Principal Volatility Component and Their Applications Yu-Pin Hu National Chi Nan University

March 2012 (This is a joint work with Ruey S. Tsay) Yu-Pin Hu National Chi Nan University

Principal Volatility Component and Their Applications

Outline I

The Background of This Research

I

Motivation: Does there exist no-ARCH portfolios among the seven exchange rates?

I

The Definitions of ARCH Dimension and Transformation

I

The Principal Volatility Components

I

Estimation and Testing

I

Data Analysis Yu-Pin Hu National Chi Nan University

Principal Volatility Component and Their Applications

The data: Seven exchange rates I

We consider seven exchange rates of currency against US dollars, including British Pounds, Norwegian Kroner, Swedish Kroner, Swiss Francs, Canadian Dollars, Singapore Dollars, and Australian Dollars.

I

We collected weekly log returns of the exchange rates from 29 March 2000 to 26 October 2011. Each series has 604 observations. (yt = log(pt ) − log(pt−1 ))

Yu-Pin Hu National Chi Nan University

Principal Volatility Component and Their Applications

Figure 1: The returns of the exchange rates

British Ponds 0.08 0.06 0.04 0.02 0 -0.02 -0.04

Norwegian Kroner 0.08 0.06 0.04 0.02 0 -0.02 -0.04 -0.06 -0.08

Swedish Kroner 0.08 0.06 0.04 0.02 0 -0.02 -0.04 -0.06 -0.08

Swiss Francs 0.1 0.08 0.06 0.04 0.02 0 -0.02 -0.04 -0.06 -0.08

Canadian Dollars 0.08 0.06 0.04 0.02 0 -0.02 -0.04 -0.06 -0.08

Singapore Dollars 0.04 0.03 0.02 0.01 0 -0.01 -0.02 -0.03

Australian Dollars 0.15 0.1 0.05 0 -0.05 -0.1

Background - 1: Why we care about the volatility movement of asset returns? I

The weekly log return of an asset price is defined as yt = log(pt ) − log(pt−1 ), where pt is the price at time t.

I

Usually, the mean movement of yt has no or less story. That is, if we fit an AR(p) model of yt as E(yt |Ωt−1 ) = φ0 + φ1 yt−1 + · · · + φp yt−p , the R2 of the model will be quite small, and the estimations of the AR coefficients are usually insignificant. Yu-Pin Hu National Chi Nan University

Principal Volatility Component and Their Applications

Background - 2: Why we care about the volatility movement of asset returns? I

However, the empirical studies show that the volatility movement of yt can be predicted by the past information (Var(yt |Ωt−1 )).

I

By letting E(yt |Ωt−1 ) = 0, Engle (1982) invented the ARCH model to describe the behavior of Var(yt |Ωt−1 ): q 2 2 + · · · + αp yt−p , yt = t σt , σt = α0 + α1 yt−1 where t is an i.i.d. N (0, 1) sequence. Yu-Pin Hu National Chi Nan University

Principal Volatility Component and Their Applications

Background - 3: Why we care about the volatility movement of asset returns? I

The ARCH model and its generalized form (GARCH) model capture the volatility movement of the univariate asset return successfully.

I

One can use the variance predicted by the models to control the financial risk. For example, estimate the value at risk (VaR), calculate the pricing of an option or an future, etc.

Yu-Pin Hu National Chi Nan University

Principal Volatility Component and Their Applications

Background - 4: Why we care about the volatility movement of asset returns? I

In practice, we should analyze multivariate asset returns.

I

The vector AR(p) model is the framework to understand the mean movement, yt = φ0 + φ1 yt−1 + · · · + φp yt−p + error, where yt is a k-dimensional series, and φi is a k × k matrix for 1 ≤ i ≤ p . Yu-Pin Hu National Chi Nan University

Principal Volatility Component and Their Applications

Background - 5: Why we care about the volatility movement of asset returns? I

Until now, analyzing the volatility movement of an multivariate series still is a difficult issue.

I

The constrain of the positive-definite conditional covariance for the Σt = E(yt yt0 |Ωt−1 ) will make the estimation of the multivariate “ARCH” model very complex.

