Granger Causality in Quantiles and the Stock Return

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Granger Causality in Quantiles and the Stock Return

The Stock Return-Volume Relations: Causality

in Quantiles and Patterns of Causal Effects

Chia-Chang Chuang

Department of International Busienss

National Taipei College of Business

Chung-Ming Kuan

Institute of Economics

Academia Sinica

Hsin-yi Lin

Department of Economics

National Chengchi University

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Introduction

Return-Volume Relations

C.-M. Kuan, RV-Causal.1

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Introduction

Return-Volume Relations

C.-M. Kuan, RV-Causal.1

• Important for understanding operational efficiency and information dynam-

ics in asset markets.

⋄ Sequential information arrival model: Copeland (1976), Jennings, Starks,

and Fellingham (1981), Jenngins and Barry (1983)

⋄ Mixture of distributions model: Clark (1973), Epps and Epps (1976),

Tauchen and Pitts (1983).

⋄ Other equilibrium models on the information content and heterogene-

ity of investors: Harris and Ravis (1993), Blume, Easley, and O’Hara

(1994), Wang (1994).

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• P. 676 of Ying (1966):

C.-M. Kuan, RV-Causal.2

“any model of the stock market which separates prices from volumes or

vice versa will inevitably yield incomplete if not erroneous results”

• Empirical Studies:

⋄ Contemporaneous relations: Early studies include Granger and Mor-

genstern (1963) and Ying (1966); see Karpoff (1987) for a review and

also Gallant, Rossi, and Tauchen (1992).

⋄ Dynamic (causal) relations: Gallant, Rossi, and Tauchen (1992) and

others based on various causality tests.

⋄ Mixed empirical evidences.

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Testing Causal Relations

Verifying causal relations based on tests of Granger causality.

Causality in mean and linear causality test: Granger (1969, 1980).

C.-M. Kuan, RV-Causal.3

Causality in variance: Granger et al. (1986), Chuang and Ng (1996).

• Nonlinear causality: Hiemstra and Jones (1994).

Causality in risk: Hong, Liu, and Wong (2006).

• Drawbacks:

⋄ Non-causality in a particular moment does not imply non-causality in

other distribution characteristics.

⋄ Diks and Panchenko (2005): The test of Hiemstra and Jones (1994)

is problematic because the relation being tested is not even implied by

Granger non-causality.

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This Paper

• Consider causality in quantiles.

C.-M. Kuan, RV-Causal.4

• Joint tests of non-causality across quantiles: Based on quantile regressions

(Koenker and Bassett, 1978) and test of joint significance of parameters

(Koenker and Machado, 1999).

• Empirical findings:

⋄ Significant two-way Granger causality between return and volume.

⋄ Quantile causal effects of volume on return exhibit a V shape across

quantiles, but not conversely.

⋄ There are more lagged volumes (returns) that can cause return (vol-

ume) in quantiles, and the effects of lagged volumes are mainly due to

tail quantiles.

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Granger Causality

C.-M. Kuan, RV-Causal.5

Granger non-causality in distribution: x does not Granger cause y if

Fy t (η|(Y, X )t−1) = Fy t (η|Yt−1), ∀η ∈ IR.

Hong, Liu, and Wong (2006) consider non-causality in risk with η = −VaR.

• Non-causality in mean: IE(yt|(Y, X )t−1) = IE(yt|Yt−1), ∀η ∈ IR.

⋄ Model: α0 + � p

i=1 αiyt−i + � q

j=1 βjxt−j.

⋄ Null hypothesis of no causality: βj = 0, j = 1, . . . , q.

• Non-causality in distribution is equivalent to non-causality in quantiles:



Qτ(yt�

(Y, X )t−1) = Qτ(yt�Yt−1),

∀τ ∈ (0, 1).

Lee and Yang (2006) consider non-causality in a given quantile.

