Economics Letters 155 (2017) 104–110
Contents lists available at ScienceDirect
Economics Letters
journal homepage: www.elsevier.com/locate/ecolet
Tests for serial correlation of unknown form in dynamic least squares
regression with wavelets✩
Meiyu Li, Ramazan Gençay *
Department of Economics, Simon Fraser University, 8888 University Drive, Burnaby, BC, V5A 1S6, Canada
article
info
Article history:
Received 12 March 2017
Accepted 18 March 2017
Available online 29 March 2017
JEL classification:
C1
C2
C12
C22
C26
a b s t r a c t
This paper extends the multi-scale serial correlation tests of Gençay and Signori (2015) for observable time
series to unobservable errors of unknown forms in a linear dynamic regression model. Our tests directly
build on the variance ratio of the sum of squared wavelet coefficients of residuals over the sum of squared
residuals, utilizing the equal contribution of each frequency of a white noise process to its variance and
delivering higher empirical power than parametric tests. Our test statistics converge to the standard
normal distribution at the parametric rate under the null hypothesis, faster than the nonparametric test
using kernel estimators of the spectrum.
© 2017 Elsevier B.V. All rights reserved.
Keywords:
Dynamic least squares regression
Serial correlation
Conditional heteroscedasticity
Maximum overlap discrete wavelet
transformation
1. Introduction
Existing residual based serial correlation tests for dynamic
models fall into two categories, the parametric tests in the time
domain and the nonparametric tests in the frequency domain. The
residual based parametric tests estimate autocorrelation coefficients directly, delivering easy implementation and desirable finite
sample performance (Hayashi, 2000; Box and Pierce, 1970; Godfrey, 1978; Wooldridge, 1990, 1991), whereas their power relies on
the choice of lag length heavily. In contrast, the residual based nonparametric tests compare the kernel-based spectrum estimator
with the white noise spectrum and have more power (Hong, 1996;
Hong and Lee, 2003). Nevertheless, the nonparametric estimation
of the underlying spectrum sacrifices the convergence rate, which
is slower than the parametric rate and often associated with poor
finite sample performance (Chen and Deo, 2004a, b).
Based on the fact of the equal contribution of each frequency
of a white noise process to its variance, Gençay and Signori
✩ We gratefully acknowledge Bertille Antoine for providing helpful comments
and suggestions. All errors are attributed to the authors.
Corresponding author.
E-mail addresses: [email protected] (M. Li), [email protected] (R. Gençay).
*
http://dx.doi.org/10.1016/j.econlet.2017.03.021
0165-1765/© 2017 Elsevier B.V. All rights reserved.
(2015) proposes a family of wavelet-based serial correlation tests
for observable time series, GS and GSM, utilizing only wavelet
coefficients of the observed time series data. This paper extends
the GS and GSM tests into a linear regression framework, particularly when lagged dependent variables are included in regression
equations. Our modified tests are robust to models with conditionally heteroscedastic errors of unknown form. Inheriting the test
design of Gençay and Signori (2015) by using the additive variance
decomposition of the wavelet and the scaling coefficients, instead
of any nonparametric estimation of the underlying spectrum, our
test statistics converge to the normal distribution at the parametric
rate under the null hypothesis (faster than the nonparametric test)
and display higher power than the parametric test. In addition,
contrary to the sensitiveness of finite sample performance to the
choice of lag length for the parametric test, and the choice of
bandwidth for the nonparametric test, our tests are rather stable
when different wavelet decomposition levels are utilized.
This paper proceeds as follows. Section 2 illustrates the modeling environment for our residual-based tests. Section 3 proposes
The corresponding modified test statistics. Section 4 reports comprehensive Monte Carlo simulations for finite sample performance
of our tests in comparison to commonly used tests. Section 5
contains concluding remarks of this paper.
M. Li, R. Gençay / Economics Letters 155 (2017) 104–110
2. Model setting
Consider the dynamic regression model
yt = Yt′ α + Wt′ γ + ut = Xt′ β + ut
(1)
where: Yt = (yt −1 , . . . , yt −P ) with P ≥ 1; α = (α1 , . . . , αP ) having
the values such that the roots of z P − α1 z P −1 − · · · − αP = 0 are
all strictly inside the unit circle; Wt is an L-vector of regressors
which may include lagged exogenous variables and constants;
Xt′ ≡ (Yt′ , Wt′ ); and β ′ ≡ (α ′ , γ ′ ). Let K = P + L denotes the
number of regression coefficients, and T denotes the number of
observations available for estimation. All of the data generation
processes are subject to the following two assumptions.