I

Practically, there is no model can handel the k-dimensional series well for k ≥ 5 ( k ≥ 3). Yu-Pin Hu National Chi Nan University

Principal Volatility Component and Their Applications

Background - 6: Why we care about the volatility movement of asset returns? I

This research proposes an easy and useful method to look at the multivariate volatility movement: Dimension-reduction of the conditional covariance Σt = E(yt yt0 |Ωt−1 ) without model assumption. Yu-Pin Hu National Chi Nan University

Principal Volatility Component and Their Applications

Motivation - 1: The data of seven exchange rates I

We consider seven exchange rates of currency against US dollars, including British Pounds, Norwegian Kroner, Swedish Kroner, Swiss Francs, Canadian Dollars, Singapore Dollars, and Australian Dollars.

I

We collected weekly log returns of the exchange rates from 29 March 2000 to 26 October 2011. Each series has 604 observations.

Yu-Pin Hu National Chi Nan University

Principal Volatility Component and Their Applications

Figure 1: The returns of the exchange rates

British Ponds 0.08 0.06 0.04 0.02 0 -0.02 -0.04

Norwegian Kroner 0.08 0.06 0.04 0.02 0 -0.02 -0.04 -0.06 -0.08

Swedish Kroner 0.08 0.06 0.04 0.02 0 -0.02 -0.04 -0.06 -0.08

Swiss Francs 0.1 0.08 0.06 0.04 0.02 0 -0.02 -0.04 -0.06 -0.08

Canadian Dollars 0.08 0.06 0.04 0.02 0 -0.02 -0.04 -0.06 -0.08

Singapore Dollars 0.04 0.03 0.02 0.01 0 -0.01 -0.02 -0.03

Australian Dollars 0.15 0.1 0.05 0 -0.05 -0.1

Motivation - 2: The data of seven exchange rates I

We are interested in the volatility movements of the returns of the exchange rates. A VAR(5) model is adopted to remove the serial correlation to have the residual series.

I

According to the Ljung-Box test and Engle’s LM test of the squared series, each residual series performs ARCH effects.

I

The ARCH effect means that the conditional variance is not a constant value or matrix. Yu-Pin Hu National Chi Nan University

Principal Volatility Component and Their Applications

Figure 2: The residual fitted by VAR(5) model

British Ponds 0.05 0.04 0.03 0.02 0.01 0 -0.01 -0.02 -0.03 -0.04

Norwegian Kroner 0.06 0.04 0.02 0 -0.02 -0.04 -0.06

Swedish Kroner 0.06 0.04 0.02 0 -0.02 -0.04 -0.06

Swiss Francs 0.08 0.06 0.04 0.02 0 -0.02 -0.04 -0.06 -0.08

Canadian Dollars 0.06 0.04 0.02 0 -0.02 -0.04 -0.06

Singapore Dollars 0.03 0.02 0.01 0 -0.01 -0.02

Australian Dollars 0.12 0.1 0.08 0.06 0.04 0.02 0 -0.02 -0.04 -0.06

Table 1: Results of ARCH-effect tests: The residual fitted by the VAR(5) model Ljung-Box test

LM test

Lag=5

Lag=10

Lag=5

Lag=10 Lag=5

48.5 0.0

74.5 0.0

41.1 0.0

47.72 0.0

113.66 151.97 0.0 0.0

79.4 0.0

90.5 0.0

59.9 0.0

63.5 0.0

119.32 160.31 0.0 0.0

27.5 0.0

42.8 0.0

24.4 0.0

31.2 0.0

63.8 0.002

149.40 0.0

48.5 0.0

55.9 0.0

44.27 0.0

56.1 0.0

81.03 0.0

211.8 0.0

109.5 0.0

111.9 0.0

192.32 0.0

26.26 0.0

31.5 0.0

123.7 0.0

63.2 0.0

66.68 0.0

174.0 0.0

Extension of -LM test Lag=10

British Pounds Statistics P-value Norwegian Kroner Statistics P-value Swedish Kroner Statistics P-value Swiss Frances Statistics P-value Canadian Dollars Statistics P-value Singapore Dollars Statistics P-value Australian Dollars Statistics P-value

184.9 0.0 31.7 0.0 82.2 0.0

234.5 0.0 41.3 0.0 92.7 0.0

217.8 0.0 184.7 0.0 201.9 0.0

Motivation - 3: The data of seven exchange rates I

Does there exist no-ARCH effect portfolios between these seven exchange rates?

