1. INTRODUCTION
We provide a new and improved approach to the asymptotics of
hypothesis testing in time series models with “arbitrary,”
i.e., unspecified, serial correlation and heteroskedasticity. Our results
are general enough to apply to stationary models estimated by generalized
method of moments (GMM). Heteroskedasticity and autocorrelation consistent
(HAC) estimation and testing in these models involves calculating an
estimate of the spectral density at zero frequency of the estimating
equations or moment conditions defining the estimator. We focus on the
class of nonparametric spectral density estimators1
An alternative to the nonparametric approach has been
advocated by den Haan and Levin (1997, 1998). Following Berk (1974)
and others in the time series statistics literature, they propose
estimating the zero frequency spectral density parametrically using vector
autoregression (VAR) models. They show that this parametric approach can
achieve essentially the same generality as the nonparametric approach if
the VAR lag length increases with the sample size at a suitable
rate.
that were originally proposed and analyzed in the time
series statistics literature. See Priestley (
1981) for the standard textbook treatment. Important
contributions to the development of these estimators for covariance matrix
estimation in econometrics include White (
1984),
Newey and West (
1987), Gallant (
1987), Gallant and White (
1988), Andrews (
1991),
Andrews and Monahan (
1992), Hansen (
1992), Robinson (
1998), de
Jong and Davidson (
2000), and Jansson (
2002).
We stress at the outset that we are not proposing new estimators or
statistics; rather we focus on improving the asymptotic distribution
theory for existing techniques. Our results provide a framework that can
be used to assess the impact of the choice of HAC estimator on the
resulting test statistic.
Conventional asymptotic theory for HAC estimators is well established
and has proved useful in providing practical formulas for estimating
asymptotic variances. The ingenious “trick” is the assumption
that the variance estimator depends on a fraction of sample
autocovariances, with the number of sample autocovariances going to
infinity, but the fraction going to zero as the sample size grows. Under
this condition it has been shown that well-known HAC estimators of the
asymptotic variance are consistent. Then, the asymptotic distribution of
estimated coefficients can essentially be derived assuming the variance is
known. That is, sampling bias and variance of the variance estimator do
not appear in the first-order asymptotic distribution theory of test
statistics regarding parameters of interest. Although this is an extremely
productive simplifying assumption that leads to standard asymptotic
distribution theory for tests, the accuracy of the resulting asymptotic
theory is often less than satisfactory. In particular there is a tendency
for HAC robust tests to overreject (sometimes substantially) under the
null hypothesis in finite samples; see Andrews (1991), Andrews and Monahan (1992), and the July 1996 special issue of Journal
of Business & Economic Statistics for evidence.
There are two main sources of finite-sample distortions. The first
source is inaccuracy via the central limit theorem approximation to the
sampling distribution of parameters of interest. This becomes a serious
problem for data that have strong or persistent serial correlation. The
second source is the bias and sampling variability of the HAC estimate of
the asymptotic variance. This second source is the focus of this paper.
Simply appealing to a consistency result for the asymptotic variance
estimator as is done under the standard approach does not capture these
important small-sample properties.
The assumption that the fraction of the sample autocovariances used in
calculating the HAC variance estimator goes to zero as the sample size
goes to infinity is a clever technical assumption that substantially
simplifies asymptotic calculations. However, in practice there is a given
sample size, and some fraction of sample autocovariances is used to
estimate the asymptotic variance. Even if a practitioner chooses the
fraction based on a rule such that the fraction goes to zero as the sample
size grows, it does not change the fact that a positive fraction is being
used for a particular data set. The implications of this simple
observation have been eloquently summarized by Neave (1970, p. 70) in the context of spectral density
estimation:
When proving results on the asymptotic behavior of estimates
of the spectrum of a stationary time series, it is invariably assumed that
as the sample size T tends to infinity, so does the truncation
point M, but at a slower rate, so that M/T
tends to zero. This is a convenient assumption mathematically in that, in
particular, it ensures consistency of the estimates, but it is unrealistic
when such results are used as approximations to the finite case where the
value of M/T cannot be zero.
Based on this observation, Neave (1970)
derived an asymptotic approximation for the sampling variance of spectral
density estimates under the assumption that M/T is a
constant and showed that his approximation was more accurate than the
standard approximation.
In this paper, we effectively generalize the approach of Neave (1970) for zero frequency nonparametric spectral
density estimators (HAC estimators). We derive the entire asymptotic
distribution (rather than just the variance) of these estimators under the
assumption that M = bT where b ∈
(0,1] is a constant. We label asymptotics obtained under this nesting
of the bandwidth “fixed-b asymptotics.” In contrast,
under the standard asymptotics b goes to zero as T
increases. Therefore, we refer to the standard asymptotics as
“small-b asymptotics.” We show that under
fixed-b asymptotics, the HAC robust variance estimators converge
to a limiting random matrix that is proportional to the unknown asymptotic
variance and has a limiting distribution that depends on the kernel and
b. Under the fixed-b asymptotics, HAC robust test
statistics computed in the usual way are shown to have limiting
distributions that are pivotal but depend on the kernel and b.
This contrasts with small-b asymptotics where the effects of the
kernel and bandwidth are not captured.
Fixed-b asymptotics is a natural generalization of ideas
explored by Kiefer, Vogelsang, and Bunzel (2000), Bunzel, Kiefer, and Vogelsang (2001), Kiefer and Vogelsang (2002a), Kiefer and Vogelsang (2002b), and Vogelsang (2003). Those papers analyzed HAC robust tests for the
case where the HAC bandwidth is set equal to the sample size, i.e.,
b = 1. By considering values of b < 1 in this paper,
we follow the traditional approach where the bandwidth controls the
downweighting on higher order sample autocovariances. An alternative has
been proposed in recent work by Phillips, Sun, and Jin (2003) and Phillips, Sun, and Jin (2005) where downweighting is achieved by
exponentiating kernels that use bandwidth equal to the sample size. They
show that if the exponent increases with the sample size at a suitable
rate, then consistent and asymptotically normal HAC variance estimators
can be obtained. In an analysis analogous to fixed-b asymptotics,
they also develop an asymptotic theory under the assumption that the
exponent is fixed as the sample size increases. Fixed-exponent asymptotics
captures the choice of kernel and exponent in much the same way that
fixed-b asymptotics captures the choice of kernel and
bandwidth.
