Impulse Response and Variance Decomposition in the Presence of Time-Varying Volatility With Applications to Price Discovery

8/2/2019 Impulse Response and Variance Decomposition in the Presence of Time-Varying Volatility With Applications to Price

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Impulse Response and Variance Decomposition inthe Presence of Time-Varying Volatility: with

Applications to Price Discovery

Chor-yiu SIN

Department of Economics

Hong Kong Baptist University, Hong Kong

May 18, 2005First draft. Comments are welcome.

Abstract

Macroeconomic or financial data are often modelled with cointegration andtime-varying volatility. Noticeable examples include stock prices of the sameunderlying asset. Interestingly, most if not all studies in price discovery donot consider the possible time-varying volatility. As a result, the impulse re-

sponses are not inflated or deflated by the time-varying volatility. Intuitively,the impacts of an i.i.d. innovation is bigger when the volatility (conditionalvariance) is larger. On the other hand, the shocks derived from a Choleski de-composition also depend on the time-varying volatility. In this paper, we firstgeneralize the conventional price discovery impulse response function (IRF)to its time-varying counterpart. Time-varying information share (IS) and thegeneral variance decomposition (VD) are defined accordingly. Using the as-ymptotic theories developed in Li, Ling and Wong (2001) and Sin and Ling(2004), we extend Phillips (1998) to cases that consider time-varying volatil-ity. Larger sample sizes are required as time-varying volatility and some otherparameters need to be estimated.

Key Words: Error-correction model; Exchange-traded fund, Impulse re-sponse; Information share; Price discovery; Shock; Variance decomposition

JEL Codes: C32, C51, G14

Acknowledgments: This research is partially supported by the Hong Kong Research GrantCouncil competitive earmarked grant HKBU2014/02H. The usual disclaimers apply.

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1 Introduction

Throughout this paper, we consider an m

dimensional autoregressive (AR) process

of{yt}, which is generated by

yt = J(L)yt1 + t, (1.1)

t = V1/2t1 t, (1.2)

where L is the lag operator. J(L) =s

k=1 JkLk1, Jks are constant matrices. t =

(1t, . . . , mt) is a sequence of independently and identically distributed (i.i.d.) ran-

dom mx1-vectors with zero mean and identity covariance matrix, Vt1 is measurable-

Ft1, where Ft = {s, s = t, t1, . . .}. As a result, E(t|Ft1) = 0 and E(tt|Ft1) =Vt1.

The system (1.1) is initialized at t = s + 1, . . . , 0 and we may let these initialvalues be any (degenerate or non-degenerate) random vectors. It is often convenient

to set the initial conditions so that the I(0) component of (1.1) is stationary and we

will proceed as if this has been done. The presence of deterministic components in

(1.1) does not affect our conclusions in any substantive way, so we will proceed as

if they are absent just to keep the derivations as simple as possible.

One may see that (1.1) is exactly the partially nonstationary vector autoregres-

sive model employed in Phillips (1998) (see Section 2 below). Instead of assuming

an i.i.d. {t}, (1.2) allows possible time-varying heteroskedasticity.In this paper, we adopt the constant-correlation multivariate GARCH first sug-

gested by Bollerslev (1990). More precisely, we assume that Vt1 = Dt1Dt1,

where is a symmetric square matrix of constant correlations and Dt1 =

diag(

h1t1, . . . ,

hmt1) is a diagonal matrix of conditional standard deviations,

where:

hit1 = ai0 +q

j=1

aij2itj +

pk=1

bikhit1k, i = 1, . . . , m . (1.3)

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Despite the fact that many papers in the literature of price discovery, such as Bar-

clay and Hendershott (2003), Covrig and Melvin (2002), Engle and Patton (2004),

Hasbrouck (2003), Huang (2002), and Yan and Zivot (2004), consider a system ofthis kind, few if not none of them estimate model (1.1) with the consideration of

GARCH. Following the lines in Ling and Li (1997) and Tse (2002), Sin (2004) de-

rives the asymptotic theory of testing for multivariate ARCH when the conditional

mean is an ECM. In addition and unsurprisingly, GARCH is found in many of the

time series data, especially those that are related to financial markets.

