Max-Correlation Test and Max-Causality Test for …motegi/slides_UNC_seminar_v5.pdfMax-Correlation Test and Max-Causality Test for Economic Time Series Kaiji Motegi1 1Graduate School

Max-Correlation Test and Max-Causality Testfor Economic Time Series

Kaiji Motegi1

1Graduate School of Economics, Kobe University

UNC Econometrics SeminarFebruary 17, 2017

Motegi (Economics, Kobe U.) Max-Correlation and Max-Causality Tests February 17, 2017 1 / 49

Introduction

Y

X

1. Are past values of Y useful for

predicting future Y?

-- White Noise Tests

2. Do past values of X serve as extra

information for predicting future Y?

-- Granger causality Tests

Present

Past Future

time

time


Introduction

White Noise

Granger

Causality

Hill and Motegi (2017a, WP)

Max-Correlation Test

Hill and Motegi (2017b, WP)

Stock Market Efficiency

Ghysels, Hill, and Motegi (2016, JoE)

Mixed Frequency Granger Causality

Ghysels, Hill, and Motegi (2017, R&R, JBES)

Max-Causality Test

Methodology Application


Table of Contents

1 Max-Correlation White Noise Test

1 Literature Review of White Noise Tests

2 Max-Correlation Test

3 Monte Carlo Simulations

2 Testing for Weak Form Efficiency of Stock Markets

1 Periodicity of Confidence Bands

2 Case Study (S&P 500)

3 Max-Causality Test (Overview)

4 Future Work


White Noise Tests: Motivation

We propose a new white noise test based on the largest sampleautocorrelation.

Two major fields where white noise tests are of use:

1 Testing for weak form efficiency of stock markets (cf. Fama,1965; Fama, 1970). Under weak form efficiency, stock returnsshould be white noise.

2 Residual diagnostics.

Establishing a formal white noise test is a relatively new andchallenging topic.

1 Infinitely many zero restrictions – H0 : ρ(h) = 0 for any h ≥ 1.2 No more than uncorrelatedness – weaker than IID or MDS.


White Noise Tests: Literature Review

Consider univariate covariance stationary {y1, . . . , yn}.Assume that E[yt] = 0 for notational simplicity.

Population quantities are

γ(0) = E[y2t ], γ(h) = E[ytyt−h], ρ(h) =γ(h)

γ(0).

Sample quantities are

γn(0) =1

n

n∑t=1

y2t , γn(h) =1

n

n∑t=h+1

ytyt−h, ρn(h) =γn(h)

γn(0).

White noise hypothesis is

H0 : ρ(h) = 0 for h ≥ 1.



Consider the Q-test (or portmanteau test):

Qn = nL∑

h=1

wn(h)ρ2n(h)

d→ χ2L.

wn(h) = (n+ 2)/(n− h) in Ljung and Box (1978).

Two reasons why the Q-test is not a white noise test:

1 It cannot capture autocorrelations beyond lag L.2 Asymptotic χ2 property requires the asymptotic independence

of {ρn(1), . . . , ρn(L)}, which holds when {yt} is IID.



The issue of fixed lag length L is addressed by Hong’s (1996)spectral test.

A version of Hong’s test based on the Bartlett kernel is:

Nn =1√2Ln

Ln∑h=1

wn(h)[nρ2n(h)− 1

] d→ N(0, 1),

where Ln → ∞ as n→ ∞ and Ln = o(n).

Asymptotic normality still requires serial independence.

Shao’s (2010, 2011) dependent wild bootstrap (DWB)enables us to construct a formal white noise test.



1 Set a block size bn.2 Generate iid {ξ1, ξ2, . . . , ξn/bn}. Define an auxiliary variable:

ω = [ ξ1, . . . , ξ1︸︷︷︸bn terms

, ξ2, . . . , ξ2︸︷︷︸bn terms

, . . . , ξn/bn , . . . , ξn/bn︸︷︷︸bn terms

]′.

3 Compute bootstrapped autocorrelations:

ρ(dw)n (h) =

1

γn(0)× 1

n

n∑t=h+1

ωt[ytyt−h − γn(h)], h = 1, . . . ,Ln,

and N(dw)n = (2Ln)

−1/2∑Ln

h=1wn(h)[n{ρ(dw)n (h)}2 − 1].

4 Repeat Steps 2-3 M times and compute the bootstrappedp-value p

(dw)n,M = 1

M

∑Mi=1 I(|N

(dw)n,i | ≥ |Nn|).