Although the fixed-b assumption is a better reflection of
practice in reality, that alone does not justify the new asymptotic
theory. In fact, our asymptotic theory leads to two important innovations
for HAC robust testing.
First, because the fixed-b asymptotics explicitly captures
the choice of bandwidth and kernel, a more accurate first-order asymptotic
approximation is obtained for HAC robust tests. Finite-sample simulations
reported here and in the working paper (Kiefer and
Vogelsang, 2005) show that in many situations the fixed-b
asymptotics provides a more accurate approximation than the standard
small-b asymptotics. There is also theoretical evidence by
Jansson (2004) showing that fixed-b
asymptotics is more accurate than small-b asymptotics in Gaussian
location models for the special case of the Bartlett kernel with
b = 1. Jansson (2004) proves that
fixed-b asymptotics delivers an error in rejection probability
that is O(T−1 log(T)). This
contrasts with small-b asymptotics where the error in rejection
probability is no smaller than
O(T−1/2) (see Velasco and Robinson, 2001). Again focusing on
Gaussian location models, recent work by Phillips, Sun, and Jin (2004) has shown that the error in rejection
probability is O(T−1) for exponentiated
kernels under fixed-exponent asymptotics. Phillips et al. (2004) conjecture that this result likely extends to
traditional kernels under fixed-b asymptotics. If this conjecture
is true, then fixed-b asymptotics has an error in rejection
probability of smaller order than the standard normal approximation
(small-b).
Second, fixed-b asymptotic theory permits a local asymptotic
power analysis for HAC robust tests that depends on the kernel and
bandwidth. We can theoretically analyze how the choices of kernel and
bandwidth affect the power of HAC robust tests. Such an analysis is not
possible under the standard small-b asymptotics because local
asymptotic power does not depend on the choice of kernel or bandwidth.
Because of this fact, the existing HAC robust testing literature has
focused instead on minimizing the asymptotic truncated mean square error
(MSE) of the asymptotic variance estimators when choosing the kernel and
bandwidth. For the analysis of HAC robust tests, this is not a completely
satisfying situation, as noted by Andrews (1991,
p. 828).2
Additional discussion of this point
is given by Cushing and McGarvey (1999, p.
80).
An obvious alternative to asymptotic approximations is the bootstrap.
Recently it has been shown by Hall and Horowitz (1996), Götze and Künsch (1996), Inoue and Shintani (2004), and others that higher order refinements to the
small-b asymptotics are feasible when bootstrapping the
distribution of HAC robust tests using blocking. Using finite-sample
simulations we compare fixed-b asymptotics with the block
bootstrap. Our results suggest some interesting properties of the
bootstrap and indicate that fixed-b asymptotics may be a useful
analytical tool for understanding variation in bootstrap performance
across bandwidths. One result is that the bootstrap without blocking
performs almost identically to the fixed-b asymptotics even when
the data are dependent (in contrast the small-b asymptotics
performs relatively poorly). When blocking is used, the bootstrap can
perform slightly better or slightly worse than fixed-b
asymptotics depending on the choice of block length. It may be the case
that the block bootstrap delivers an asymptotic refinement over the
fixed-b first-order asymptotics if the block length is chosen in
a suitable way. A higher order fixed-b asymptotic theory is
required before such a result can be formally established.
The remainder of the paper is organized as follows. Section 2 lays out
the GMM framework and reviews standard results. Section 3 reports some
small-sample simulation results that illustrate the inaccuracies that can
occur when using the small-b asymptotic approximation. Section 4
introduces the new fixed-b asymptotic theory. Section 5 analyzes
the performance of the new asymptotic theory in terms of size distortions
and local asymptotic power. The impact of the choice of bandwidth and
kernel is analyzed, and comparisons are made with the traditional
small-b asymptotics and the block bootstrap. Section 6 gives
concluding comments. Proofs are provided in an Appendix.
The following notation is used throughout the paper. The symbol ⇒
denotes weak convergence, Bj(r)
denotes a j vector of standard Brownian motions (Wiener
processes) defined on
denotes a j vector of standard Brownian bridges, and
[rT] denotes the integer part of rT for
r ∈ [0,1].
2. INFERENCE IN GMM MODELS: THE STANDARD
APPROACH
We present our results in the GMM framework, noting that this covers
estimating equation methods (Heyde, 1997). Since
the influential work of Hansen (1982), GMM is
widely used in virtually every field of economics. Heteroskedasticity or
autocorrelation of unknown form is often an important specification issue,
especially in macroeconomics and financial applications. Typically the
form of the correlation structure is not of direct interest (if it is, it
should be modeled directly). What is desired is an inference procedure
that is robust to the form of the heteroskedasticity and serial
correlation. HAC covariance matrix estimators were developed for exactly
this setting.
Consider the p × 1 vector of parameters θ ∈
Θ ⊂ Rp. Let θ0
denote the true value of θ and assume θ0 is an
interior point of Θ. Let vt denote a
vector of observed data and assume that q moment conditions hold
that can be written as
where f (·) is a q × 1 vector of
functions with q ≥ p and rank(E
[∂f/∂θ′]) = p. The
expectation is taken over the endogenous variables in
vt and may be conditional on exogenous
elements of vt. There is no need in what
follows to make this conditioning explicit in the notation. Define
where
is the sample analog to (1). The GMM estimator is defined as
where WT is a q ×
q positive definite weighting matrix. Alternatively,
can also be defined as an estimating equations estimator, the solution to
the p first-order conditions associated with (2)
where
.
Of course, when the model is exactly identified and q =
p, an exact solution to
is attainable and the weighting matrix WT is
irrelevant. Application of the mean-value theorem implies that
where
denotes the q × p matrix whose ith row
is the corresponding row of
Gt(θT(i))
where
for some 0 ≤ λi,T ≤ 1 and
λT is the q × 1 vector with
ith element λi,T.
To focus on the new asymptotic theory for tests, we avoid listing
primitive assumptions and make rather high-level assumptions on the GMM
estimator
.