′
105
(2015) distribute more tightly around zero than a standard normal
distribution in dynamic models, more difficult to reject under the
null hypothesis and less powerful. For dynamic linear models, we
modify the statistics to restore the asymptotic distributions and
improve efficiency in small samples.
′
Assumption 2.1 (Ergodic Stationary). The (K + 1) dimensional
vector stochastic process {yt , Xt } is jointly stationary and ergodic.
Assumption 3.1 (Finite Fourth Moments for Regressors). E(u2t ut −j
ut −k ), E(ut ut −j ut −k xt −l,i ), E(ut ut −j xt −k,i xt −l,n ), E(ut −j xt ,i xt −k,n xt −l,s ),
E(xt ,i xt ,n xt −j,s xt −k,v ) exist and are finite for all j, k, l = 0, 2, . . . , Lm −
1 and i, n, s, v = 1, 2, . . . , K .
Theorem 3.1. If Assumptions 2.1, 2.2 and 2.3 are satisfied, then the
sample analogue Êm,T constructed with residuals ût from the model
in Eq. (1) is a consistent estimator of the wavelet variance ratio Em
∑T
wvarm (û)
Êm,T =
= ∑t =T 1
var(û)
Assumption 2.1 implies the stationarity and ergodicity of error
process, {ut }. Nevertheless, the unconditional homoscedasticity
does not exclude the models with a conditional heteroscedastic
error term. To be specific, for any stationary and ergodic process,
Assumption 2.1 allows any form of unconditional higher order
moment dependence which disappears as time lag increases, and
any form of time varying conditional higher order moments. We
consider tests for the following null hypothesis:
Assumption 2.3 (Stronger form of Predeterminedness). All regressors are predetermined such that: E(ut |ut −1 , ut −2 , . . . , Xt , Xt −1 ,
. . .) = 0.1
Throughout this paper, all the statistics constructed with
wavelets are based on the Maximum Overlap Discrete Wavelet
3
Transform (MODWT)
Define the mth level of wavelet coef∑Lfilter.
m −1
m
ficients as wm,t =
l=0 h̃m,l ut −l mod T , where Lm := (2 − 1)(L −
1) + 1, L is the length of the initial MODWT filter, and {h̃m,l } is the
mth level MODWT filter.
Assumption 2.3 indicates that cov(Xt −j , ut ) = 0, while it allows
cov(Xt +j , ut ) ̸ = 0 for some j > 0. The correlation between the
future lagged dependent variable and current error term makes the
asymptotic variance estimator in the GS and GSM tests inconsistent
when the realizations of error terms {ut } are replaced with residuals {ût } directly. As a result, the test statistics of Gençay and Signori
1 Assumption 2.3 is a sufficient condition for g
≡ [ut −j ut , ut Xt′ ]′ to be a
jt
martingale difference sequence, stronger than we need. We maintain this stronger
assumption for the convenience of interpretation.
2 The notation a − b mod T stands for ‘a − b modulo T ’. If j is an integer such
that 1 ≤ j ≤ T , then j mod T ≡ j. If j is another integer, then j mod T ≡ j + nT
where nT is the unique integer multiple of T such that 1 ≤ j + nT ≤ T . The periodic
boundary conditions have trivial impact on the finite sample distribution due to the
short length of most discrete wavelet filters.
3 For details of MODWT filter, see Gençay et al. (2001) and Percival and Walden
(2000).
2m
.
Further, if Assumption 3.1 is satisfied, then
√
LGm =
(
Ts4
Êm,T −
m m Ψ̂m Ĉm Hm
4H ′
Ĉ ′
1
)
2m
d
−
→ N(0, 1),
where {h̃m,l } is the wavelet filter used in the construction of Êm,T and
⎡
h̃m,1
h̃m,2
h̃m,3
⎡
⎤
h̃m,2
h̃m,3
h̃m,3
···
···
..
.
···
..