Block 2 Block 3

ξ1

y3y4, y4y5, y5y6 y6y7, y7y8, y8y9 y57y58, y58y59, y59y60y1y2, y2y3

・・・

・・・

Block 1 ・・・ Block 20

Compute bootstrapped autocorrelation

Repeat M times

Compute bootstrapped p-value

Generate i.i.d. random numbers ξ1, …, ξ20 ~ N(0, 1)

Examle:

h = 1; n = 60; bn = 3.

Preserved dependence within each block.

No dependence across different blocks.

Asymptotically correct size and consistency

under weak dependence (e.g. GARCH, bilinear).

ξ2ξ3 ξ20



We propose a max-correlation test statistic:

Tn =√n max {|ρn(1)|, . . . , |ρn(Ln)|} ,

where Ln → ∞ as n→ ∞ and Ln = o(n).

We use DWB for p-value computation.

’Sum of Squares’ versus ’Maximum’.

The max-approach is likely more robust against autocorrelationsat remote lags (e.g. seasonality).

Example: Lags 12, 24, 36, . . . in monthly data.



Classical max-approach (e.g. Berman, 1964) attempts to derivean asymptotic distribution of the largest autocovariance.

Xiao and Wu (2014) prove that

an

[√n max1≤h≤Ln |γn(h)− γ(h)|√∑∞

h=0 γ(h)2

− bn

]

converges to the Gumbel distribution, where an, bn ∼√2 lnn.

Xiao and Wu (2014) use Horowitz, Lobato, Nankervis, and Savin’s

(2006) blocks-of-blocks bootstrap for p-value computation (without

proving its validity under their assumptions).



Innovation #1

The previous max-approach exploits extreme value theory inorder to derive the asymptotic distribution of max-covariance.

We bypath it by proving the asymptotic validity of DWB.

It suffices to prove that the actual test statistic andbootstrapped test statistics converge to the same distribution(no matter what it is).

Our approach allows for more general memory/momentproperties than the previous approach (e.g. NED vs. mixing).

Innovation #2

Xiao and Wu (2014) allow for observed data only.

We allow for filtered data (i.e. residuals).


First-Order Expansion for Filtered Series

Suppose that the true DGP is ARMA(1,1):

yt = 0.5yt−1 + νt + 0.5νt−1, νt ∼ WN(0, 1).

Fit AR(1) model: yt = θyt−1 + ut.

Least squares estimator: θn =∑n

t=2 ytyt−1/∑n

t=2 y2t−1.

Filtered series (i.e. residual): ut(θn) = yt − θnyt−1.

Since ut = νt + 0.5νt−1, population autocorrelation of {ut} atlag 1 is

ρ(1) = E[utut−1]/E[u2t ] = 0.4.

Sample autocorrelation of {ut(θn)} at lag 1 is

ρn(1) =1n

∑nt=2 ut(θn)ut−1(θn)

1n

∑nt=2 u

2t−1(θn)

.



By the mean value theorem,

ρn(1) = ρ(1)−E[ut−1yt−1][E[y

2t−1]]

−1E[utyt−1]

E[u2t ]+ op(1)

= ρ(1)− D(1)×A× E[mt]

γ(0)+ op(1)

= 0.4− 1.5× 0.429× 0.5

1.25+ op(1)

= 0.4− 0.257+ op(1)

= 0.143 + op(1).

ρn(1) underestimates ρ(1), which lowers the test power.

Bias correction is desired in order to raise power.



Define sample counterparts:

1 Dn(h) =1n

∑nt=h+1 ut−h(θn)yt−1.

2 An =(1n

∑nt=2 y

2t−1

)−1.

3 mt(θn) = ut(θn)yt−1.

Define a bias-corrected cross term of filtered series:

En,t,h(θn) = ut(θn)ut−h(θn)− Dn(h)× An ×mt(θn)

and its sample mean:

gn(h, θn) =1

n

n∑t=h+1

En,t,h(θn).

Execute DWB for {En,t,h(θn)− gn(h, θn)}.Motegi (Economics, Kobe U.) Max-Correlation and Max-Causality Tests February 17, 2017 16 / 49

Dependent Wild Bootstrap for Filtered Series

1 Set a block size bn.2 Generate iid {ξ1, ξ2, . . . , ξn/bn}. Define an auxiliary variable:

ω = [ξ1, . . . , ξ1︸︷︷︸bn terms

, ξ2, . . . , ξ2︸︷︷︸bn terms


]′.