In the next section, we first generalize the conventional price discovery impulse

response function (IRF) to its time-varying counterpart. Time-varying information

share (IS) and time-varying component share (CS) are defined accordingly. A sys-

tematic and practical procedure for price discovery dynamics, following the lines

in Gonzalo and Ng (2001) and Yan and Zivot (2004), can also be found there. In

Section 3, we extend Phillips (1998)s asymptotic theories to cases that consider

time-varying volatility, following the lines in Li, Ling and Wong (2001) and Sin and

Ling (2004). Conclusions can be found in the last section . The proofs are relegatedto either Appendix A or Appendix C.

Throughout, for any square matrix , min() stands for the minimum eigenvalue

of . T stands for the number of observations. L denotes convergence in distri-bution, Op(1) denotes a series of random numbers that are bounded in probability,

and op(1) denotes a series of random numbers converging to zero in probability.

2 Price Discovery Dynamics: Definitions and A

Practical Procedure

Following Phillips (1998), we first re-write (1.1) in levels and differences format as:

yt = Ayt1 + (L)yt1 + t, A = J(1),

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(L) =s1k=1

kLk1, k =

sl=k+1

Jl, (2.1)

where A = Im + , and are mxr matrices of full column rank r, r

m.

Thus the moving average (MA) representation of the system is:

yt =t1k=0

MCkMtk =t1k=0

ktk, (2.2)

where k =: MCkM, M = [Im, 0, . . . , 0], and

C =

J1 Js1 JsIm 0mxm 0mxm

0mxm Im 0mxm

.

As in Gonzalo and Ng (2001) and Yan and Zivot (2004), we first consider the set

of permanent and transitory shocks Gtk. In contrast to the existing literature

and in view of the time-varying volatility (see (1.2)-(1.3) in Section 1), we write

Gtk = GV1/2tk1tk. Note E[GV

1/2tk1tk

tkV

1/2tk1G

| Ftk1] = GVtk1G.Therefore, in order to orthogonalize the Gtk and interpret the coefficients of the

resulting MA representation in (2.2), for each t

k, we let tk = P

1tk1Gtk, and

Ptk1 is the lower triangular Choleski decomposition of GVtk1G such that:

GVtk1G = Ptk1P

tk1, and E[tk

tk | Ftk1] = Im. (2.3)

Therefore the corresponding MA representation of yt is:

yt =t1k=0

kG1Ptk1tk. (2.4)

Thus the systems impulse responses are given by the elements of the sequence of

matrices kG1Ptk1. It is clear that the impacts of one unit increase in each ele-

ment oftk depends on Ptk1. Denote the currentPt as Pn (n may be interpreted

as now). One may be interested in the impacts of n+1 on yn+1, yn+2, yn+3, . . .,

which are given by 0G1Pn, 1G

1Pn, 2G1Pn, . . . respectively.

Decompose G =

G1G2

, where G1t is the dx1 permanent shock and G2t is

the rx1 transitory shock. When d = 1, which is the case in the literature of price

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discovery where there is one efficient price for stock prices of the same underlying

asset, we can define shock js information share (IS), where j = 1, . . . , m. See, for

instance, Hasbrouck (1995) and Yan and Zivot (2004).Consider t = n + 1. As n+1 = G

1Pnn+1, the permanent shock G1n+1 =

G1G1Pnn+1, where E[G

1G1Pnn+1

n+1P

nG1G1 | Fn] = G1VnG1 but E[n+1n+1 |

Fn] = Im. Therefore, for n = 1, 2, . . ., we define the following time-varying IS ofshock j:

ISjn =([G1G

1Pn]j)2

G1VnG1. (2.5)

In general, for any mx1 vector , we can consider the following variance de-composition(VD) w.r.t. shock j for n = 1, 2, . . .,

([G1Pn]j)2

Vn. (2.6)

In the literature, there are at least two sets of permanent and transitory shocks,

they are: G =

or G =

, where is an mxd- matrix of full column

rank such that

= 0dxr. The former G is proposed by Warne (1993) whilethe latter is proposed by Gonzalo and Granger (1995). As far as permanent shock

is concerned, both sets coincide with the one by Stock and Watson (1988) who

consider the common (random walk) factor of a cointegrated system. Though the

latter G is not necessarily invertible (and our analysis will then break down, see Sub-

section 3.2.1 of Levtchenkova, Pagan and Robertson, 1999), it has the structural

interpretation that the transitory shock t does not have a long-run impact on yt.