Lists of sufficient conditions for these to hold can be found in Hansen
(1982) and Newey and McFadden (1994). Our assumptions are as follows:
Assumption 1.
.
Assumption 2.
where
.
Assumption 3.
uniformly in r ∈ [0,1] where
G0 = E [∂f
(vt,θ0)/∂θ′].
Assumption 4. WT is positive semidefinite
and p lim WT =
W∞ where W∞ is a
matrix of constants and
G0′W∞
G0 is positive definite.
These assumptions hold for many models in economics, and with the
exception of Assumption 2 they are fairly standard. Assumption 2 requires
that a functional central limit theorem hold for
T1/2gt(θ0).
This is stronger than the central limit theorem for
T1/2gT(θ0)
that is required for asymptotic normality of
.
However, consistent estimation of the asymptotic variance of
requires an estimate of Ω. Conditions for consistent estimation of
Ω are typically stronger than Assumption 2 and often imply Assumption
2. For example, Andrews (1991) requires that
f (vt,θ0) is a mean
zero fourth-order stationary process that is α-mixing. Phillips and
Durlauf (1986) show that Assumption 2 holds
under the weaker assumption that f
(vt,θ0) is a mean zero, 2 +
δ–order stationary process (for some δ > 0) that is
α-mixing. Thus our assumptions are slightly weaker than those usually
given for asymptotic testing in HAC-estimated GMM models.
Under our assumptions
is asymptotically normally distributed, as recorded in the following
lemma, which is proved in the Appendix.
LEMMA 1. Under Assumptions 1–4, as T →
∞,
where Λ*Λ*′ =
G0′W∞ΛΛ′W∞
G0 and V =
(G0′W∞
G0)−1Λ*Λ*′(G0′W∞
G0)−1.
Under the standard approach, a consistent estimator of V is
required for inference. Let
denote an estimator of Ω. Then V can be estimated by
The HAC literature builds on the spectral density estimation
literature to suggest feasible estimators of Ω and to find conditions
under which such estimators are consistent. The widely used class of
nonparametric estimators of Ω take the form
with
where k(x) is a kernel function k :
R → R satisfying k(x) =
k(−x), k(0) = 1,
|k(x)| ≤ 1, k(x)
continuous at
.
Often k(x) = 0 for |x| > 1 so
M “trims” the sample autocovariances and acts as a
truncation lag. Some popular kernel functions do not truncate, and
M is often called a bandwidth parameter in those cases. For
kernels that truncate, the cutoff at |x| = 1 is
arbitrary and is essentially a normalization. For kernels that do not
truncate, a normalization must be made because the weights generated by
the kernel k(x) and bandwidth M are the same as
those generated by kernel k(ax) with bandwidth
aM. Therefore, there is an interaction between bandwidth and
kernel choice. We focus on kernels that yield positive definite
for the obvious practical reasons although many of our theoretical
results hold without this restriction.
Standard asymptotic analysis proceeds under the assumption that
M → ∞ and M/T → 0 as
T → ∞, in which case
is a consistent estimator. Thus, b → 0 as T →
∞ and hence the label small-b asymptotics. Because in
practical settings b is strictly positive, the assumption that
b shrinks to zero has little to do with econometric practice;
rather it is an ingenious technical assumption allowing an estimable
asymptotic approximation to the asymptotic distribution of
to be calculated. The difficulty in practice is that any choice of
M for a given sample size, T, can be made consistent
with the preceding rate requirement. Although the rate requirement can
been refined if one is interested in minimizing the MSE of
(e.g., M must increase at rate T1/3 for
the Bartlett kernel), these refinements do not deliver specific choices
for M. This fact has long been recognized in the spectral density
and HAC literatures and data dependent methods for choosing M
have been proposed. See Andrews (1991) and Newey
and West (1994). These papers suggest choosing
M to minimize the truncated MSE of
.
However, because the MSE of
depends on the serial correlation structure of f
(vt,θ0), the practitioner
must estimate the serial correlation structure of f
(vt,θ0) either
nonparametrically or with an “approximate” parametric model.
Although data dependent methods are a significant improvement over the
basic case for empirical implementation, the practitioner is still faced
with either a choice of approximating parametric model or the choice of
bandwidth in a preliminary nonparametric estimation problem. Even these
bandwidth rules do not yield unique bandwidth choices in practice because,
e.g., if a bandwidth MA satisfies the
optimality criterion of Andrews (1991), then the
bandwidth MA + d where d is
a finite constant is also optimal. See den Haan and Levin (1997) for details and additional practical
challenges.
As the previous discussion makes clear, the standard approach
addresses the choice of bandwidth by determining the rate by which
M must grow to deliver a consistent variance estimator so that
the usual standard normal approximation can be justified. In contrast, for
a given data set and sample size, we take the choice of M as
given and provide an asymptotic theory that reflects to some extent how
that choice of M affects the sampling distribution of the HAC
robust test.
3. MOTIVATION: FINITE-SAMPLE PERFORMANCE OF
STANDARD ASYMPTOTICS
Although the poor finite-sample performance of HAC robust tests in the
presence of strong serial correlation is well documented, it will be
useful for later comparisons to illustrate some of these problems briefly
in a simple environment. Consider the most basic univariate time series
model with ARMA(1,1) errors,
Define the HAC robust t-statistic for μ as
where
is computed using
.
We set μ = 0 and generated data according to (7) for a wide variety
of parameter values and computed empirical rejection probabilities of the
t-statistic for testing the null hypothesis that μ ≤ 0
against the alternative that μ > 0. Note that because the test
statistic has a symmetric distribution, the performance of a two-sided
version of the test is qualitatively similar. We report results for the
sample size T = 50, and 5,000 replications were used in all
cases. To illustrate how well the standard normal approximation works as
the bandwidth varies in this finite sample, we computed rejection
probabilities for the t-statistic implemented using M =
1,2,3,…,49,50. We set the asymptotic significance level to 0.05 and
used the usual standard normal critical value of 1.645 for all values of
M. To conserve on space, we report results for six error
configurations: independent and identically distributed (i.i.d.) errors
(ρ = φ = 0), AR(1) errors with ρ = 0.7,0.9, MA(1) errors with
φ = −0.5,0.5 and ARMA(1,1) errors with ρ = 0.8,φ =
−0.4. We also give results where
is implemented with AR(1) prewhitening. We report results for the popular
Bartlett and quadratic spectral (QS) kernels. Results for other kernels
are similar.