3 Compute bootstrapped autocorrelations:

ρ(dw)n (h) =

1

γn(0)× 1

n

n∑t=h+1

ωt

[En,t,h(θn)− gn(h, θn)

], h = 1, . . . ,Ln,

and test statistics T (dw)n =

√nmax1≤h≤Ln |ρ

(dw)n (h)|.

4 Repeat Steps 2-3 M times and compute the bootstrappedp-value p

(dw)n,M = 1

M

∑Mi=1 I(T

(dw)n,i ≥ Tn).


Monte Carlo Simulations

Two tests:

1 Max-correlation test with DWB.

2 Hong’s test with DWB.

Sample size is n = 500.

Ln = 5,[0.5× n

ln(n)

],[

nln(n)

].

L500 = 5, 40, 80.


Simulation Results (Selected) – Size

Simple yt = et, Mean Filter, et ∼ IID.Ln = 5 Ln = 40 Ln = 80

1%, 5%, 10% 1%, 5%, 10% 1%, 5%, 10%Max .008, .047, .114 .004, .029, .078 .000, .027, .068Hong .007, .061, .117 .000, .013, .050 .000, .005, .022

AR(2) yt = 0.3yt−1 − 0.15yt−2 + et, AR(2) Filter, et ∼ IID.Ln = 5 Ln = 40 Ln = 80

1%, 5%, 10% 1%, 5%, 10% 1%, 5%, 10%Max .018, .064, .126 .004, .028, .075 .003, .030, .074Hong .016, .068, .135 .001, .012, .039 .000, .005, .022


Simulation Results (Selected) – Power

AR(2) yt = 0.3yt−1 − 0.15yt−2 + et, AR(1) Filter, et ∼ GARCH(1,1).Ln = 5 Ln = 40 Ln = 80

1%, 5%, 10% 1%, 5%, 10% 1%, 5%, 10%Max .356, .650, .776 .175, .403, .565 .127, .329, .469Hong .321, .632, .758 .002, .096, .235 .000, .013, .073

Simple yt = et, Mean Filter, et ∼ MA(48).Ln = 5 Ln = 40 Ln = 80

1%, 5%, 10% 1%, 5%, 10% 1%, 5%, 10%Max .013, .082, .154 .007, .055, .141 .764, .925, .957Hong .017, .064, .136 .002, .030, .117 .013, .210, .448


Max-Correlation Test: Summary

Shao’s (2010, 2011) dependent wild bootstrap enables us toconstruct a formal white noise test.

We propose a max-correlation test statistic.

Max-correlation test has sharper size and higher power thanHong’s test.

The advantage of max-correlation test is particularly prominentunder remote autocorrelations (seasonality).


Weak Form Efficiency: Introduction

A rejection of the white noise hypothesis might serve as helpful

information for arbitragers, because it indicates the presence of

non-zero autocorrelations at some lags.

IID

MDS

White Noise

xx

xx

Testing for Weak Form Efficiency of Stock Markets

= Testing for Unpredictability of Stock Returns (cf. Fama, 1970)

Many applicationscf. Lim and Brooks (2009)

REASON: It is hard to establish a formal white noise test.

BREAKTHROUGH: Shao’s (2010, 2011) dependent wild bootstrap.

Only few applications

This paper tests for the white noise hypothesis of stock returns,

using the dependent wild bootstrap.



Adaptive Market Hypothesis

(cf. Lo, 2004; Lo, 2005)

Dependent Wild Bootstrap

(Shao, 2010; Shao, 2011)

Full Sample

Shao (2010): Temperature

Shao (2011): Stock returns

Rolling Window

New!

PERIODICITY IN

CONFIDENCE BANDS

REMEDY: Randomizing a block size across

bootstrap samples and windows.

REASON: Fixed block size produces similar

bootstrapped autocorrelations in every windows.



S&P 500

White noise hypothesis is often rejected

during Iraq War and the subprime crisis.

We observe significantly negative

autocorrelations during crisis periods.

Campbell, Grossman, and Wang (1993)

cf. Fama and French (1988)


Review of Dependent Wild Bootstrap

Consider full sample analysis as a benchmark.

Consider univariate covariance stationary {y1, . . . , yn}.Assume E[yt] = 0 for notational simplicity.