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Tests of Non-Causality in Quantiles

Quantile Regressions

• A linear model for τ-th conditional quantile:

C.-M. Kuan, RV-Causal.6

yt = α0(τ) + y ′ t−1,pα(τ) + x ′ t−1,qβ(τ) + et(τ) = z ′ t−1θ(τ) + et(τ),

• The estimator ˆ θT(τ) is obtained from the following optimization problem:

min

θ

T�

t=1

(τ − 1 {yt


Tests of Non-Causality in Quantiles

• The quantile regression estimator of β(τ) is:

√ � �

T ˆβT(τ) D

1/2 1/2

− β(τ) −→ [τ(1 − τ)] Ω N (0, Iq),

C.-M. Kuan, RV-Causal.7

where Ω = ΨD(τ) −1 M zzD(τ) −1 Ψ ′ with Ψ = [0 1q], a q × k selection

matrix, M zz := limT →∞ T −1 � T

t=1 zt−1z ′ t−1,

D(τ) := lim

T →∞

1

T

T�

t=1

ft(F −1

t (τ))zt−1z ′ t−1,

and ft and Ft are, respectively, the conditional density and distribution of

yt given zt−1.

• For a given τ, the Wald statistic of β(τ) = 0 is

WT(τ) := T ˆ β T(τ) ′ � Ω −1 ˆβT(τ)/[τ(1 − τ)].

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• Noting Bq(τ) d = [τ(1 − τ)] 1/2N (0, Iq), we have

√ � �

T ˆβT(τ) D 1/2

− β(τ) −→ Ω Bq(τ),

where Bq is vector of q independent Brownian bridges. Hence,

� �


� Bq(τ)

�2


WT(τ) ⇒ ��

� , τ ∈ T ,

� τ(1 − τ) �

C.-M. Kuan, RV-Causal.8

which would be a well defined limit provided that T is a closed interval in

(0, 1) (e.g., T = [ɛ, 1 − ɛ]); see Andrews (1993).

• Koenker and Machado (1999): Choose n points (0 < τ1 < . . . < τn < 1)

and compute

sup -WT = sup WT(τi).

i=1,...,n

Its limit can be well approximated by the limit above with T = [τ1, τn].

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C.-M. Kuan, RV-Causal.9

• We may also test non-causality in any interquantile range T = [τℓ, τu]:

β(τ) = 0, for all τ in T = [τℓ, τu].

• Critical values: For s = τ/(1 − τ),

Then,

B(τ)/ � τ(1 − τ) d = W (s)/ √ s.

IP



⎩ sup

τ∈[τ ℓ,τ u]

� �


� Bq(τ)



��


� τ(1 − τ) �

2

⎫ �


< c = IP


sup

s∈[1,s 2/s 1]

� �



W q(s) �

� √ �

s �

2

< c

with s1 = τℓ/(1 − τℓ), s2 = τu/(1 − τu). The critical values for various

s2/s1 can be found in DeLong (1981) and Andrews (1993) or computed

by simulation.


,

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Empirical Study

Data

C.-M. Kuan, RV-Causal.10

• Four market indices: NYSE, S&P 500, FTSE and TOPIX, from the be-

ginning of 1990 through June 30, 2006, with 4135, 4161, 4166 and 4083

observations, respectively.

Returns: rt = 100 × (ln(pt) − ln(pt−1)). They fluctuate around their

respective mean levels and exhibit volatility clustering and excess kurtosis.

• Volume: Detrended by regressing ln vt on 1, t and t 2 , as in Gallant, Rossi,

and Tauchen (1992).

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Empirical Models

• Model with q lags:

q�

yt = α0 + αi(τ)yt−i +

i=1

q�

βj(τ)xt−j + et,

j=1

with yt being rt or vt (detrended volume). For q = 1,

yt = α0 + α1(τ)yt−i + β1(τ)xt−1 + et.

C.-M. Kuan, RV-Causal.11

• Estimate 91 quantile regressions with τ = 0.05, 0.06, . . . , 0.95, 0.95.

• Sup-Wald tests: Null hypotheses are: β(τ) = 0 for all τ in T = [0.05, 0.95],

[0.1, 0.9], . . . [0.45, 0.55].

• Lag determination: Check significance of the last coefficient of a model

with q lags, i.e., βq(τ) = 0, τ in T = [0.05, 0.95], [0.1, 0.9], . . . [0.45, 0.55].

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Causal Effects of Volume on Return

C.-M. Kuan, RV-Causal.12

• The LS estimates are all insignificant, suggesting no causality in mean.