The results are depicted in Figures 1 and
2. In each figure, the line with the label,
N(0,1), plots rejection probabilities when the critical value 1.645 is
used and there is no prewhitening. In the case of prewhitening, the label
is N(0,1), PW. The figures also depict plots of rejection probabilities
using the fixed-b asymptotic critical values, but for now we only
focus on the results using the standard small-b asymptotics.
Empirical null rejection probabilities: Bartlett kernel, T =
50, 5% nominal level.
Empirical null rejection probabilities: QS kernel, T = 50, 5%
nominal level.
Consider first the case of i.i.d. errors. When M is small
rejection probabilities are only slightly above 0.05 as one might expect
for i.i.d. data. However, as M increases, rejection probabilities
steadily rise. When M = T, rejection probabilities are
nearly 0.2 for the Bartlett kernel and exceed 0.25 for the QS kernel.
Somewhat different patterns occur for AR(1) errors. When M is
small, there are nontrivial overrejection problems. When prewhitening is
not used, the rejections fall as M increases but then rise again
as M increases further. When prewhitening is used, rejection
probabilities essentially increase as M increases. Prewhitening
reduces the extent of overrejection but does not remove it. The patterns
for MA(1) errors are similar to the i.i.d. case, and the patterns for
ARMA(1,1) errors are combinations of the AR(1) and MA(1) cases.
These plots clearly illustrate a fundamental finite-sample property of
HAC robust tests: the choice of M in a given sample matters and
can greatly affect the extent of size distortion depending on the serial
correlation structure of the data. And, when M is not very small,
the choice of kernel also matters for the overrejection problem. Given
that the sampling distribution of the test depends on the bandwidth and
kernel, it would seem to be a minimal requirement that the asymptotic
approximation reflect this dependence. Whereas the standard
small-b asymptotics is too crude for this purpose, our results in
the next section show that the fixed-b asymptotics naturally
captures the dependence on the bandwidth and kernel and captures the
dependence in a simple and elegant manner.3
An alternative approach to the fixed-b asymptotics is
to consider higher order asymptotic approximations in the small-b
framework. For example, one could take the Edgeworth expansions from the
recent work by Velasco and Robinson (2001) and
obtain a second-order asymptotic approximation for HAC robust tests that
depends on the bandwidth and kernels. Although this is a potentially
fruitful approach, it is complicated by the need to estimate the second
derivative of the spectral density.
Before moving on, it is useful to discuss in some detail the reason
that rejection probabilities have the nonmonotonic pattern for AR(1)
errors. When M is small,
is biased but has relatively small variance. Because the bias is
downward, that leads to overrejection. As M initially increases,
bias falls and variance rises. The bias effect is more important for
overrejection (see Simonoff, 1993), and the
extent of overrejection decreases. According to the conventional, but
wrong, wisdom, the story would be that as M increases further
bias continues to fall but variance increases so much that overrejections
become worse.
4 This common misperception is
an unfortunate result of folklore in the econometrics literature that
states that “as M increases, bias in
decreases but variance increases.” This statement is quite
misleading although the source is easy to pinpoint. A careful reader of
Priestley (1981) will repeatedly see the phrase
“as M increases, bias in
decreases but variance increases.” This phrase is completely
correct if, as in Priestley (1981), one is
discussing the properties of spectral density estimators at nonzero
frequencies or, in the case of known mean zero data, at the zero
frequency. However, this phrase does not apply to zero frequency
estimators computed using demeaned data as is the case in GMM models. This
fact is well known and has been nicely illustrated by Figure 2 in Ng and
Perron (1996) where plots of the exact bias and
variance of
are given for AR(1) processes.
This is not what is happening. In
fact, as
M increases further, bias eventually increases and
variance falls. It is the increase in bias that leads to the large
overrejections when
M is large. The reason that bias increases
and variance shrinks as
M gets large is easy to explain. When
M is large, high weights are placed on high-order sample
autocovariances. In the extreme case where full weight is placed on all of
the sample autocovariances, it is well known that
,
and this occurs because
is computed using residuals that have a sample average of zero.
Obviously,
is an estimator with large bias and zero variance. Thus, as M
increases,
is pushed closer to the full weight case.
4. A NEW ASYMPTOTIC THEORY
4.1. Distribution of
in the Fixed-b Asymptotic Framework
We now develop a distribution theory for
in the fixed-b asymptotic framework.
5 The results in this section are a fixed-b asymptotic
analysis of nonparametric zero frequency spectral density estimators.
Fixed-b asymptotic results for spectral density estimators at
nonzero frequencies have been obtained by Hashimzade and Vogelsang (2003).
We proceed under the asymptotic
nesting that
M =
bT where
b ∈ (0,1]
is fixed. The limiting distribution of
in the fixed-b asymptotic framework can be written in terms of
Qi(b), an i ×
i random matrix that takes on one of three forms depending on the
second derivative of the kernel. The following definition gives the forms
of Qi(b).
DEFINITION 1. Let the i × i random matrix
Qi(b) be defined as follows. Case
(i): if k(x) is twice continuously differentiable
everywhere,
Case (ii): if k(x) is continuous,
k(x) = 0 for |x| ≥ 1, and
k(x) is twice continuously differentiable everywhere
except for |x| = 1,
where k−′(1) =
limh→0[(k(1) − k(1
− h))/h], i.e.,
k−′(1) is the derivative of
k(x) from the left at x = 1. Case (iii): if
k(x) is the Bartlett kernel
The moments of Qi(1) have been derived by
Phillips et al. (2003) for the case of positive
definite kernels. Hashimzade, Kiefer, and Vogelsang (2003) have generalized those results for
Qi(b) while also relaxing the
positive definite requirement. The moments are
where Ii is the i ×
i identity matrix, κii is the standard
commutation matrix,6
The standard notation
for the commutation matrix is usually Kii
(see Magnus and Neudecker, 1999, p. 46). We are
using alternative notation because Kij is
used for a different matrix in our proofs.
and the range of all
integrals is 0 to 1. Using these moment results, Hashimzade et al. (
2003) prove that lim
b→0
E(
Qi(
b)) =
Ii and lim
b→0
var(vec(
Qi(
b))) =
0. As an
illustrative example, consider the Bartlett kernel where
We first consider the asymptotic distribution of
for the case of exactly identified models.