Population quantities:

γ(0) = E[y2t ], γ(h) = E[ytyt−h], ρ(h) =γ(h)

γ(0).

Sample quantities:

γn(0) =1

n

n∑t=1

y2t , γn(h) =1

n

n∑t=h+1

ytyt−h, ρn(h) =γn(h)

γn(0).



White noise hypothesis: ρ(h) = 0 for all h ≥ 1.

As a starting point, fix h and consider testing for ρ(h) = 0.

How can we construct a confidence band for ρn(h), assuminglittle more than serial uncorrelatedness?



1 Set a block size bn (typically bn =√n).

2 Generate iid {ξ1, ξ2, . . . , ξn/bn}. Define an auxiliary variable:

ω = [ ξ1, . . . , ξ1︸︷︷︸bn terms

, ξ2, . . . , ξ2︸︷︷︸bn terms


]′.

3 Compute a bootstrapped autocorrelation:

ρ(dw)n (h) =

1

γn(0)× 1

n

n∑t=h+1

ωt[ytyt−h − γn(h)].

4 Repeat Steps 2-3 M times and sort ρ(dw)n,(1)(h) < · · · < ρ

(dw)n,(M)(h).

5 The 95% confidence band is C(h) = [ρ(dw)n,(0.025M)(h), ρ

(dw)n,(0.975M)(h)].

6 If ρn(h) ∈ C(h), then we do not reject ρ(h) = 0.If ρn(h) /∈ C(h), then we reject ρ(h) = 0.


Hidden Pitfall: Periodic Confidence Bands

Now consider rolling window analysis.

Suppose that window size is n = 60 and block size is bn = 3.

In window #1 (y1, . . . , y60), we have

[ y1y2, y2y3︸︷︷︸Block 1 (×ξ1)

, y3y4, y4y5, y5y6︸︷︷︸Block 2 (×ξ2)

, y6y7, y7y8, y8y9︸︷︷︸Block 3 (×ξ3)

, . . . , y57y58, y58y59, y59y60︸︷︷︸Block 20 (×ξ20)

].

In window #4 (y4, . . . , y63), we have

[ y4y5, y5y6︸︷︷︸Block 1 (×ξ1)

, y6y7, y7y8, y8y9︸︷︷︸Block 2 (×ξ2)

, . . . , y57y58, y58y59, y59y60︸︷︷︸Block 19 (×ξ19)

, y60y61, y61y62, y62y63︸︷︷︸Block 20 (×ξ20)

].

Similar bootstrapped autocorrelations appear in windows #1, #4, #7, . . . .

=⇒ Periodicity with bn = 3 cycles.


Hidden Pitfall: Periodic Confidence Bands

2 4 6 8 10 12-0.5

0

0.5

y1, . . . , y71i.i.d.∼ N(0, 1).

Window size is n = 60. There are 71− 60 + 1 = 12 windows.

Block size is bn = 3.

We plot ρn(1) and 95% confidence band for each window.


Remedy: Randomized Block Size

Block size is bn = [c√n].

Conventional choice that c = 1 produces periodicity.

We propose to draw c ∼ U(1− δ, 1 + δ) independently acrossrolling windows and bootstrap samples.

Randomness across windows removes periodicity.

Randomness across bootstrap samples reduces the volatility ofconfidence bands.

We choose δ = 0.5 (i.e. c ∼ U(0.5, 1.5)).



Illustrative Example:

y1, y2, . . . , y400i.i.d.∼ N(0, 1).

Window size is n = 240.

There are 400− 240 + 1 = 161 windows.

Block size is bn = [c√n] = [c

√240] = [c× 15.5].

We choose either c = 1 or c ∼ U(0.5, 1.5).

We plot ρn(1) and 95% confidence band for each window.



50 100 150-0.2

-0.1

0

0.1

0.2

a. c = 1

50 100 150-0.2

-0.1

0

0.1

0.2

b. c ∼ U

When c = 1, confidence bands have clear periodicity.

When c ∼ U(0.5, 1.5), the periodicity evaporates dramatically.


Stock Price Data: S&P 500

Jan05 Jan10 Jan150

100

200

300

Subprime

Shanghai

AA+

Iraq

a. Level

Jan05 Jan10 Jan15-0.2

-0.1

0

0.1

0.2

Subprime AA+Shanghai

Iraq

b. Log Return

Daily data of S&P 500.

January 1, 2003 – October 29, 2015 (3230 days).