.
h̃m,Lm −1
h̃m,Lm −1
0
h̃m,Lm −1
0
0
.
···
⎢
⎢
⎢
Hm
≡⎢
⎢
((Lm −1)×1)
⎣
..
.
h̃m,Lm −1
0 ⎥
⎥
0 ⎥
⎥
..
.
..
⎥
⎦
0
⎤
h̃m,0
⎢ h̃m,1
⎢
⎢
× ⎢ h̃m,2
⎢ .
⎣ ..
Assumption 2.3 is stronger than the assumption of white noise
and needed for the derivation of the null distribution of our test
statistic. If Assumptions 2.1, 2.2 and 2.3 are satisfied, the OLS coefp
ficient estimator in Eq. (1) is consistent, namely β̂ −
→ β (Hayashi,
2000 Proposition 2.1(a), page 124). Throughout the paper, we assume Assumptions 2.1 and 2.2 hold and impose periodic boundary
conditions on all time series, like {ut }, where ut ≡ ut mod T .2
3. Asymptotic null distribution
1
p
−
→
2
t =1 ût
′
Assumption 2.2 (Rank Condition). The K × K matrix E(Xt Xt ) is nonsingular (and hence finite), which is denoted by ΣXX .
2
ŵm
,t
⎥
⎥
⎥
⎥,
⎥
⎦
h̃m,Lm −2
⎤
⎡
ĉ1
Ĉm
((Lm −1)(K +1)×(Lm −1))
..
⎢
≡⎣
⎥
⎦,
.
ĉLm −1
⎡
T
1∑
s2 ≡
1
T −K
T
∑
û2t , SXX =
t =1
T
1∑
T
T
t =1
T
Xt Xt′ , µ̄j ≡
t =1
−1
−SXX
µ̄j
,
⎤
Xt ût ût −j ⎥
t =1
T
û2t Xt Xt′
⎥
⎥,
⎥
⎦
t =1
T
1∑
T
]
1
′ 2
1∑
Xt û2t ût −k
T
1∑
≡
((K +1)×1)
û2t ût −j ût −k
⎢
⎢ T t =1
the (j, k) block of Ψ̂m ≡ ⎢
T
⎢ ∑
((K +1)×(K +1))
⎣ 1
T
[
ĉj
Xt ût −j .
t =1
Combining these multiple-scale tests to gain power against a
wide range of alternatives, we derive the asymptotic joint distribution of these tests as follows.
Theorem 3.2. If Assumptions 2.1, 2.2, 2.3 and 3.1 are satisfied, then
the vector
√
Ts4
4
(
1
Ê1,T −
21
, Ê2,T −
where the mth column of
0
.
((LN −Lm )(K +1)×1)
1
22
, . . . , ÊN ,T −
Υ
((LN −1)(K +1)×N)
is
1
2N
)′
d
−
→ N(0, Υ ′ ΨN Υ ),
C m Hm
((Lm −1)(K +1)×1)
followed by
106
M. Li, R. Gençay / Economics Letters 155 (2017) 104–110
Further, a feasible multivariate test statistic is
4
LGMN =
Ts
(
1
Ê1,T −
4
21
, . . . , ÊN ,T −
)
1
2N
(Υ̂ ′ Ψ̂N Υ̂ )−1
(
)′
1
1
d
× Ê1,T − 1 , . . . , ÊN ,T − N −
→ χN2 .
2
2
Large sample inference on the values of the vector (LG1 , . . . ,
LGN ) can be handily implemented using the χ 2 distribution. Indeed, it is a standard result (Bierens, 2004 Theorem 5.9, page 118)
that for a multivariate normal N-dimensional vector X and the
corresponding non-singular N × N variance–covariance matrix Σ ,
X ′ Σ −1 X is distributed as a χN2 .
Theorems 3.1 and 3.2 accommodate models with conditional
heteroscedastic error terms, such as ARCH and GARCH, due to the
estimation of Ψ . The fourth moments are easily estimated with
sample analogues but may slow the convergence of our tests for
very small samples. When errors satisfy conditional homoscedasticity, a more efficient test statistic is available to accelerate the
convergence rate in finite samples.
Assumption 3.2 (Stronger form of Conditional Homoscedasticity).
E(u2t |ut −1 , ut −2 , . . . , Xt , Xt −1 , . . .) = σ 2 > 0.