THEOREM 1. (Exactly identified models). Suppose that q =
p. Let M = bT where b ∈ (0,1] is fixed.
Let Qp(b) be given by Definition 1
for i = p. Then, under Assumptions 1–4, as T →
∞,
Several useful observations can be made regarding this theorem. Under
fixed-b asymptotics,
converges to a matrix of random variables (rather than constants) that is
proportional to Ω through Λ and Λ′. This contrasts with
the small-b asymptotic approximation where
is approximated by the constant Ω. Because, as b → 0
p lim Qp(b) =
Ip, it follows from Lemma 1 that p
limb→0
ΛQp(b)Λ′ = Ω.
Thus, the fixed-b asymptotics coincides with the standard
small-b asymptotics as b goes to zero. The advantage of
the fixed-b asymptotic result is that the limit of
depends on the kernel through k′′(x) and
k−′(1) and on the bandwidth through
b but is otherwise nuisance parameter free. Therefore, it is
possible to obtain a first-order asymptotic distribution theory that
explicitly captures the choice of kernel and bandwidth. Under
fixed-b asymptotics, any choice of bandwidth leads to
asymptotically pivotal tests of hypotheses regarding θ0
when using
to construct standard errors (details are given subsequently). Note that
Theorem 1 generalizes results obtained by Vogelsang (2003) where the focus was b = 1.
When q > p and the model is overidentified, the
limiting expressions for
are more complicated, and asymptotic proportionality to Ω
no longer holds. This was established for the special case of b =
1 by Vogelsang (2003). This does not imply,
however, that valid testing is not possible when using
in overidentified models because the required asymptotic proportionality
does hold for
,
the middle term in
.
The following theorem provides the relevant result.
THEOREM 2. (Overidentified models). Suppose that q >
p. Let M = bT where b ∈ (0,1] is fixed.
Let Qp(b) be given by Definition 1
for i = p. Define Λ* =
G0′W∞Λ. Under
Assumptions 1–4, as T → ∞,
This theorem shows that
is asymptotically proportional to Λ*Λ*′ and otherwise only
depends on the random matrix Qp(b).
It directly follows that
is asymptotically proportional to V, and asymptotically pivotal
tests can be obtained.
4.2. Inference
We now examine the limiting null distributions of tests regarding
θ0 under fixed-b asymptotics. Consider the
hypotheses
where r(θ) is an m × 1 vector (m
≤ p) of continuously differentiable functions with first
derivative matrix R(θ) =
∂r(θ)/∂θ′. Applying the delta
method to Lemma 1 we obtain
where VR =
R(θ0)VR(θ0)′.
Using (8) one can construct the standard HAC robust Wald test of the null
hypothesis or a t-test in the case of m = 1. To remain
consistent with earlier work, we consider the F-test version of
the Wald statistic defined as
When m = 1 the usual t-statistic can be computed
as
Often, the significance of individual parameters is of interest, which
leads to t-statistics of the form
where
is the ith diagonal element of the
matrix. To avoid any confusion, note that these statistics are being
computed in exactly the same way as under the standard approach. Only the
asymptotic approximation to the sampling distribution is different.
Note that some kernels, including the Tukey–Hanning, allow
negative variance estimates. In this case some convention must be adopted
in calculating the denominator of the test statistics. Equally arbitrary
conventions include reflection of negative values through the origin or
setting negatives to a small positive value. Although our results apply to
kernels that are not positive definite, we see no merit in using a kernel
allowing negative estimated variances absent a compelling argument in a
specific case. Nevertheless, we have experimented with the
Tukey–Hanning and trapezoid kernels, and results not reported here
do not support their consideration over a kernel guaranteeing positive
variance estimates.
The following theorem provides the asymptotic null distributions of
F and t.
THEOREM 3. Let b ∈ (0,1] be a constant and
suppose M = bT. Let Qi(b)
be given by Definition 1 for i = m. Then, under Assumptions
1–4 and H0, as T → ∞,
Theorem 3 shows that under fixed-b asymptotics,
asymptotically pivotal tests are obtained and the asymptotic distributions
reflect the choices of kernel and bandwidth. This contrasts with
asymptotic results under the standard approach, where F would
have a limiting χm2/m
distribution and t a limiting N(0,1) distribution
regardless of the choice of M and k(x). As
discussed in Section 4, as b → 0, p lim
Qm(b) =
Im and the fixed-b asymptotics
reduces to the standard small-b asymptotics. Therefore, if a
traditional bandwidth rule is used in conjunction with the
fixed-b asymptotics, in large samples the two asymptotic theories
will coincide. However, because the value of b is strictly
greater than zero in practice, it is natural to expect fixed-b
asymptotics to deliver a more accurate approximation. The simulation
results reported in Section 5 and in the working paper (Kiefer and Vogelsang, 2005), indicate that this is
true in some Gaussian models. Finite-sample results reported by Ravikumar,
Ray, and Savin (2004) indicate that the
fixed-b asymptotic approximation can substantially reduce size
distortions in tests of joint hypotheses, especially when the number of
hypotheses being tested is large.
A theoretical comparison of the accuracy of the fixed-b
asymptotics with the small-b asymptotics is not currently
available because existing methods in higher order asymptotic expansions,
such as Edgeworth expansions, do not directly apply to the
fixed-b asymptotic nesting given the nonstandard nature of the
distribution theory. Obtaining such theoretical results appears difficult,
although for the Gaussian local model and the special case of the Bartlett
kernel with b = 1 Jansson (2004) has
shown that the error in rejection probability of F is
O(T−1 log(T)). Phillips et al.
(2004) conjecture that this result can be
strengthened to O(T−1) and can be
generalized to include b < 1 and other kernels besides the
Bartlett kernel.
4.3. Asymptotic Critical Values
The limiting distributions given by Theorem 3 are nonstandard.