Autocorrelations at Lag 1

Jan05 Jan10-0.5

0

0.5

a. c = 1

Jan05 Jan10-0.5

0

0.5

Iraq Subprime AA+

b. c ∼ U

Window size is n = 240 (roughly a year).

Block size is bn = [c√n] = [c× 15.5].


Cramer-von Mises White Noise Test

White noise requires that ρ(h) = 0 for all h ≥ 1.

Following Shao (2011), we use the Cramer-von Mises statistic:

Cn = n

∫ π

0

{n−1∑h=1

γn(h)ψh(λ)

}2

dλ, ψh(λ) =sin(hλ)

hπ.

Bootstrapped p-values are computed based on the dependentwild bootstrap (with a randomized block size).

We observe similar results after using Hill and Motegi’s (2017a)max-correlation test and Andrews and Ploberger’s (1996)sup-LM test.


P-Values of Cramer-von Mises Test

Jan05 Jan100

0.5

1

P-values over rolling windows.

S&P has significant autocorrelations during Iraq War and thesubprime mortgage crisis.


Weak Form Efficiency: Summary

1. Outline

2. Contributions

3. Empirical Finding

Test for white noise hypothesis of stock returns

Perform rolling window analysis with dependent wild bootstrap

Find that a fixed block size results in periodic confidence bands

Reveal that the periodicity stems from repeated block structures

Propose randomizing a block size to remove the periodicity

White noise hypothesis is rejected for S&P during crisis periods

-- Significantly negative autocorrelations at lag 1


Granger Causality Tests: Motivation

Consider two time series {x1, . . . , xn} and {y1, . . . , yn}.

Definition: {xt} does not Granger cause {yt} if

E[ yt+1 | yt, yt−1, . . . ]︸︷︷︸Univariate prediction

= E[ yt+1 | yt, yt−1, . . . , xt, xt−1, . . . ]︸︷︷︸Prediction with extra information of x

.

Interpretation: Knowing past and present values of x does notimprove the prediction accuracy of y.

How to test for Granger non-causality from {xt} to {yt}?


Max-Causality Test

Classical approach: formulate a regression model

yt = α0 + α1yt−1 + · · ·+ αpyt−p + β1xt−1 + · · ·+ βqxt−q + ut,

Then perform a Wald test with respect to β21 + · · ·+ β2

q = 0 .

Our approach (cf. Ghysels, Hill, and Motegi, 2017):

Model 1 : yt = α0 + α1yt−1 + · · ·+ αpyt−p + β1xt−1 + ut,

......

......

Model q : yt = α0 + α1yt−1 + · · ·+ αpyt−p + βqxt−q + ut.

Then test for max{|β1|, . . . , |βq|} = 0 .

We show via local power analysis and Monte Carlo simulationsthat our test achieves sharper size and higher power.


Mixed Frequency Granger Causality Test

MIDAS allows for

mixed frequency

� Higher precision

Classical approach requires

single frequency

� Lower precision

InflationAnnounced monthly

GDPAnnounced quarterly

InflationQuarterly average

Temporal aggregation (= Loss of information)

GDPQuarterly

Ghysels, Hill, and Motegi (2016, 2017) expand Granger causality tests

into the mixed data sampling (MIDAS) literature (cf. Ghysels, Santa-

Clara, and Valkanov, 2004).



x1,t x2,t x3,t x1,t+1 x2,t+1 x3,t+1

yt yt+1

Quarter t Quarter t+1

Classical approach only uses

MIDAS treats as if they were three distinct

quarterly variables.

time

time



Classical approach: formulate a regression model

yt = α0 +

p∑k=1

αkyt−k +

q∑k=1

βkxt−k + ut

= α0 +

p∑k=1

αkyt−k +

q∑k=1

βk(13x1,t−k + 1

3x2,t−k + 13x3,t−k) + ut.

Then perform a Wald test with respect to∑q

k=1 β2k = 0 .

Our approach (cf. Ghysels, Hill, and Motegi, 2016):

yt = α0 +

p∑k=1

αkyt−k +

q∑k=1

(β1kx1,t−k + β2kx2,t−k + β3kx3,t−k) + ut.

Then perform a Wald test with respect to∑q

k=1

∑3j=1 β

2jk = 0 .

If β1k = β2k = β3k, our model reduces to the classical model.


MF Causality Tests: Application

We analyze monthly oil prices, monthly inflation,

and quarterly GDP in the U.S.