I

Although each series displays ARCH effects, is it possible to find linear combinations of these seven variables performing no-ARCH effect?

Yu-Pin Hu National Chi Nan University

Principal Volatility Component and Their Applications

ARCH Dimension and Transformation - 1 I

Suppose yt = (yi,t , · · · , yk,t )0 is a k-dimensional series with conditional mean zero, and a time-varying conditional covariance Σt = E(yt yt0 |Ωt−1 ),

where Ωt−1 is the σ-field generated by {yt−1 , yt−2 , · · · }.

Yu-Pin Hu National Chi Nan University

Principal Volatility Component and Their Applications

ARCH Dimension and Transformation - 2 I

Suppose most of elements of Σt are time-varying. We aim to look for a simplified structure of Σt .

I

A k × k matrix M = (M20 , M10 )0 exists such that M yt = ((M2 yt )0 , (M1 yt )0 )0 and E(M yt (M yt )0 |Ωt−1 ) = M Σt M 0 =

∆t C2 C20 C1

! ,

where ∆t is an r × r time-varying conditional covariance of M2 yt (0 ≤ r ≤ k). Yu-Pin Hu National Chi Nan University

Principal Volatility Component and Their Applications

ARCH Dimension and Transformation - 3 I

The ARCH dimension of yt is r if a (k − r) × k matrix M1 exists such that cov(M1 yt , yt |Ωt−1 ) = M1 Σt is a constant matrix.

I

The M1 is referred to as the no-ARCH transformation.

I

Engle and Kozicki (1993) and Engle and Susmel (1993) discussed a pair-wised method to identify the common feature in volatility.

Yu-Pin Hu National Chi Nan University

Principal Volatility Component and Their Applications

Principal Volatility Components - 1 I

In order to capture the no-ARCH transformation, we consider the Generalized Cross-Kurtosis Covariance (Li, 1992, pHd method) cov(yt yt0 , xt−1 ),

I

where xt−1 ∈ Ωt−1 is an univariate random variable. The generalized cross-kurtosis covariance satisfies cov{M yt (M yt )0 , xt−1 } = M cov(yt yt0 , xt−1 )M 0 . Yu-Pin Hu National Chi Nan University

Principal Volatility Component and Their Applications

Principal Volatility Components - 2 Example 1. Let yt be a k-dimensional series with the ARCH dimension r:     y1,t f 1,t    .   .  yt =  ..  = Hk×r Ak×(k−r)  ..  , yk,t

fk,t

q 2 i,t is an ARCH(1) where for 1 ≤ i ≤ r, fi,t = 1 + φfi,t−1 process and i,t follows an i.i.d. N (0, 1); and for r + 1 ≤ i ≤ k, fi,t follows an i.i.d. N (0, 1).

Yu-Pin Hu National Chi Nan University

Principal Volatility Component and Their Applications

Principal Volatility Components - 3 Example 1.

2 By letting xt−1 = yi,t−1 for 1 ≤ i ≤ k,

2 cov(yt yt0 , yi,t−1 )=  2 2 , yi,t−1 ) cov(f1,t    0 H A    0 0

Yu-Pin Hu National Chi Nan University

0 ..

0 .

0 0

0 2 2 cov(fr,t , yi,t−1 ) 0

0



0 0 0(k−r)

    

H0 A0

Principal Volatility Component and Their Applications

! .

Principal Volatility Components - 4 I

Let E be the intersection of the null space of cov(yt yt0 , xt−1 ) for all xt−1 ∈ Ωt−1 .