Theorem 3.3. If Assumptions 2.1, 2.2, 2.3 and 3.2 are satisfied, then
√
LGhm
=
(
T
′ (I
4Hm
Lm −1 − Φ̂m )Hm
Êm,T −
1
)
2m
d
−
→ N(0, 1),
where {h̃m,l } is the wavelet filter used in the construction of Êm,T and
∑
−1
Φ̂m ≡ (φ̂jk ), φ̂jk ≡ µ̄′j SXX
µ̄k /s2 , µ̄j ≡ 1/T Tt=1 Xt ût −j .
In addition, the vector
√ (
T
4
Ê1,T −
1
1
2
2
2N
, . . . , ÊN ,T −
2
d
−
→ N(0, H ′ (ILN −1 − ΦN )H),
where the mth column of
H
((LN −1)×N)
is
followed by
0
.
Hm
((LN −Lm )×1)
((Lm −1)×1)
Further, a feasible multivariate test statistic is
LGMNh =
T
4
(
Ê1,T −
1
1
2
2N
, . . . , ÊN ,T −
1
)
(H ′ (ILN −1 − Φ̂N )H)−1
)′
(
1
1
d
→ χN2 .
× Ê1,T − 1 , . . . , ÊN ,T − N −
2
2
4. Monte Carlo simulations
Consider the following linear dynamic regression model,
M: Yt = 1 + 0.5Yt −1 + 0.5W1,t + 1.5W2,t + ut , where the
exogenous variables W1,t = 0.01 + 0.4W1,t −1 + ν1,t and W2,t =
0.2W1,t −1 + 0.01W2,t −1 + ν2,t , with innovations
[
All the wavelet based test statistics from T1 to T4 are computed
with the MODWT Haar wavelet. The robustness of our modified
tests to different wavelet filters is discussed at the end of Section 4.
In addition, both T5 and T6 are computed with a quadratic norm
and the Daniell kernel function, k(z) = sin(π z)/π z, z ∈ (−∞, ∞),
which maximizes the power of the kernel test over a wide class of
the kernel functions when p → ∞ (Hong, 1996, 2010).
To study the empirical size of each test statistic, we consider
sample sizes of 50, 100, and 500 observations. For each of the following null hypotheses and sample sizes, we compute the rejection
rates from 5000 replications.
i.i.d
(1) Normal process ut ∼ N(0, 9).
)′
1
, Ê2,T −
1
T4: The estimated Gençay and Signori (2015) tests EGSm and
EGSMN computed with estimated heteroscedastic variance via
Newey–West estimator at scale m = 1, 2, 3 or level N = 2.
T5: The Hong (1996) test Kph with bandwidth p. Same as Hong
(1996), we use three rates: p1 = [ln(n)], p2 = [3n0.2 ] and
p3 = [3n0.3 ], where [a] denotes the integer closest to a.
T6: The Hong and Lee (2003) test Kp with same bandwidth p as
Kph . Kp is computed following the formula suggested in Hong
(2010).
T7: The Breusch (1978) and Godfrey (1978) test LMk with lag
lengths of k = 5, 10, 20.
T8: The Wooldridge (1990, 1991) test Wk with lag lengths of k =
5, 10, 20. Wk is computed following the two step regressions
suggested in Hong (2010): (i) Regress (ût −1 , . . . , ût −k ) on Xt and
save the estimated k × 1 residual vector v̂t ; (ii) Regress 1 on v̂t ût
and obtain SSR. Without serial correlation, N − SSR, robust to
conditional heteroscedasticity, follows χk2 asymptotically.
]
([ ] [
ν1,t
0
1
∼N
,
ν2,t
0
0.3
0.3
3
])
.
We examine finite sample performance of our test statistics in
comparison to some commonly used residual based tests for serial
correlations, which are detailed as follows.
T1: The modified conditional homoscedasticity constrained tests
LGhm and LGMNh at scale m = 1, 2, 3 or level N = 2.
T2: The modified conditional heteroscedasticity robust tests LGm
and LGMN at scale m = 1, 2, 3 or level N = 2.
