,
so that, for example, y
1i
denotes the i
th
element of y
1
.
We assume that l > k. This means that the model is either just identified or over-
identified. The disturbances are assumed to be serially uncorrelated. When they are
homoskedastic, they have a contemporaneous covariance matrix
Σ ≡
σ
2
1
ρσ
1
σ
2
ρσ
1
σ
2
σ
2
2
.
However, we will often allow them to be heteroskedastic with unknown (but bounded)
variances σ
2
1i
and σ
2
2i
and correlation coefficient ρ
i
that may depend on W
i
, the row
vector of instrument al variables for observation i.
The usual t statistic for β = β
0
can be written as
t
s
(
ˆ
β, β
0
) =
ˆ
β − β
0
ˆσ
1
||P
W
y
2
− P
Z
y
2
||
−1
, (3)
where
ˆ
β is the g eneralized IV, or 2SLS, estimate of β, P
W
and P
Z
are the matrices
that project orthogo nally on to the subspaces S(W ) and S(Z), respectively, and || · ||
denotes the Euclidean length of a vector. I n equation (3),
ˆσ
1
=
1
−
n
ˆ
u
1
⊤
ˆ
u
1
1/2
=
1
−
n
(y
1
−
ˆ
βy
2
− Z
ˆ
γ)
⊤
(y
1
−
ˆ
βy
2
− Z
ˆ
γ)
1/2
(4)
is the usual 2SLS estimate of σ
1
. Here
ˆ
γ denotes the I V estimate of γ, and
ˆ
u
1
is the
usual vector of IV residuals. Many regression packages divide by n − k − 1 instead of
by n. Since ˆσ
1
as defined in (4) is not necessarily biased downwards, we do not do so.
– 2 –
When homoskedasticity is not assumed, the usual t st atistic (3) should be replaced by
the heteroskedasticity-robust t statistic
t
h
(
ˆ
β, β
0
) =
ˆ
β − β
0
s
h
(
ˆ
β)
, (5)
where
s
h
(
ˆ
β) ≡
P
n
i=1
ˆu
2
1i
(P
W
y
2
− P
Z
y
2
)
2
i
1/2
P
W
y
2
− P
Z
y
2
2
. (6)
Here ( P
W
y
2
−P
Z
y
2
)
i
denotes the i
th
element of the vector P
W
y
2
−P
Z
y
2
. Expression
(6) is what most regression packages routinely print as a heteroskedasticity-consistent
standard error for
ˆ
β. It is evidently the square root of a sandwich variance estimate.
The basic idea o f bootstrap testing is to compare the observed value of some test
statistic, say ˆτ , wit h the empirical distribution of a number o f bootstrap test statistics,
say τ
∗
j
, for j = 1, . . . , B, where B is the number of bootstrap replications. The
bootstrap statistics are generated using the bootstra p DGP, which must satisfy the
null hypothesis tested by the bootstrap statistics. When α is the level of the test, it is
desirable that α(B + 1) should be an integer, and a commonly used value of B is 999.
See Davidson and MacKinnon ( 2000) for more on how to cho o se B appropriately. If
we are prepared to assume that τ is symmetrically distributed around the origin, then
it i s reasonable to use the sy mmetric bootstrap P value
ˆp
∗
s
(ˆτ) =
1
B
B
X
j=1
I
|τ
∗
j
| > |ˆτ|
. (7)
We reject the null hypothesis whenever ˆp
∗
s
(ˆτ ) < α.
For test statistics that a r e always positive, such as the AR and K statistics that will be
discussed in the next section, we can use (7) without taking absolute values, and this
is really the only sensible way to proceed. In the case of IV t statistics, however, the
probability of rejecting in one direction can be very much greater than the probability
of rejecting in the other, because
ˆ
β is often biased. In such cases, we can use the
equal-tail bootstrap P value
ˆp
∗
et
(ˆτ) = 2 min
1
B
B
X
j=1
I(τ
∗
j
≤ ˆτ),
1
B
B
X
j=1
I(τ
∗
j
> ˆτ)
. (8)
Here we actually perform two tests, one against val ues in the lower tail of the distr ibu-
tion and the other against values in the upper tail, and reject if either of them yields
a bootstrap P value less than α /2.
Bootstrap testing can be expected to work well when the quantity bootstrapped is
approximately pivotal, that is, w hen its distribution changes little as the DGP varies
– 3 –