I

Theorem 1: Suppose yt is a k-dimensional stationary series. The ARCH dimension of yt is r, if and only if, the dimension of E is k − r and E is spanned by the no-ARCH transformation M1 .

Yu-Pin Hu National Chi Nan University

Principal Volatility Component and Their Applications

Principal Volatility Components - 5 I

Multivariate-ARCH(∞) families: vec(E(yt yt0 |Ωt−1 ))

= C0 +

∞ X

0 Λh vec(yt−h yt−h ).

h=1 I

Consider the generalized cross-kurtosis covariances cov(yt yt0 , xt−1 ) when xt−1 = yi,t−h yj,t−h . Let Γm ≡

m X k X k X

cov 2 (yt yt0 , yi,t−h yj,t−h ).

h=1 i=1 j=i Yu-Pin Hu National Chi Nan University

Principal Volatility Component and Their Applications

Principal Volatility Components - 6 I

Theorem 2: Suppose yt is a k-dimensional stationary series generated by the MARCH(∞) families and Γ∞ exists. The ARCH dimension of yt is r, if and only if, rank(Γ∞ ) = r and the no-ARCH transformation is the matrix M1 satisfying M1 Γ∞ = 0.

Yu-Pin Hu National Chi Nan University

Principal Volatility Component and Their Applications

Principal Volatility Components - 7 I

The principal component analysis (eigenvalues and eigenvectors) of Γ∞ contains all the information of the ARCH dimension and transformation.

I

Let M be a matrix composed by the eigenvectors of Γ∞ . The variables transformed by M are referred to as the Principal Volatility Components, i.e. M yt .

I

Roughly, the Principal Volatility Components are arranged according to the magnitude of their ARCH effects.

Yu-Pin Hu National Chi Nan University

Principal Volatility Component and Their Applications

Estimation and Testing - 1 I

We estimate Γ∞ by ˆn = Γ

n−1 X k X k X

ω?(

h=1 i=1 j=i

h )cov c 2 (yt yt0 , yi,t−h yj,t−h ), mn

ˆ −1/2 yt .) where ω ? (.) is the smoothing function. (Σ I

I

ˆ = (M ˆ 0, M ˆ 0 )0 is the matrix composed by the Suppose M 2 1 ˆ ˆ 1 is a eigenvectors of Γn . Under the regular conditions, M consistent estimation of the no-ARCH transformation when r is given. ˆ yt . The Principal Volatility Component is estimated by M Yu-Pin Hu National Chi Nan University

Principal Volatility Component and Their Applications

Estimation and Testing - 2 Testing the ARCH dimension: I

Firstly, we estimate the no-ARCH transformation ˆ 1 , where M ˆ 1 is an s × k matrix, and obtain an M ˆ 1 yt . s-dimensional series eˆt = M

I

Then we develop a test to diagnose the ARCH effect of eˆt , i.e. whether the correlation between 2 eˆ2t and yt−j is zero?

Yu-Pin Hu National Chi Nan University

Principal Volatility Component and Their Applications

Estimation and Testing - 3 I

I

We use the Ling-Li statistic (Ling and Li, 1997) to test the ARCH effects. ˆ 1 yt , where M ˆ 1 is a s × k matrix. Define Let eˆt = M n X ρˆj,s = 1/n ˆt xˆt−j t=j+1

where ˆt = qP n

eˆ0t Vˆ −1 eˆt − s

e0t Vˆ −1 eˆt t=1 (ˆ

xˆt−j

− s)2 /n

0 yt−j Sˆ−1 yt−j − k . = qP n 0 ˆ−1 2 /n (y S y − k) t t t=1

Yu-Pin Hu National Chi Nan University

Principal Volatility Component and Their Applications

Estimation and Testing - 4 I

The statistic considered is 0 ˆ −1 ˆ ˆ d,s Φd,s Rd,s Td,s = nR

ˆ d,s = (ˆ ˆ d,s = [φij ]d×d is the where R ρ1,s , · · · , ρˆd,s )0 and Φ √ ˆ estimated covariance of nRd,s ; and φjj = 1 − (j/n) when 1 ≤ j ≤ d φij = (1 − nj )cov(ˆ c xt , xˆt−(j−i) ) when j > i, P where cov(ˆ c xt , xˆt−h ) = 1/n nt=h+1 (ˆ xt − x¯)(ˆ xt−h − x¯). I