T3: The original Gençay and Signori (2015) tests GSm and GSMN
computed with constrained homoscedastic variance at scale
m = 1, 2, 3 or level N = 2.
i.i.d
(2) ARCH(2) process ut = σt ϵt , ϵt ∼ N(0, 1), σt2 = 0.001 +
0.2ϵt2−1 + 0.4ϵt2−2 .
i.i.d
(3) N(0,1)-GARCH(1, 1) process ut = σt ϵt , ϵt ∼ N(0, 1), σt2 =
0.001 + 0.1ϵt2−1 + 0.7σt2−1 .
i.i.d
(4) t5 -GARCH(1, 1) process ut = σt ϵt , ϵt ∼ t5 , σt2 = 0.001 +
0.1ϵt2−1 + 0.7σt2−1 , where t5 stands for a Student’s t with 5
degrees of freedom.
i.i.d
(5) EGARCH(1, 1) process ut = σt ϵt , ϵt ∼ N(0, 1), ln(σt2 ) =
0.001 + 0.4|ϵt −1 | + 0.1ϵt −1 + 0.7 ln(σt2−1 ).
Hypothesis (1) permit us to examine size performances under
the conditional homoscedastic white noise errors. For dynamic
models, the larger the variance of the innovations, the more severe
the underrejections of the tests T3 and T4 are. To highlight the
impact of lagged dependent variables on these inconsistent tests,
we choose a large variance. Hypotheses (2), (3), (4) and (5) allow us
to focus on the circumstances of conditional heteroscedastic white
noise errors.
For conditional homoscedastic white noise errors, Table 1 illustrates the rejection rate of all tests under the null hypothesis
(1) at 10%, 5% and 1% levels of significance. The empirical sizes
of the LGhm and LGMNh tests match their nominal sizes even with
sample size as small as 50 observations. The LGm and LGMN tests
converge relatively slower, with sizes being viable with sample
size 100, and accurate with sample size 500. All of our modified
statistics are stable for varied wavelet decomposition levels and
model specifications. The performance of the LMk test is acceptable
but relies heavily on the choice of lag lengths. Analogous results
hold for Wk with slight underrejections in all circumstances. Both
Kph and Kp significantly underreject at all levels of significance,
indicating the slow convergence rate and right skewness in finite
samples. In addition, the performance of the two kernel tests are
vulnerable to the choice of bandwidth.
3.66
10.82
7.18
5.58
9.32
11.74
11.38
10.34
10.92
8.12
3.26
GS1
GS2
GS3
GSM2
LGh1
LGh2
LGh3
LGM2h
LM5
LM10
LM20
2.80
4.50
5.34
10%
Level
h
Kp1
h
Kp2
h
Kp3
50
T
1.74
2.56
3.44
5.52
2.96
0.72
4.76
6.26
6.76
5.06
1.18
5.64
3.46
2.50
5%
0.74
1.10
1.34
0.90
0.16
0.00
0.78
1.48
2.36
1.44
0.10
1.22
0.84
0.62
1%
3.96
5.60
6.48
10.20
9.16
5.14
9.56
10.54
10.22
9.58
3.54
10.02
7.10
5.26
10%
100
2.24
3.26
4.02
4.68
3.86
2.04
5.04
5.36
5.52
4.96
1.34
4.98
3.16
2.30
5%
0.78
1.48
1.66
0.82
0.52
0.10
1.00
1.22
1.88
1.30
0.14
1.14
0.76
0.42
1%
M : Yt = 1 + 0.5Yt −1 + 0.5W1,t + 1.5W2,t + ut and ut ∼ N(0, 9)
i.i.d.
5.22
6.60
7.90
9.68
10.10
9.48
9.52
9.94
10.08
9.68
3.52
9.80
7.38
4.72
10%
500
3.30
4.24
4.66
4.68
5.08
4.50
4.96
4.80
5.32
4.58
1.38
4.66
3.12
1.98
5%
1.40
1.60
1.88
0.82
0.94
0.82
1.02
0.90
1.10
1.08
0.08
0.90
0.62
0.34
1%
Kp1
Kp2
Kp3
W5
W10
W20
LG1
LG2
LG3
LGM2
EGS1
EGS2
EGS3
EGSM2
3.60
4.76
5.42
9.24
4.42
0.00
14.42
15.54
15.18
18.20
7.98
19.82
16.68
23.84
10%
50
2.08
2.76
3.30
3.40
0.86
0.00
8.00
8.92
8.20
10.18
4.20
12.48
11.28
16.98
5%
0.82
1.16
1.24
0.14
0.02
0.00
1.60
2.38
1.60
2.08
1.32
4.18
5.50
8.68
1%
4.16
5.66
6.50
10.54
7.46
2.90
11.74
12.72
11.72
12.82
6.04
15.84
15.28
15.14
10%
100
Table 1
Rejection probabilities in percentages of tests at 10%, 5% and 1% levels of significance under the null hypothesis (1). All size simulations are based on 5000 replications.