• For NYSE, S&P 500 and FTSE, there exist opposite quantile causal effects

of volume on the two sides of the return distribution, and such effects are

stronger for tail quantiles. The effects of volume on TOPIX return are

mainly positive and arise from the quantiles above the 3rd quartile.

• The sup-Wald tests reject non-causality for all interquantile ranges of

NYSE, S&P 500 and FTSE. For TOPIX, causality is resulted from the

quantiles beyond the 3rd quartile.

• Putting lagged volume on vertical axis and return on the horizontal axis,

the return-volume relations exhibit a V shape across quantiles, cf. Kar-

poff (1987), Gallant, Rossi, and Tauchen (1992), and Blume, Easley, and

O’Hara (1994).

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Causal Effects of Return on Volume

C.-M. Kuan, RV-Causal.13

• The LS estimates of β1 are significantly negative for NYSE, S&P 500 and

FTSE but significantly positive for TOPIX, suggesting causality in mean.

• For NYSE, the estimated β1(τ) are significantly negative for almost all

quantiles; for S&P 500, the estimates are significantly negative only for

upper quantiles (τ ≥ 0.45), and their magnitude tends to increase with τ.

• For FTSE, these estimates are mostly insignificant and are significantly

negative only at a few right tail quantiles (τ in [0.88, 0.93]).

• For TOPIX, the estimates are significantly positive for τ ≥ 0.18 and in-

crease with τ.

• The sup-Wald tests also clearly reject the null of non-causality in all in-

terquantile ranges for all indices but FTSE.

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Models with Multiple Lags

• Regression of rt on lagged rt and vt:

C.-M. Kuan, RV-Causal.14

⋄ The sup-Wald tests: At 5% (1%) level, the selected models are a lag

6 (lag 3) model for NYSE, a lag 5 (lag 2) model for S&P 500, and a

lag 1 model for FTSE and TOPIX.

⋄ The LS estimates again suggest no causality in mean, even when more

lags are included.

⋄ There are opposite quantile causal effects of lagged volumes for NYSE

and S&P 500. Such effects are weaker than those in the lag 1 model

and exhibit significance mainly in tail quantiles.

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• Regression of vt on lagged vt and rt:

C.-M. Kuan, RV-Causal.15

⋄ The sup-Wald tests: At 1% level, the selected models are a lag 3 model

for NYSE, a lag 2 model for S&P 500 and TOPIX, and a lag 1 model

for FTSE.

⋄ For NYSE and S&P 500, the LS estimates are significantly negatvie.

⋄ The negative quantile causal effects of lagged returns on volume for

NYSE and S&P 500 are similar to those in the lag 1 model. For

TOPIX, rt−1 has a positive quantile causal effect, but rt−2 has a neg-

ative effect.

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Concluding Remarks

C.-M. Kuan, RV-Causal.16

• We find clear causality in quantiles between index return and volume.

• Two-way causality between return and volume for NYSE, S&P 500 and

TOPIX, but only one way (volume Granger causing return) in FTSE.

• For NYSE and S&P 500, the causal effects of volume on return exhibit

a V shape across quantiles. This shows the dispersion of the conditional

return distribution increases with volume, which is another way to show

volume has a positive effect on return volatility.

• For NYSE and S&P 500, the V shape causality pattern also holds for

return and other lagged volumes, but the causal effects are weaker.

• For NYSE and S&P 500, there is negative association between volume and

lagged returns.

• The causal effects in TOPIX are quite different from those in other indices.

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C.-M. Kuan, RV-Causal.17

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C.-M. Kuan, RV-Causal.28

Figure 1: Volume series: NYSE (upper left), S&P 500 (upper right), FTSE (lower left), TOPIX (lower

right).

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C.-M. Kuan, RV-Causal.29

Figure 2: Detrended volume series: NYSE (upper left), S&P 500 (upper right), FTSE (lower left),

TOPIX (lower right).

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−1.0 −0.5 0.0 0.5 1.0

0.2 0.4 0.6 0.8

C.-M. Kuan, RV-Causal.30

0.2 0.4 0.6 0.8

Figure 3: ˆ β 1(τ) in the regressions of return on volume: NYSE (right) and S&P 500 (right).