Note that if yt has no-ARCH effect, then ˆ d,s becomes a diagonal cov(xt , xt−h ) = 0 for h 6= 0 and Φ matrix. That is the case considered by Ling and Lin. Yu-Pin Hu National Chi Nan University

Principal Volatility Component and Their Applications

Estimation and Testing - 5

I

Theorem 3: (Ling-Li statistic) Suppose yt is a k-dimensional stationary series with ARCH dimension r = k − s. Under some conditions, 0 ˆ −1 ˆ ˆ d,s Td,s = nR Φd,s Rd,s → χ2d .

Yu-Pin Hu National Chi Nan University

Principal Volatility Component and Their Applications

Estimation and Testing - 6 I

Theorem 4: (Generalized Ling-Li statistic, Hong, 1996) Suppose yt is a k-dimensional stationary series with ARCH dimension r = k − s. Under some conditions, P 2 n n−1 ρ2j,s − Mn (ω) j=1 ω (j/pn )ˆ GTpn ,s = → N (0, 1), {2∆Vn (ω)}1/2 where ∆ = 1 + 2

P∞

h=1

cov 2 (xt , xt−h ) (∆ = 1 in Hong, 1996),

Pn−1

Mn (ω) = j=1 (1 − j/n)ω 2 (j/pn ) and Pn−2 Vn (ω) = j=1 (1 − j/n){1 − (j + 1)/n}ω 4 (j/pn ).

Yu-Pin Hu National Chi Nan University

Principal Volatility Component and Their Applications

Estimation and Testing - 7 I

We adopt Td,s or GTpn ,s to decide the ARCH dimension by testing the hypotheses sequentially: For s = k, k − 1, · · · , 1, H0 : ARCH dimension = k − s vs. H1 : ARCH dimension > k − s.

I

The ARCH dimension is rˆ = k − s? , where s? is the first one that H0 is accepted. Yu-Pin Hu National Chi Nan University

Principal Volatility Component and Their Applications

Data Analysis -1: The seven exchange rates I

Recalling the data of seven exchange rates (k = 7). Each return series contains 604 observations. A VAR(5) model is adopted to remove the serial correlation to have residual series yt .

I

Each residual series displays ARCH effects. We ask ”whether there exists no-ARCH portfolios?” Yu-Pin Hu National Chi Nan University

Principal Volatility Component and Their Applications

Data Analysis -2: The seven exchange rates I

ˆ m for m = 10 to estimate the We adopt Γ ˆ , where transformation M ˆ m = Pm Pk Pk (1 − h/n)2 cov Γ c 2 (yt yt0 , yi,t−h yj,t−h ). h=1 i=1 j=i

I

The estimation of principal volatility component ˆ yt . is M

Yu-Pin Hu National Chi Nan University

Principal Volatility Component and Their Applications

Figure 3: The Principal volatility components First Component 4 3 2 1 0 -1 -2 -3 -4

Second Component 5 4 3 2 1 0 -1 -2 -3 -4

Third Component 4 3 2 1 0 -1 -2 -3 -4

Fourth Component 5 4 3 2 1 0 -1 -2 -3 -4

Fifth Component 10 8 6 4 2 0 -2 -4 -6

Sixth Component 6 4 2 0 -2 -4 -6

Seventh Component 8 6 4 2 0 -2 -4 -6 -8

Table 2: The loadings of the Principal Volatility Components. (The eigenvectors) Ist

2nd

3rd

4th

0.23

-0.36

-0.14

-0.65

Norwegian Kroner 0.19

0.75

-0.17

-0.15

-0.05

0.02

0.31

Swedish Kroner

0.21

-0.51

0.32

0.31

-0.40

0.35

0.23

Swiss Francs

0.22

0.07

0.23

0.03

0.17

-0.19

-0.62

Canadian Dollars

-0.57

0.06

0.40

-0.25

0.65

0.08

0.04

Singapore Dollars

-0.65

-0.17

-0.76

-0.40

-0.20

0.85

-0.6

Australian Dollars

0.23

-0.02

-0.20

0.46

0.57

-0.23

-0.02

Eigenvalues (Percentage)

4%

7%

9%

14 %

17 %

37 %

British Pond

11%

5th

6th

0.05

-0.22

7th 0.20

Data Analysis -3: The seven exchange rates I

The results of ARCH dimension test using Generalized Ling-Li statistic indicate that there is only one no-ARCH portfolio.

