Recent results of Cribari-Neto and Zarkos (1999) show that bootstrap methods can be successfully used to estimate a heteroskedasticity robust covariance matrix estimator. In this paper, we show that the wild bootstrap estimator can be calculated directly, without simulations, as it is just a more traditional estimator. Their experimental results seem to conflict with those of MacKinnon and White (1985); we reconcile these two results.wild bootstrap, heteroskedasticity

Emmanuel Flachaire

English

Research Papers in Economics

Bootstrapping Heteroskedasticity
Consistent Covariance Matrix Estimator
by
Emmanuel Flachaire
Eurequa, University Paris I Panth´ eon-Sorbonne
December 2001
Abstract
Recent results of Cribari-Neto and Zarkos (1999) show that bootstrap methods can
be successfully used to estimate a heteroskedasticity robust covariance matrix estima-
tor. In this paper, we show that the wild bootstrap estimator can be calculated directly,
without simulations, as it is just a more traditional estimator. Their experimental re-
sults seem to conﬂict with those of MacKinnon and White (1985); we reconcile these
two results.
JEL classiﬁcation: C1
Keywords: wild bootstrap, heteroskedasticity1 Introduction
In this paper, we show that the wild bootstrap covariance matrix estimator can be calculated
directly, without simulation, since it is simply a traditional heteroskedasticity consistent
covariance matrix estimator ﬁrst proposed by Eicker (1963) and White (1980), see section 2.
Simulation results in MacKinnon and White (1985) had suggested that the jackknife is a
reliable heteroskedasticity-consistent variance estimator. Simulation results in Cribari-Neto
and Zarkos (1999) suggest that this estimator did not perform as well as its competitors.
This conclusion appears to show a conﬂict between diﬀerent simulation results and could
lead us to the wrong conclusion that the jackknife estimator is not reliable. In section 3, we
reconcile these experimental results.
2 HCCME and bootstrap estimators
Consider the linear heteroskedastic model:
yt = Xtβ + ut E(ut|X) = 0, E(u
2
t|X) = σ
2
t (1)
where yt is a dependent variable, Xt a regressor row matrix 1 × k, β and σt are respectively
the unknown parameters and the unknown variances of the error terms; ut is error term.
We consider only unconditional heteroskedasticity, which means that the σ2
t may depend
on the exogeneous regressors, but not, for instance, on lagged dependent variables. The
Heteroskedasticity Consistent Covariance Matrix Estimator, or HCCME, is
(X
⊤X)
−1X
⊤ˆ ΩX(X
⊤X)
−1 (2)
where the n × n diagonal matrix ˆ Ω has typical diagonal element a2
t ˆ u2
t, where ˆ ut is the OLS
residual. We refer to the basic version of the HCCME, proposed by Eicker (1963) and White
(1980), as HC0 where at = 1. Bias is reduced by multiplying the ˆ ut by the square root of
n/(n−k), thereby by multiplying the elements of ˆ Ω by n/(n−k); this procedure, analogous
to the use in the homoskedastic case of the unbiased OLS estimator of the error variance,
gives rise to form HC1 of the HCCME where at = (n/(n − k))1/2. In the homoskedastic
case, the variance of ˆ ut is proportional to 1−ht, where ht ≡ Xt(X⊤X)−1X⊤
t , the tth element
of the orthogonal projection matrix on to the span of the columns of X. This suggests
at = (1 − ht)−1/2 in order to obtain ˆ Ω. If this is done, we obtain form HC2 of the HCCME.
Finally, arguments based on the jackknife lead MacKinnon and White to propose form
HC3, for which at = (1 − ht)−1. MacKinnon and White (1985), and Chesher and Jewitt
(1987), show that, in terms of size distortion, HC0 is outperformed by HC1, which is in turn
outperformed by HC2 and HC3. The last two cannot be ranked in general, although HC3
has been shown in a number of Monte carlo experiments to be superior in typical cases.
Heteroskedasticity of unknown form cannot be mimicked in the bootstrap distribution.
The wild bootstrap gets round this problem by using a bootstrap data generating process,
or bootstrap DGP, of the form
y
⋆
t = Xtˆ β + atˆ utε
⋆
t (3)
2where ˆ β is a vector of parameter estimates, at takes the forms HC0, HC1, HC2 or HC3 and
the ε⋆
t are mutually independent drawings, completely independent of the original data, from
some auxiliary distribution such that E(ε⋆
t) = 0 and E(ε⋆2
t ) = 1. The heteroskedasticity con-
sistent covariance matrix estimate of the OLS parameter estimate under the wild bootstrap
DGP (3) is the wild bootstrap covariance matrix estimator. Cribari-Neto and Zarkos (1999)
suggest to approximate the wild bootstrap estimator by the following steps:
1. Estimate the model (1) by Ordinary Least Squares (OLS): ˆ β and ˆ ut are respectively
the parameter estimates and the residuals.
2. Generate a bootstrap sample, based on the bootstrap data generating process: y⋆
t =
Xtˆ β+atˆ utε⋆
t, where ε⋆
t is drawn from a probability distribution F with zero expectation
and variance equal to one. Compute the OLS bootstrap parameter estimate ˆ β⋆ =
(X⊤X)−1X⊤y⋆. Repeat this step B times.
3. The variance of the B bootstrap estimates is an approximation of the wild bootstrap
covariance matrix estimate of ˆ β.
We can now give the principal result of this section.
Theorem 1 For a ﬁxed number of observations and under the data generating process (3),
the wild bootstrap covariance matrix estimator is equal to the Heteroskedasticity Consistent
Covariance Matrix Estimator (2).
Proof: The wild bootstrap data generating process is y⋆
t = Xtˆ β + atˆ utε⋆
t. For a ﬁxed
number of observations n, the error terms are independent: E(ε⋆
tε⋆
s|X) = 0 with t  = s.
The ordinary least squares parameter estimate is equal to: ˆ β⋆ = (X⊤X)−1X⊤y⋆ and its
covariance matrix is
(X
⊤X)
−1X
⊤Ω
⋆X(X
⊤X)
−1 (4)
where E(u⋆u⋆⊤|X) = Ω⋆ and u⋆ = atˆ utε⋆
t. The oﬀ-diagonal element of Ω⋆ for row t and
column s, where t  = s, is equal to:
Ω
⋆
t,s = E(u
⋆
tu
⋆
s|X) = atˆ ut asˆ us E(ε
⋆
tε
⋆
s|X) = 0 (5)
The diagonal element t is equal to:
Ω
⋆
t,t = E(u
⋆2
t |X) = a
2
t ˆ u
2
t E(ε
2
t|X) = a
2
t ˆ u
2
t (6)
It is clear that, for a ﬁxed number n of observations, (4) is equal to (2). ￿
This theorem implies that the wild bootstrap covariance matrix estimator is equal to the
HCCME proposed by Eicker (1963), White (1980) and MacKinnon and White (1985). If we
approximate the wild bootstrap covariance matrix estimator by simulations, we can observe
numerical diﬀerences because a ﬁnite number B of simulations introduces an experimental
error. Consequently, the HCCME estimator should always be preferred to the wild bootstrap
estimator obtained by simulations.
33 Experimental results
Experimental results of Cribari-Neto and Zarkos (1999) show that the bias of the HC2 form
of the wild bootstrap estimator is less than that of the jackknife estimator. These results
are for the bias of the variance estimator, but MacKinnon and White (1985) results show
that tests based on the jackknife estimator of the variance tend to be accurate, its bias
notwithstanding. The two sets of results concern fundamentally diﬀerent objects. In this
section we directly compare the size and the bias of diﬀerent tests. The results lead us to
reconcile the conﬂict claimed by Cribari-Neto and Zarkos. In our Monte-Carlo experiments,
we consider a simple linear regression model:
yt = α + xtβ + ut ut ∼ N(0,1) (7)
The true parameters are: α0 = 1 and β0 = 0, the error terms are independent and identically
distributed following the standard Normal distribution. The sample size is small, n = 20,
and the regressor is supposed ﬁxed1.
In subsection 3.1, we experimentally check the theoretical results obtained in the preced-
ing section. In subsection 3.2, we study the bias of the estimators, as in Cribari-Neto and
Zarkos (1999). In 3.3, we study the size distortion of the t-tests based on these estimators,
as in MacKinnon and White (1985).
3.1 Bootstrap and HCCME estimators
We check the theoretical results of the preceding section: for a ﬁxed number of observations
n, the wild bootstrap bootstrap estimator converges to the HCCME estimator when the
number of bootstrap samples B increases to inﬁnity. In our experiment, we draw one sample
(y,x) using (7), we calculate the HCCME estimator and the wild bootstrap estimator of
ˆ β with at = (1 − ht)−1/2 (HC2 form) and we increase the number of bootstrap samples:
B = 199;499;999;9,999;999,999. We calculate two wild bootstrap estimators: one with
the resampling distribution F1 ∼ N(0,1), and the other with F2, a bi-atomic distribution
yielding the value −(
√
5 − 1)/2 with probability p = (
√
5 + 1)/2
√
5, and (
√
5 + 1)/2 with
probability 1 − p, see Liu (1988).
Design B=199 B=499 B=999 B=9,999 B=999,999
F1,HC2 0.06088 0.05557 0.05465 0.05545 0.05637
F2,HC2 0.04592 0.05515 0.05539 0.05516 0.05632
Table 1: Wild bootstrap estimator
With our data, the HC2 form of the HCCME estimator is equal to 0.0562873, the two
wild bootstrap estimators are presented in Table 1. These results make clear that, for a
ﬁxed number of observations n = 20, the wild bootstrap estimators converge to the HCCME
estimator as B → ∞.
1x = -2.2824, -0.435864, 2.27108, -1.05705, -1.10142, 0.648927, 0.143281, -0.25922, 1.87924, -1.32969,