See Definition 1 of Gonzalo and Granger (1995). Further, this property is preserved

with the transformation of G1Pn, as one can show by the arguments similar to

those in Sub-section 2.3 of Gonzalo and Ng (2001). See also Sub-section 3.1 of Yan

and Zivot (2004).

Both G can be obtained from the following error-correction model (ECM), which

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is an alternative representation of (2.1):

yt = yt1 +

s1

l=1

lytl + t. (2.7)

A practical procedure for estimating the impulse responses can be summarized

as:

1. Giving {yt : t = 1, . . . , T }, estimate the parameters of the ECM in (2.7) and the{Vt1 : t = 1, . . . , T } in (1.2).2. Refer to (2.2). Construct k := M

CkM, k = 0, 1, . . ..

3. Construct G. The set of permanent and transitory shocks are constructed as

{GV1/2t1 t : t = 1, . . . , T }4. For n = 1, 2, . . ., obtain a lower triangular matrix Pn such that PnP

n = GVnG

.

5. For n = 1, 2, . . ., the impacts of n+1 = P1

n Gn+1 on yn+1, yn+2, yn+3, . . ., are

estimated respectively by 0G1Pn, 1G

1Pn, 2G1Pn, . . .. See Theorem 3.4 for

the asymptotic properties.

6. For n = 1, 2, . . ., the IS of shock j is estimated by ISjn =([G

1G1Pn]j)

2

G1VnG1

, j =

1, . . . , m. See Corollary 3.8 for the asymptotic properties.7. For n = 1, 2, . . ., the general VD of shock j is estimated by V Djn =

([G1Pn]j)2

Vn,

j = 1, . . . , m. See Theorem 3.7 for the asymptotic properties.

The above procedure differs from that in Gonzalo and Ng (2001) or Yan and Zivot

(2004) in two respects. Instead of looking at the impacts on yn+1, yn+2, yn+3, . . .,

we look at those on yn+1, yn+2, yn+3, . . .. Secondly and more importantly, due to the

time-varying Pn, the impulse responses are also time-varying.

In the balance of this paper, we focus on the volatility model specified in (1.3).

We first denote the parameter in the ECM (2.7) as = [1, 2], where 1 = vec[]

(the nonstationary parameter) and 2 = vec[, 1, . . . , s1] (the stationary

parameter). Then we denote the parameter in the GARCH equation (1.3) as

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= [1, 2], where 1 = [a

0, a

1, . . . , a

q, b

1, . . . , b

p], aj = [a1j,...,amj]

, bl = [b1l,...,

bml], j = 0, 1,...,q,l = 1,...,p, and 2 (), which is obtained from vec() by

eliminating the supradiagonal and the diagonal elements of (see Magnus, 1988,p.27). Conditional on the initial values yt = 0 for t 0, the log-likelihood function,as a function of the true parameter, can be written as:

l(, ) =n

t=1

lt and lt = 12

tV1t1t

1

2ln |Vt1|, (2.8)

where Vt1 = Dt1Dt1, Dt1 = diag(

h1t1, . . . ,

hmt1). Further denote ht1 =

(h1t1, . . . , hmt1), Ht1 = (h

11t1, . . . , h

1mt1)

, and t1 = [(yt1, y

t1, . . . , y

ts+1]

.

By Sections 3 and 4 of Sin and Ling (2004), the score function, as a function ofthe true parameter, can be expressed as:

1lt = 1

21ht1( w(ttV1t1)) Ht1 + (yt1 )V1t1t,

2lt = 1

22ht1( w(ttV1t1)) Ht1 + (t1 Im)V1t1t,

lt =

121ht1( w(ttV1t1)) Ht1(1 1D1t1ttD1t11)

.