I

Further analysis of the principal volatility components confirms the findings.

Yu-Pin Hu National Chi Nan University

Principal Volatility Component and Their Applications

Table 3: The results of ARCH dimension tests Residual fitted by the VAR(5) model Generalized Ling-Li statistics Pn=5

r= k-s = 6 Statistics P-value r= k-s = 5 Statistics P-value r= k-s = 4 Statistics P-value r= k-s = 3 Statistics P-value r= k-s = 2 Statistics P-value r= k-s = 1 Statistics P-value r= k-s = 0 Statistics P-value

Pn=10

Estimation of Δ

-7.4 0.77

-1.0 0.84

2.32

3.3 0.0006

4.2 0.0

2.32

10.8 0.0

12.7 0.0

2.32

26.1 0.0

28.5 0.0

2.32

29.3 0.0

32.5 0.0

2.32

42.4 0.0

46.4 0.0

2.32

55.4 0.0

60.43 0.0

2.32

Table 4: Results of ARCH-effect tests: The Principal volatility components, i.e. the transformed series Mˆ yt

Ljung-Box test

LM test

Lag=5

Lag=10

Lag=5

8.4 0.14

13.7 0.18

9.5 0.09

14.4 0.16

35.4 0.45

81.8 0.16

Statistics 16.7 P-value 0.005

25.4 0.0045

15.3 0.009

20.5 0.024

74.3 0.0

111.6 0.0

Statistics 13.8 P-value 0.016

16.9 0.09

13.1 0.022

15.1 0.13

82.6 0.0

126.7 0.0

Statistics 48.7 P-value 0.0

84.2 0.0

33.9 0.0

47.7 0.0

135.3 0.0

178.9 0.0

Statistics 67.9 P-value 0.0

79.5 0.0

56.3 0.0

59.5 0.0

128.3 0.0

156.8 0.0

Statistics 33.7 P-value 0.0

80.6 0.0

25.9 0.0

52.5 0.0

114.1 0.0

253.7 0.0

130.7 0.0

162.2 0.0

180.2 0.0

293.0 0.0

Extension of -LM test Lag=10 Lag=5

Lag=10

st

1 Component Statistics P-value 2nd Component

3rd Component

4th Component

5th Component

6th Component

7th Component Statistics 192.9 P-value 0.0

231.8 0.0

Data Analysis - 4: The seven exchange rates I

We have tried different values of the parameters ˆ m and GTp ,s . Results are quite to estimate Γ n robust to the parameters.

I

In conclusion, there is a linear combination with no-ARCH effect among the seven exchange rates.

Yu-Pin Hu National Chi Nan University

Principal Volatility Component and Their Applications

Interpretation in Economics - 1 I

The exchange rates are driven by three main factors: (1): The GDP of countries involved. (2): The interest rates. (3): The news events.

I

Due to the globalization and exchange rate trading, the GDP and interest rate effects are exploited by trading. This means their effects should be market neutral. Yu-Pin Hu National Chi Nan University

Principal Volatility Component and Their Applications

Interpretation in Economics - 2 I

However, the news effect is hard to control. But its effect should be short-lived. This is the reason that we find a no-ARCH portfolio in weekly data.

I

In our other empirical experiences, we find that there does not exist no-ARCH portfolio in daily data from 2000 to 2011 (the news effect exists).

Yu-Pin Hu National Chi Nan University

Principal Volatility Component and Their Applications

Thank you!

Yu-Pin Hu National Chi Nan University

Principal Volatility Component and Their Applications