2.52
3.54
3.82
4.70
3.06
0.70
6.26
6.08
5.98
6.66
2.92
9.28
10.40
9.44
5%
0.92
1.44
1.54
0.58
0.30
0.00
1.26
1.30
1.00
1.48
0.44
2.66
4.94
3.50
1%
500
5.06
6.48
7.66
9.42
9.08
7.90
9.94
10.04
10.68
10.48
4.22
11.78
9.94
7.64
10%
5%
3.26
3.90
4.64
5.04
4.38
3.62
5.28
5.18
5.40
5.06
1.50
6.48
4.98
3.58
1%
1.26
1.58
1.80
0.94
0.86
0.60
1.16
1.02
1.00
1.08
0.08
1.36
1.32
0.84
M. Li, R. Gençay / Economics Letters 155 (2017) 104–110
107
108
M. Li, R. Gençay / Economics Letters 155 (2017) 104–110
Table 2
Rejection probabilities in percentages of tests at 10%, 5% and 1% levels of significance under the null hypothesis (2), (3), (4) and (5). All size simulations are based on 5000
replications.
M : Yt = 1 + 0.5Yt −1 + 0.5W1,t + 1.5W2,t + ut
N(0, 1) − GARCH(1, 1)
ARCH(2)
T
100
500
100
Level
10%
5%
1%
10%
5%
1%
10%
5%
1%
10%
5%
1%
LG1
13.24
7.04
1.40
10.36
5.00
1.00
12.92
6.40
1.58
10.20
4.98
1.00
LG2
12.56
5.92
0.78
11.10
5.50
1.00
12.44
6.54
1.48
10.16
5.04
0.88
LG3
12.94
6.42
1.22
10.52
5.08
0.94
12.18
6.14
1.20
10.46
5.60
0.98
LGM2
13.66
6.86
1.06
10.92
5.18
0.94
13.82
7.60
1.46
10.36
5.30
1.10
Kp1
9.48
6.30
2.92
9.00
6.12
2.86
10.68
7.32
3.58
9.32
6.54
2.90
Kp2
9.50
6.00
2.36
8.80
6.14
2.52
10.34
7.06
3.14
9.38
6.50
2.88
Kp3
9.08
5.82
2.32
9.04
5.62
2.34
10.30
6.92
2.96
10.02
6.38
2.42
W5
8.02
3.00
0.34
9.08
3.78
0.44
9.64
4.14
0.54
10.46
5.22
1.16
W10
6.02
1.88
0.08
8.30
3.64
0.44
7.20
2.68
0.30
10.00
4.60
0.84
W20
2.86
0.66
0.00
7.76
3.38
0.46
3.04
0.66
0.00
8.46
3.60
0.42
t5 − GARCH(1, 1)
500
EGARCH(1, 1)
LG1
12.32
5.82
1.06
10.56
5.16
0.76
12.40
6.52
1.46
9.58
4.58
0.96
LG2
12.18
6.24
1.04
10.40
5.30
1.00
12.22
6.56
1.16
10.28
5.56
1.26
LG3
11.38
5.80
1.24
10.18
4.74
0.82
12.00
6.34
1.26
10.26
5.58
1.18
LGM2
12.80
5.92
1.10
10.86
5.20
0.92
14.00
6.52
1.04
9.82
5.18
1.10
Kp1
8.82
6.12
2.88
9.36
6.22
2.78
8.06
5.50
2.70
8.10
5.42
2.46
Kp2
8.62
5.58
2.28
9.06
5.62
2.56
8.08
5.34
2.64
8.12
5.22
2.08
Kp3
8.44
5.20
2.02
8.56
5.26
2.10
8.44
5.56
2.38
8.44
5.00
1.72
W5
8.50
3.30
0.42
9.98
4.50
0.66
9.08
3.72
0.44
9.58
4.50
0.66
W10
6.20
2.60
0.16
9.00
4.00
0.56
6.64
2.32
0.22
9.28
4.28
0.64
W20
3.14
0.74
0.06
7.20
3.16
0.48
2.96
0.50
0.00
7.98
3.20
0.50
Table 3
Size adjusted power in percentages of tests at 1% level of significance under the alternative hypothesis (a), AR(2)-N(0,1) process. Simulations are carried out for a set of
alternatives obtained varying α1 and α2 in interval [−0.4, 0.4] in increments of 0.1. All rejection rates are based on 5000 replications with 1% nominal size for sample size
of 100 observations.