−1.0 −0.5 0.0 0.5 1.0

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−0.8 −0.6 −0.4 −0.2 0.0 0.2 0.4

0.2 0.4 0.6 0.8

C.-M. Kuan, RV-Causal.31

0.2 0.4 0.6 0.8

Figure 4: ˆ β 1(τ) in the regressions of return on volume: FTSE (left) and TOPIX (right).

−0.6 −0.4 −0.2 0.0 0.2 0.4

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45

40

35

30

25

20

15

10

5

0

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95

120

105

90

75

60

45

30

15

0

C.-M. Kuan, RV-Causal.32

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95

Figure 5: Sup-Wald tests for the regression of return on volume: NYSE (left) and S&P 500 (right).

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35

30

25

20

15

10

5

0

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95

14

12

10

8

6

4

2

0

C.-M. Kuan, RV-Causal.33

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95

Figure 6: Sup-Wald tests for the regression of return on volume: FTSE (left) and TOPIX (right).

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−0.025 −0.015 −0.005 0.000

0.2 0.4 0.6 0.8

C.-M. Kuan, RV-Causal.34

0.2 0.4 0.6 0.8

Figure 7: ˆ β 1(τ) in the regressions of volume on return: NYSE (right) and S&P 500 (right).

−0.020 −0.010 0.000 0.005

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−0.02 −0.01 0.00 0.01

0.2 0.4 0.6 0.8

C.-M. Kuan, RV-Causal.35

0.2 0.4 0.6 0.8

Figure 8: ˆ β 1(τ) in the regressions of volume on return: FTSE (left) and TOPIX (right).

−0.01 0.00 0.01 0.02 0.03 0.04 0.05

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18

16

14

12

10

8

6

4

2

0

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95

18

16

14

12

10

8

6

4

2

0

C.-M. Kuan, RV-Causal.36

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95

Figure 9: Sup-Wald tests for the regression of return on volume: NYSE (left) and S&P 500 (right).

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14

12

10

8

6

4

2

0

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95

30

27

24

21

18

15

12

9

6

3

0

C.-M. Kuan, RV-Causal.37

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95

Figure 10: Sup-Wald tests for the regression of return on volume: FTSE (left) and TOPIX (right).

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Figure 11: Sup-Wald tests for lag selection: NYSE (left) and S&P 500 (right).

C.-M. Kuan, RV-Causal.38

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−0.8 −0.6 −0.4 −0.2 0.0 0.2 0.4

−0.5 0.0 0.5

0.2 0.4 0.6 0.8

0.2 0.4 0.6 0.8

−0.5 0.0 0.5

0.2 0.4 0.6 0.8

Figure 12: ˆ β 1(τ), ˆ β 2(τ) and ˆ β 3(τ) in the regressions of return on volume: NYSE.

C.-M. Kuan, RV-Causal.39

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−0.5 0.0 0.5

0.2 0.4 0.6 0.8

C.-M. Kuan, RV-Causal.40

0.2 0.4 0.6 0.8

Figure 13: ˆ β 1(τ) and ˆ β 2(τ) in the regressions of return on volume: S&P 500.

−1.0 −0.5 0.0 0.5 1.0

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−0.020 −0.015 −0.010 −0.005 0.000

−0.025 −0.015 −0.005 0.005

0.2 0.4 0.6 0.8

0.2 0.4 0.6 0.8

−0.025 −0.015 −0.005 0.005

0.2 0.4 0.6 0.8

Figure 14: ˆ β 1(τ), ˆ β 2(τ) and ˆ β 3(τ) in the regressions of volume on return: NYSE.

C.-M. Kuan, RV-Causal.41

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−0.020 −0.015 −0.010 −0.005 0.000 0.005

0.2 0.4 0.6 0.8

C.-M. Kuan, RV-Causal.42

0.2 0.4 0.6 0.8

Figure 15: ˆ β 1(τ) and ˆ β 2(τ) in the regressions of volume on return: S&P 500.

−0.025 −0.015 −0.005 0.000

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0.00 0.01 0.02 0.03 0.04 0.05

0.2 0.4 0.6 0.8

C.-M. Kuan, RV-Causal.43

0.2 0.4 0.6 0.8

Figure 16: ˆ β 1(τ) and ˆ β 2(τ) in the regressions of volume on return: TOPIX.

−0.025 −0.015 −0.005

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