MIDAS and the classical approach produce

remarkably different results.

The MIDAS-based results seem more reasonable

(e.g. significant causality from oil prices to inflation).

Ghysels, Hill, and Motegi (2016)


MF Causality Tests: Application

Ghysels, Hill, and Motegi (2017)

We analyze weekly interest rate spread

and quarterly GDP in the U.S.

MIDAS detects a longer period of significant causality

from spread to GDP than the classical approach.


Future Work

1 Expanding max tests into general hypothesis testing.

Linear regression

Nonlinear regression

Parameters on the boundary (e.g. no ARCH effects)

2 Elaborating max-variance ratio tests for random walk.

3 Testing for Granger causality between atmosphere and ocean?

Lagged causality due to seasonality?

Mixed frequency data?


References

Andrews, D. W. K. and W. Ploberger (1996). Testing for SerialCorrelation against an ARMA(1, 1) Process. Journal of the AmericanStatistical Association, 91, 1331-1342.

Berman, S. M. (1964). Limit Theorems for the Maximum Term inStationary Sequences. Annals of Mathematical Statistics, 35,502-516.

Campbell, J. Y., S. J. Grossman, and J. Wang (1993). TradingVolume and Serial Correlation in Stock Returns. Quarterly Journal ofEconomics, 108, 905-939.

Fama, E. F. (1965). The Behavior of Stock-Market Prices. Journalof Business, 38, 34-105.

Fama, E. F. (1970). Efficient Capital Markets: A Review of Theoryand Empirical Work. Journal of Finance, 25, 383-417.


References

Fama, E. F. and French, K. R. (1988). Permanent and TemporaryComponents of Stock Prices. Journal of Political Economy, 96, 246-273.

Ghysels, E., J. B. Hill, and K. Motegi (2016). Testing for Granger Causalitywith Mixed Frequency Data. Journal of Econometrics, 192, 207-230.

Ghysels, E., J. B. Hill, and K. Motegi (2017). Simple Granger CausalityTests for Mixed Frequency Data. Revise-and-resubmit stage, Journal ofBusiness and Economic Statistics.

Ghysels, E., P. Santa-Clara, and R. Valkanov (2004). The MIDAS Touch:Mixed Data Sampling Regression Models. Working paper at the UNCChapel Hill and UCLA.

Hill, J. B. and K. Motegi (2017a). A Max-Correlation White Noise Test forWeakly Dependent Time Series. Working paper at the UNC Chapel Hilland Kobe University.


References

Hill, J. B. and K. Motegi (2017b). Testing for Weak Form Efficiencyof Stock Markets. Working paper at the UNC Chapel Hill and KobeUniversity.

Hong, Y. (1996). Consistent Testing for Serial Correlation ofUnknown Form. Econometrica, 64, 837-864.

Horowitz, J. L., I. N. Lobato, J. C. Nankervis, and N. E. Savin(2006). Bootstrapping the Box-Pierce Q Test: A Robust Test ofUncorrelatedness. Journal of Econometrics, 133, 841-862.

Lim, K.-P. and R. Brooks (2009). The Evolution of Stock MarketEfficiency over Time: A Survey of the Empirical Literature. Journalof Economic Surveys, 25, 69-108.

Ljung, G. M. and G. E. P. Box (1978). On a Measure of Lack of Fitin Time Series Models. Biometrika, 65, 297-303.


References

Lo, A. W. (2004). The Adaptive Market Hypothesis: MarketEfficiency from an Evolutionary Perspective. Journal of PortfolioManagement, 30, 15-29.

Lo, A. W. (2005). Reconciling Efficiency Markets with BehavioralFinance: The Adaptive Market Hypothesis. Journal of InvestmentConsulting, 7, 21-44.

Shao, X. (2010). The Dependent Wild Bootstrap. Journal of theAmerican Statistical Association, 105, 218-235.

Shao, X. (2011). A Bootstrapped-Assisted Spectral Test of WhiteNoise under Unknown Dependence. Journal of Econometrics, 162,213-224.

Xiao, H. and W. B. Wu (2014). Portmanteau Test and SimultaneousInference for Serial Covariances. Statistica Sinica, 24, 577-600.


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Max-Correlation Test and Max-Causality Test for …motegi/slides_UNC_seminar_v5.pdfMax-Correlation Test and Max-Causality Test for Economic Time Series Kaiji Motegi1 1Graduate School