M. Li, R. Gençay / Economics Letters 155 (2017) 104–110
109
Table 4
Size adjusted power in percentages of tests at 1% level of significance under the alternative hypothesis (b), AR(2)-GARCH(1,1) process. Simulations are carried out for a set
of alternatives obtained varying α1 and α2 in interval [-0.4, 0.4] in increments of 0.1. All rejection rates are based on 5000 replications with 1% nominal size for samples
of 100 observations.
Table 5
Size and power in percentages for various wavelet families at 5% level of significance in model Yt = 1 + 0.5Yt −1 + 0.5W1,t + 1.5W2,t + ut , against the following hypotheses,
i.i.d
i.i.d
AR(2)-N(0,1): ut = α1 ut −1 + α2 ut −2 + ϵt and ϵt ∼ N(0, 1); AR(2)-GARCH(1,1): ut = α1 ut −1 + α2 ut −2 + σt ϵt , ϵt ∼ N(0, 1) and σt2 = 0.001 + 0.1ϵt2−1 + 0.7σt2−1 . All the
rejection rates are based on 5000 replications with sample sizes of 100, 500 and 1000 observations.
Model
Panel A: AR(2)-N(0,1), LGM2h
T
α1
α1
α1
α1
α1
α1
α1
α1
α1
= 0, α2 = 0
= 0, α2 = 0
= 0, α2 = 0
= 0.1, α2 = 0
= 0.1, α2 = 0
= 0.1, α2 = 0
= 0, α2 = 0.1
= 0, α2 = 0.1
= 0, α2 = 0.1
Haar
D(4)
Panel B: AR(2)-GARCH(1,1), LGM2
D(6)
LA(8)
BL(14)
C(6)
T
Haar
D(4)
D(6)
LA(8)
BL(14)
C(6)
100
500
1000
5.20
5.04
4.94
5.18
5.32
4.84
4.96
5.34
4.76
4.86
5.38
4.74
4.80
5.42
4.98
5.14
5.34
4.84
100
500
1000
6.94
4.84
5.58
6.84
4.70
5.30
6.86
4.78
5.50
7.02
4.92
5.48
7.12
5.04
5.58
6.84
4.72
5.34
100
500
1000
10.64
44.10
75.88
11.54
44.40
76.04
11.62
44.28
75.94
11.60
44.24
75.78
11.74
43.94
75.04
11.62
44.40
76.08
100
500
1000
10.62
38.80
69.34
10.56
38.46
69.12
10.22
37.98
68.98
10.18
37.76
68.94
10.02
37.56
68.18
10.46
38.36
69.16
100
500
1000
10.18
38.62
69.18
9.06
33.80
63.84
8.48
30.80
59.34
8.32
29.18
56.68
7.92
26.06
51.44
9.06
33.56
63.52
100
500
1000
11.16
34.42
61.20
10.78
31.10
56.68
10.16
29.28
53.36
10.10
27.96
51.12
9.80
25.50
47.14
10.62
31.00
56.34
Table 2 illustrates the rejection rate for conditional heteroscedastic white noise errors under the null hypotheses (2), (3),
(4) and (5). We only focus on the robust test statistics LGm , LGMN ,
Kp and Wk .4 LGm and LGMN deliver accurate sizes in sample with
500 observations under all hypotheses. For a small sample, they
slightly overreject but are rather stable over different levels of
decompositions. The Kp tests have great difficulty of getting it close
to their corresponding nominal sizes at all levels of significance
with all sample sizes, underrejecting at 10% level of significance
but significantly overreject at both 5% and 1% levels of significance.