Analytical forms of the densities are not available with the exception of
t for the case of the Bartlett kernel with b = 1 (see
Abadir and Paruolo, 2002; Kiefer and Vogelsang, 2002b). However, because the
limiting distributions are simple functions of standard Brownian motions,
critical values are easily obtained using simulations. In the working
paper (Kiefer and Vogelsang, 2005) we provide
critical values for the t-statistic for a selection of popular
kernels (see the formula appendix for formulas for the kernels).
Additional critical values for the F-test will be made available
in a follow-up paper.
To make the use of the fixed-b asymptotics easy for
practitioners we provide critical value functions for the
t-statistic using the cubic equation
For a selection of well-known kernels, we computed the
cv(b) function for the percentage points 90%, 95%,
97.5%, and 99%. Critical values for the left tail follow by symmetry
around zero. The ai coefficients are given in
Table 1. They were obtained as follows. For each
kernel and the grid b = 0.02,0.04,…,0.98,1.0 critical
values were calculated via simulation methods using 50,000 replications.
Normalized partial sums of 1,000 i.i.d. N(0,1) random deviates
were used to approximate the standard Brownian motions in the respective
distributions given by Theorem 3. For each percentage point, the simulated
critical values were used to fit the cv(b) function by
ordinary least squares (OLS). The intercept was constrained to yield the
standard normal critical value so that cv(0) coincides with the
standard asymptotics. Table 1 also reports the
R2 from the regressions, and the fits are excellent in
all cases (R2 ranging from 0.9825 to 0.9996).
Fixed-b asymptotic critical value function coefficients for
t: cv(b) = a0 +
a1 b + a2
b2 + a3
b3
5. CHOICE OF KERNEL AND BANDWIDTH:
PERFORMANCE
In this section we analyze effects of the choice of kernel and
bandwidth on the performance of HAC robust tests. We focus on accuracy of
the asymptotic approximation under the null and on local asymptotic power.
As far as we know, our analysis is the first to explore theoretically the
effects of kernel and bandwidth choice on the power of HAC robust
tests.
5.1. Accuracy of the Asymptotic Approximation under the
Null and Comparison with the Bootstrap
The way to evaluate the accuracy of an asymptotic approximation to a
null distribution, or indeed any approximation, is to compare the
approximate distribution to the exact distribution. Sometimes this can be
done analytically; more commonly the comparison can be made by simulation.
We argued earlier that our approximation to the distribution of HAC robust
tests was likely to be better than the usual approximation because ours
takes into account the randomness in the estimated variance. However, as
noted, that argument is unconvincing in the absence of evidence on the
approximation's performance. We provide results for two popular
positive definite kernels: Bartlett and QS. Results for the Parzen,
Bohman, and Daniell kernel are similar and are not reported here. The
working paper (Kiefer and Vogelsang, 2005)
contains additional finite-sample simulation results that are similar to
what is reported here.
The simulations were based on the simple location model (7) with the
same design as described previously. Figures 1
and 2 provide plots of empirical rejection
probabilities using the fixed-b asymptotics. Compared to the
standard small-b asymptotics, the results are striking. In nearly
all cases, the tendency to overreject is substantially reduced. The
exceptions are for small values of M where the fixed-b
asymptotics show only small improvements. But, this is to be expected
given that the two asymptotic theories coincide as b goes to 0.
In the case of i.i.d. errors, the fixed-b asymptotics is
remarkably accurate. In the case of moving average (MA) errors,
fixed-b asymptotics gives tests with rejection probabilities
close to 0.05. In the case of autoregressive (AR) errors, overrejections
decrease as M increases. Intuitively, the fixed-b
asymptotics is capturing the variance of
and the downward bias in
that can be substantial when M is large. As M
increases, the tendency to overreject falls. The QS kernel delivers tests
with smaller size distortions than the Bartlett kernel, especially when
M is large.
7 A recent paper by
Müller (2004) provides a novel way of
theoretically quantifying the size robustness properties of HAC robust
tests as a function of the kernel and b. The robustness measure
proposed by Müller (2004) closely matches
the patterns seen in the finite-sample simulations.
The results
in Figures
1 and
2
strongly suggest that the use of fixed-
b asymptotic critical
values greatly improves the performance of HAC robust
t-tests.
Note however, that as serial correlation in the errors becomes
stronger, the tendency to overreject increases even when fixed-b
critical values are used. The reason is that the functional central limit
theorem approximation begins to erode. Therefore, there may be potential
for the bootstrap to provide further refinements in accuracy. The
fixed-b asymptotics suggests that the bootstrap could work
because the HAC robust tests are asymptotically pivotal for any kernel and
bandwidth. Ideally, one would like to establish theoretically whether or
not the bootstrap provides an asymptotic refinement. Such a theoretical
exercise is well beyond the scope of this paper because higher order
expansions for fixed-b asymptotics have not been developed.
Obtaining such expansions is an important topic that deserves
attention.
To show the potential of the bootstrap for HAC robust tests, we
expanded our simulation study to include the block bootstrap. We
implemented the block bootstrap resampling scheme following Götze and
Künsch (1996) as follows. The original
data,
y1,y2,…,yT,
are divided into T − l + 1 overlapping blocks of
length l. For simplicity, we report results for l = 1,4,
both of which factor into T = 50. The T/l
blocks are drawn randomly with replacement to form a bootstrap sample
labeled
y1*,y2*,…,yT*.
The bootstrap t-statistic is computed as
where
is computed using formula (6) with
.
We use the label “naive” because the bootstrap
t-statistic is computed using the same formula for
as was used for the original data.
8 We
considered implementing the recentered and rescaled version of the
bootstrap as proposed by Götze and Künsch (1996). Because Götze and Künsch (1996) recommend using the block length, l,
equal to the HAC bandwidth, M, it is difficult to implement this
bootstrap for the full range of M values given in Figures 1 and 2. The problem is that
for large values of M, l is so large that the resampling
is based on a very small number of blocks and it breaks down. We leave a
comparison of the naive bootstrap and the Götze and Künsch
(1996) bootstrap to future
research.