0.013618, -0.303695, 1.24507, 0.670023, 0.658823, 0.521237, -0.0656568, -0.370603, -0.0734635, -0.169986
43.2 Bias
In this subsection, we report the bias of the wild bootstrap and HCCME estimators of ˆ β.
In our experiment, we consider the following estimators: the two wild bootstrap estimators
described in the preceding subsection, the HC2 form of the HCCME, and the HC3 form of
the HCCME. MacKinnon and White (1985) investigated the performance of the jackknife
estimator, but subsequent computer simulations suggested that the HC3 form of the HCCME
behaves very similarly to the jackknife, see Davidson and MacKinnon (1993, section 16.3).
We calculate the bias as the (mean) estimated variance minus the true variance of ˆ β.
In addition, we calculate the mean squared error (MSE). The number of replications is
N = 9,999 and the number of bootstrap samples is B = 499. The results are presented in
Table 2.
Design Bias MSE
wboot, F1,HC2 0.000206 0.000828
wboot, F2,HC2 0.000180 0.000823
HCCME, HC2 0.000078 0.000809
HCCME, HC3 0.013018 0.001645
Table 2: Bias and MSE
We ﬁnd the same results as Cribari-Neto and Zarkos (1999): the bias of the HC2 form
of the wild bootstrap estimator is smaller than that of the HC3 form of the HCCME.
Furthermore, these results make clear that the same forms of the wild bootstrap and of
the HCCME estimators have similar ﬁnite sample behavior, and that the wild bootstrap
estimators (which include experimental error) are substantially worse, as theory suggests.
3.3 Size distortion
In this subsection we report the size distortion of tests based on the HC2 and HC3 forms of
the HCCME of ˆ β. We do not study the wild bootstrap estimator because we have shown that
its ﬁnite sample behavior is quite similar to that of the HCCME. We test the null hypothesis
H0 : β = 0 with the squared t-statistic, which is asymptotically χ2(1): τ = t2 = [ˆ β/ˆ S(ˆ β)]2,
where ˆ β is the ordinary least square parameter estimate and ˆ S(ˆ β)2 the second diagonal
element of the covariance matrix estimator. The realizations of the statistics ˆ τ are used to
calculate a P-value at the nominal level α = 0.05. The number of replications is N = 9,999
and the number of bootstrap samples is B = 499.
estimator Empirical level
HCCME, HC2 0.0774
HCCME, HC3 0.0547
Table 3: Empirical level (α = 0.05)
Table 3 presents the empirical level of the t-statistics based on the HC2 and HC3 forms
of the HCCME estimators. Tests based on the HC2 estimator reject the null hypothesis
5more than 5% of the time when it is true (7.74%), whereas tests based on the HC3 reject
near 5%. We ﬁnd here the same results as those of MacKinnon and White (1985): tests
based on the jackknife estimator, or HC3, are more reliable than others.
4 Conclusion
In this paper, we show that the wild bootstrap is not useful to estimate a heteroskedasticity
consistent covariance matrix. However, it is well known that the bootstrap yields better
results when applied to asymptotic pivots, and it is therefore a better idea to use HCCME
in order to construct such asymptotic pivots before using the bootstrap, see Horowitz (2000)
for an account of the relevant issues.
References
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Cribari-Neto, F. and S. G. Zarkos (1999). “Bootstrap methods for heteroskedastic regression
models: evidence on estimation and testing”. Econometric Reviews 18, 211–228.
Davidson, R. and J. G. MacKinnon (1993). Estimation and Inference in Econometrics. New
York: Oxford University Press.
Eicker, B. (1963). “Limit theorems for regression with unequal and dependant errors”. Annals
of Mathematical statistics 34, 447–456.
Horowitz, J. L. (2000). “The bootstrap”. In Handbook of Econometrics, Volume 5. J. J.
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6

A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity”.

Bootstrap methods for heteroskedastic regression models: evidence on estimation and testing”.

Bootstrap procedure under some non-i.i.d. models”.

Estimation and Inference in Econometrics.

Limit theorems for regression with unequal and dependant errors”.

Some heteroskedasticity consistent covariance matrix estimators with improved ﬁnite sample properties”.

The bias of a heteroskedasticity consistent covariance matrix estimator”.

The bootstrap”.

Bootstrapping heteroskedasticity consistent covariance matrix estimator

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Bootstrapping heteroskedasticity consistent covariance matrix estimator

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