The Wk tests underreject in all models and rely on the choice of lag
lengths heavily.
4 The simulation results for other test statistics are available upon request.
To examine the empirical size adjusted power, we consider
the following alternative hypotheses of errors in model M. All the
simulation results are based on 5000 replications with samples of
100 observations.5
i.i.d
(a) AR(2)-N(0,1) process ut = α1 ut −1 + α2 ut −2 + ϵt and ϵt ∼
N(0, 1).
(b) AR(2)-GARCH(1, 1) process ut = α1 ut −1 + α2 ut −2 + σt ϵt ,
i.i.d
ϵt ∼ N(0, 1) and σt2 = 0.001 + 0.1ϵt2−1 + 0.7σt2−1 .
Table 3 illustrates the empirical size adjusted power of
h
and LM5 against the alternative hypothesis (a)
tests LGM2h , Kp1
5 Size adjusted power is computed with, for a given sample size, the empirical
critical values obtained from Monte Carlo simulations with 50,000 replications.
110
M. Li, R. Gençay / Economics Letters 155 (2017) 104–110
AR(2)-N(0,1) process by varying α1 and α2 in interval [-0.4, 0.4] in
increments of 0.1.6 The rejection rates are computed with respect
to 1% nominal size for sample size of 100 observations. Bandwidth
p1 is chosen for the Kph test to guarantee its larger power, since a
faster p gives better size while a slower p gives better power (Hong,
1996). We choose k = 5 for the LMk test for the same reason.
The first column of the table contains the size adjusted power
of each test for various alternatives. In the second column, we
report the relative power gains of the conditional homoscedasticity
h
constrained test LGM2h with respect to the Kp1
test and the LM5 test.
Against the great majority of the alternatives, LGM2h outperforms
the LM5 test a large amount. Nevertheless, there is no clear edge
h
h
among the LGM2h test, the Kp1
test. Specifically, the Kp1
test has
higher power under slightly more number of alternatives with the
highest power gain around 46%. Yet the highest power gain of LGM2h
reaches 174% in other alternatives.
In Table 4, we repeat the previous power analysis for the tests
LGM2 , Kp1 and W5 against the alternative hypothesis (b), AR(2)GARCH(1, 1) process. Roughly speaking, the corresponding conditional heteroscedasticity robust tests inherit the relative power
performance in the previous alternative (a) process. The LGM2 test
performs neck to neck with the Kp1 test, leaving W5 dominated
significantly in all alternatives. The Kp1 test has higher power
against slightly more number of alternatives. Nevertheless, the
highest power gain of the Kp1 test is 41% while the relative power
gain of the LGM2 test achieves 310% against other alternatives.
Finally, we examine the effect of various wavelet families on
our modified tests, LGM2h and LGM2 , in Table 5. The wavelet filters investigated include Daubechies filters Haar, D(4), D(6), Least
Asymmetric filter LA(8), Best Localized filter BL(14) and Coiflet filter C(6). Overall, the size and power of wavelet tests do not exhibit
significant differences across various wavelet filters, although the
shorter filters displays larger power in small samples.
5. Conclusions
This paper proposes a test for serial correlation of unknown
form for the residuals from a linear dynamic regression model.
Our test possesses the same strong power as the nonparametric
test in the frequency domain and the convergence rate identical to
the parametric test in the time domain simultaneously. Our tests
build on the characteristics of the spectral density function of the
underlying process compared to that of the white noise process.
There is no intermediate step such as the kernel estimation of
the spectral density functions in our approach. With the additive
feature of the variance decomposition by the wavelet and the
6 Simulation results for the moving average processes are similar to those for
hypotheses (a) and (b), autoregressive processes. We omit them in the interest of
space and they are available upon request.
scaling coefficients, our test converges to the standard normal
distribution at the parametric rate, avoiding the issue of slower
convergence rate caused by nonparametric spectrum estimators.
Our test is easily implementable and calculated fast.
Acknowledgments
Ramazan Gençay gratefully acknowledges financial support
from the Natural Sciences and Engineering Research Council of
Canada and the Social Sciences and Humanities Research Council
of Canada. The software for the tests are available at www.sfu.ca/
~rgencay/.
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