Empirical rejection probabilities using the naive bootstrap critical
values are reported in Figures 1 and 2 and are labeled NBoot. There are some compelling
patterns in the figures. Contrary to theoretical results by Davison and
Hall (1993) and Götze and Künsch
(1996) suggesting otherwise, the naive bootstrap
performs quite well; it dominates the standard normal approximation and
has rejection probabilities that are very similar to the fixed-b
asymptotics. This is true even for the i.i.d. (l = 1) bootstrap
when the data are not i.i.d. In addition, the fact that the naive
bootstrap works well when using the Bartlett kernel contrasts with
conventional wisdom that claims bootstrapping Bartlett kernel tests are
not more accurate than the standard normal approximation.
It is quite striking how closely the naive bootstrap with l =
1 (the i.i.d. bootstrap) follows the fixed-b approximation
regardless of the error structure. Recent work by Goncalves and Vogelsang
(2004) has shown that this pattern is
systematic. Goncalves and Vogelsang (2004) prove
that the naive bootstrap has the same first-order asymptotic distribution
as the t-statistic under fixed-b asymptotics, including
the case when l = 1. The simulations also suggest that the naive
bootstrap could be more accurate than fixed-b asymptotics with
suitable choice of the block length. Focusing on the cases of AR(1) errors
with l = 4, notice that the naive bootstrap often has rejection
probabilities closer to 0.05 than fixed-b. Although the naive
bootstrap may provide a refinement over the fixed-b asymptotics,
such a refinement has not been formally established. Higher order
properties of the naive bootstrap under fixed-b asymptotics are a
very interesting, but potentially difficult, area where more work is
needed.
5.2. Local Asymptotic Power
Whereas the existing HAC literature has almost exclusively focused on
the MSE of the HAC estimator to guide the choice of kernel and bandwidth,
a more relevant metric is to focus on the power of the resulting tests. In
this section we compare power of HAC robust t-tests using a local
asymptotic power analysis within the fixed-b asymptotic
framework. Our analysis permits comparison of power across bandwidths and
across kernels. Such a comparison is not possible using the traditional
first-order small-b asymptotics because local asymptotic power is
the same for all bandwidths and kernels.
For clarity, we restrict attention to linear regression models. Given
the results in Theorem 1, the derivations in this section are very simple
extensions of results given by Kiefer and Vogelsang (2002b). Therefore, details are kept to a minimum.
Consider the regression model
with θ0 and xt being
p × 1 vectors. In terms of the general model we have
f (vt,θ0) =
xt(yt −
xt′θ0). Without loss of
generality, we focus on θi0, one element of
θ, and consider null and alternative hypotheses:
where c ≥ 0 is a constant. If the regression model
satisfies Assumptions 1–4, then we can use the results of Theorem 1
and results from Kiefer and Vogelsang (2002b) to
easily establish that under the local alternative, H1,
as T → ∞,
where
.
Asymptotic power curves can be computed for given bandwidths and
kernels by simulating the asymptotic distribution of t based on
(10) for a range of values for δ and computing rejection probabilities
with respect to the relevant null critical value. Using the same
simulation methods as for the asymptotic critical values, local asymptotic
power was computed for δ = 0,0.2,0.4,…,4.8,5.0 using 5%
asymptotic null critical values.
The power results are reported in two ways. Figures 3, 4, 5, 6, 7, 8, 9, and 10 plot power across
the kernels for a given value of b. Figures 11, 12, 13, 14, and 15 plot power across values of b for a given
kernel. Figures 3, 4,
5, 6, 7, 8, 9, and 10 show that for
small bandwidths, power is essentially the same across kernels. As
b increases, it becomes clear that the Bartlett kernel has the
highest power, whereas the QS and Daniell kernels have the lowest power.
If power is the criterion used to choose a test, then the Bartlett kernel
is the best choice within this set of five kernels. If we compare the
Bartlett and QS kernels, we see that the power ranking of these kernels is
the reverse of their ranking based on accuracy of the asymptotic
approximation under the null.
Local asymptotic power of t, b = 0.02.
Local asymptotic power of t, b = 0.06.
Local asymptotic power of t, b = 0.01.
Local asymptotic power of t, b = 0.3.
Local asymptotic power of t, b = 0.5.
Local asymptotic power of t, b = 0.7.
Local asymptotic power of t, b = 0.9.
Local asymptotic power of t, b = 1.0.
Local asymptotic power of t, Bartlett kernel.
Local asymptotic power of t, Parzen kernel.
Local asymptotic power of t, Bohman kernel.
Local asymptotic power of t, Daniell kernel.
Local asymptotic power of t, QS kernel.
Figures 11, 12,
13, 14, and 15 show how the choice of bandwidth affects power.
Regardless of the kernel, power is highest for small bandwidths and lowest
for large bandwidths, and power is decreasing in b. These figures
also show that power of the Bartlett kernel is least sensitive to
b, whereas power of the QS and Daniell kernels is the most
sensitive to b. Again, power rankings of b are the
opposite of rankings of b based on accuracy of the asymptotic
approximation under the null.
5.3. Size-Power Trade-Off
The finite-sample simulations and local asymptotic power calculations
clearly indicate a trade-off between size distortions and power with
regard to choice of kernel and bandwidth when using fixed-b
critical values. Smaller bandwidths lead to tests with higher power but at
the cost of greater size distortions, whereas larger bandwidths lead to
tests with smaller size distortions but lower power. Similar trade-offs
occur across kernels.9
Simulations reported
by Phillips et al. (2003) and Phillips et al.
(2005) found a similar trade-off between size
distortion and power with respect to the exponent of exponentiated
kernels.
Balancing the size-power trade-off in some systematic
way could lead to the development of useful bandwidth rules that deliver
tests with desirable properties.
10 In the
context of trend function inference analyzed using fixed-b
asymptotics, Bunzel and Vogelsang (2005) propose
a data dependent bandwidth rule that maximizes integrated power. They
focus only on power because the class of tests they consider are very
robust in terms of size distortions. Their approach does not naturally
apply to the models in this paper because their size correction technique
does not easily apply to models with stochastic regressors.
