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 Bus**in**ess

Chung-M**in**g Kuan

Institute of Economics

Academia S**in**ica

Hs**in**-yi L**in**

Department of Economics

National Chengchi University

•�

Introduction

**Return**-Volume Relations

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

•�

Introduction

**Return**-Volume Relations

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

• Important for underst**and****in**g operational efficiency **and** **in**formation dynam-

ics **in** asset markets.

⋄ Sequential **in**formation arrival model: Copel**and** (1976), Jenn**in**gs, Starks,

**and** Fell**in**gham (1981), Jenng**in**s **and** Barry (1983)

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

Tauchen **and** Pitts (1983).

⋄ O**the**r equilibrium models on **the** **in**formation content **and** heterogene-

ity of **in**vestors: Harris **and** Ravis (1993), Blume, Easley, **and** O’Hara

(1994), Wang (1994).

•�

• P. 676 of Y**in**g (1966):

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

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

vice versa will **in**evitably yield **in**complete if not erroneous results”

• Empirical Studies:

⋄ Contemporaneous relations: Early studies **in**clude **Granger** **and** Mor-

genstern (1963) **and** Y**in**g (1966); see Karpoff (1987) for a review **and**

also Gallant, Rossi, **and** Tauchen (1992).

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

o**the**rs based on various causality tests.

⋄ Mixed empirical evidences.

•�

Test**in**g Causal Relations

Verify**in**g causal relations based on tests of **Granger** causality.

• **Causality** **in** mean **and** l**in**ear causality test: **Granger** (1969, 1980).

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

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

• Nonl**in**ear 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**

o**the**r distribution characteristics.

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

is problematic because **the** relation be**in**g 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

• Jo**in**t tests of non-causality across quantiles: Based on quantile regressions

(Koenker **and** Bassett, 1978) **and** test of jo**in**t significance of parameters

(Koenker **and** Machado, 1999).

• Empirical f**in**d**in**gs:

⋄ 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 ma**in**ly due to

tail quantiles.

•�

**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 hypo**the**sis 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.

•�

Tests of Non-**Causality** **in** **Quantiles**

Quantile Regressions

• A l**in**ear 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 obta**in**ed from **the** follow**in**g optimization problem:

m**in**

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)].

•�

• Not**in**g 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 **in**dependent Brownian bridges. Hence,

� �

�

� Bq(τ)

�2

�

WT(τ) ⇒ ��

� , τ ∈ T ,

� τ(1 − τ) �

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

which would be a well def**in**ed limit provided that T is a closed **in**terval **in**

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

• Koenker **and** Machado (1999): Choose n po**in**ts (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].

•�

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

• We may also test non-causality **in** any **in**terquantile 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.

�

,

•�

Empirical Study

Data

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

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

g**in**n**in**g of 1990 through June 30, 2006, with 4135, 4161, 4166 **and** 4083

observations, respectively.

• **Return**s: rt = 100 × (ln(pt) − ln(pt−1)). They fluctuate around **the**ir

respective mean levels **and** exhibit volatility cluster**in**g **and** excess kurtosis.

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

**and** Tauchen (1992).

•�

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 be**in**g 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 hypo**the**ses are: β(τ) = 0 for all τ **in** T = [0.05, 0.95],

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

• Lag determ**in**ation: 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].

•�

Causal Effects of Volume on **Return**

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

• The LS estimates are all **in**significant, suggest**in**g no causality **in** mean.

• For NYSE, S&P 500 **and** FTSE, **the**re 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

ma**in**ly positive **and** arise from **the** quantiles above **the** 3rd quartile.

• The sup-Wald tests reject non-causality for all **in**terquantile ranges of

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

quantiles beyond **the** 3rd quartile.

• Putt**in**g 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).

•�

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, suggest**in**g 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** **the**ir magnitude tends to **in**crease with τ.

• For FTSE, **the**se estimates are mostly **in**significant **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 **in**dices but FTSE.

•�

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 aga**in** suggest no causality **in** mean, even when more

lags are **in**cluded.

⋄ 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 ma**in**ly **in** tail quantiles.

•�

• 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.

•�

Conclud**in**g Remarks

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

• We f**in**d clear causality **in** quantiles between **in**dex return **and** volume.

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

TOPIX, but only one way (volume **Granger** caus**in**g 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 **in**creases with volume, which is ano**the**r 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** o**the**r lagged volumes, but **the** causal effects are weaker.

• For NYSE **and** S&P 500, **the**re is negative association between volume **and**

lagged returns.

• The causal effects **in** TOPIX are quite different from those **in** o**the**r **in**dices.

•�

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).

•�

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).

•�

−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

•�

−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

•�

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).

•�

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).

•�

−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

•�

−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

•�

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).

•�

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

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

•�

−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

•�

−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

•�

−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

•�

−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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