Indeed, for the exponentiated Bartlett kernel, Phillips et al. (
2004) propose a data dependent rule for the choice of
exponent that minimizes a loss function that is a weighted sum of type I
and type II errors. They develop higher order expansions within the
fixed-exponent asymptotic framework and use these expansions to
analytically quantify their loss function. The authors conjecture that
their results can be extended to traditional kernels via fixed-
b
asymptotics. Thus it seems promising that fixed-
b asymptotics can
be used to develop data dependent bandwidth rules that balance size
distortion and power.
6. CONCLUSIONS
We have provided a new approach to the asymptotic theory of HAC robust
testing. We consider tests based on the popular nonparametric kernel
estimates of the standard errors. We are not proposing new tests, but
rather we propose a new asymptotic theory for these well-known tests. Our
results are general enough to apply to stationary models estimated by GMM.
In our approach, b, the ratio of bandwidth to sample size, is
held constant when deriving the asymptotic behavior of the relevant
covariance matrix estimator (i.e., zero frequency spectral density
estimator). Thus we label our asymptotic framework
“fixed-b” asymptotics. In standard asymptotics,
b is sent to zero and can be viewed as a
“small-b” asymptotic framework. Fixed-b
asymptotics improves upon two well-known problems with the standard
approach. First, as has been well documented in the literature, the
standard asymptotic approximation of the sampling behavior of tests is
often poor. Second, the kernel and bandwidth choice do not appear in the
approximate distribution, leaving the standard theory silent on the choice
of kernel and bandwidth with respect to properties of the tests. Our
theory leads to approximate distributions that explicitly depend on the
kernel and bandwidth. The new approximation performs much better and gives
insight into the choice of kernel and bandwidth with respect to test
behavior. Once the higher order results of Phillips et al. (2004) have been extended to the fixed-b
framework, data dependent bandwidth rules that balance size distortion and
power can be developed. In addition, fixed-b asymptotics should
be useful in explaining the performance of the naive bootstrap when
applied to HAC robust tests.
The new approximations should be used for HAC robust test statistics
for any choice of kernel and bandwidth. The fixed-b approximation
is an unambiguous improvement over the standard normal approximation in
most cases considered. We show that size distortions are reduced when
large bandwidths are used but so is asymptotic power. Generally there is a
trade-off in bandwidth and kernel choice between size (the accuracy of the
approximation) and power. Among a group of popular kernels, the QS kernel
leads to the least size distortion, whereas the Bartlett kernel leads to
tests with highest power (and generally acceptable size distortion when
large bandwidths are used).
APPENDIX: Proofs
We first define some relevant functions and derive preliminary results
before proving the lemma and theorems. Define the functions
Notice that
Because k(r) is an even function around r =
0, DT*(−r) =
DT*(r). If
k*′′(r) exists then
limT→∞
DT*(r) =
k*′′(r) by the definition of the second
derivative. If k*′′(r) is continuous, then
DT*(r) converges to
k*′′(r) uniformly in r. Define the
stochastic process
for 1/T ≤ r ≤ 1 and
XT(r) = 0 for 0 ≤ r <
1/T (i.e., set g0(θ0) =
0). It directly follows from Assumptions 2–4 that
Proof of Lemma 1. Setting t = T,
multiplying both sides of (4) by
,
and using the first-order condition
gives
Solving (A.2) for
and scaling by T1/2 gives
Because
by Assumptions 3 and 4, it follows from (A.1) that
We now prove Theorem 2. The proof of Theorem 1 follows using similar
arguments and is omitted.
Proof of Theorem 2. Define the random process
for
.
Plugging in for
using (4) gives
using (A.3). It directly follows from Assumptions 3 and 4 and (A.1)
that
Straightforward algebra gives
Using algebraic arguments similar to those used by Kiefer and
Vogelsang (2002b), it is straightforward to show
that
To simplify notation, define
.
Using (A.5) and the facts that
which follow from (3), it directly follows that
The rest of the proof is divided into three cases.
Case 1: k(x) is twice continuously differentiable.
Using (A.6) it follows that
using the continuous mapping theorem. The final expression is obtained
using k*′′(x) =
(1/b2)k′′(x/b).
Case 2: k(x) is continuous, k(x) =
0 for |x| ≥ 1, and k(x) is
twice continuously differentiable everywhere except for
|x| = 1. Let 1([bull ]) denote the indicator
function. Noting that Δ2Kij =
0 for |i − j| >
[bT] + 1,
Δ2Kij =
−k([bT]/bT) =
−k*([bT]/T) for
|i − j| = [bT] +
1, and Δ2Kij =
[k([bT]/bT) −
k(([bT]/bT) −
(1/bT))] +
k([bT]/bT) =
[k*([bT]/T) −
k*(([bT]/T) −
(1/T))] +
k*([bT]/T) for |i
− j| = [bT], break up the double
sum following the second = in (A.6) to obtain
Using the fact that CT,i =
Ci,T = 0, it follows that
By definition
in which case we can write
Therefore it follows that
where the second two terms of (A.8) are
op(1) because
Note that the op(1) terms become
identically zero when bT is an integer. Using (A.9) the
right-hand side of (A.7) simplifies to
Using this expression we can write
Let k_*′(b) denote the first
derivative of k*(x) from the left at x =
b. By definition
Therefore, by the continuous mapping theorem
Case 3: k(x) is the Bartlett kernel. It is easy to
calculate that Δ2Kij =
2/bT for |i − j| = 0,
Δ2Kij =
−(1/bT) + 1 −
([bT]/bT) for |i −
j| = [bT],
Δ2Kij = −(1 −
([bT]/bT)) for |i −
j| = [bT] + 1, and
Δ2Kij = 0 otherwise. Using
similar algebraic calculations as were done in Case 2, it is
straightforward to show that
Proof of Theorem 3. We only give the proof for F as
the proof for t follows using similar arguments. Applying the
delta method to the result in Lemma 1 and using the fact that
Bq(1) is a vector of independent standard
normal random variables gives
where Λ*j*j*j*j is the matrix square root of
R(θ0)(G0′W∞
G0)−1Λ*Λ*′(G0′W∞
G0)−1R(θ0)′.
Using the results in Theorem 2, it directly follows that
where we use the fact that
Using (A.11) and (A.12) it directly follows that
which completes the proof. █