Given some initial values 1, 2, , we perform a one-step iteration:

1 = 1 (Tt=1

R1t|1,2,)1(Tt=1

1lt|1,2,), (2.9)

2 = 2 (Tt=1

R2t|1,2,)1(Tt=1

2lt|1,2,), (2.10)

= (Tt=1

St|1,2,)1(Tt=1

lt|1,2,), (2.11)

where, in terms of the true parameter,

R1t = (yt1y

t1

V

1

t1) (1ht1)D2

t1(

1

+ Im)D2

t1(

1ht1)/4,

R2t = (t1t1 V1t1) (2ht1)D2t1(1 + Im)D2t1(2ht1)/4;

and, in terms of the true parameter, St = (Sijt)22 with

S11t = (1ht1)D2t1(1 + Im)D2t1(1ht1)/4,

S12t = (1ht1)D2t1m(Im 1)NmLm,

S22t = 2LmNm[1 1]NmLm,

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= (1, 1, . . . , 1) and w() is a vector containing the diagonal elements of the square

matrix . m, Nm and Lm are constant matrices (see Magnus, 1988, pp.109, 48 and

96).In practice, we may repeat the iterative procedure in (2.9)-(2.11), in order to

get an estimator closer to the quasi-maximum likelihood estimator (QMLE) of the

log-likelihood function (LF) specified in (2.8), though the asymptotic distribution is

not altered.

In the next section, using the asymptotic theories developed in Li, Ling and

Wong (2001), Sin and Ling (2004) and Phillips (1998), we derive the asymptotic

properties of the IRF, the general VD, and the IS.

3 Price Discovery Dynamics: Asymptotic Prop-

erties

In order to study the (asymptotic) confidence interval, and some other (asymptotic)

properties of the IRF, the IS and the general VD, we first note that for k = 1, 2, . . .:

kG1Pn kG1Pn

= (k k)G1Pn + k(G1 G1)Pn + kG1(Pn Pn). (3.1)

In Lemma 3.1, we first derive the asymptotic approximation of

T(k k).Lemma 3.1 is followed by Lemma 3.2(a), which derives the asymptotic approxima-

tion of

T(G G), where G =

. When G =

, since

T( ) =

Op(T1/2), we only need to derive the asymptotic approximation of

T( ).

The result can be found in Lemma 3.2(b). Lemmas 3.2(a)-3.2(b) are followed by

Lemma 3.3, which derives the asymptotic approximation of

T(hn hn). All theapproximations are in terms ofT(11),

T(22) and

T( ). The proofs

of Lemmas 3.1-3.3 can be found in Appendix A.

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Equipped with these three lemmas (as well as those in Appendix B), we are

able to derive an asymptotic approximation of T(kG1

Pn kG1

Pn) in The-orem 3.4. Corollary 3.5 contains a special case in which the volatility is constant

across time. This is the case thoroughly discussed in Gonzalo and Ng (2001) and

Yan and Zivot (2004). Though as far as we know, the asymptotic approximation

derived in this corollary is new. In Corollary 3.6, we show that the asymptotic dis-

tribution derived in Phillips (1998) turns out to be a special case in Theorem 3.4,

with the volatility being a constant on the one hand, and no permanent-transitory

decomposition is considered (that is, G = Im) on the other hand.

Similarly, we are able to derive an asymptotic approximation of the general

T( V Djn V Djn) in Theorem 3.7. Corollary 3.8 documents the special case

T(ISjn ISjn), in which = G1 = ; while Corollary 3.9 documents thespecial case that is pre-determined. The proof of Theorem 3.7 can be found in

Appendix C.

We first state the following assumptions:

Assumption 3.1. The determinantal equation | Im J(L)L |= 0 has roots onor outside the unit circle. 2

Assumption 3.2. | ((1) Im) |= 0, where is defined around (2.6)while, in a similar token, is anmxd matrix of full column rank that is orthogonal

to . That is, = 0dxr. 2

Assumption 3.3. Fori = 1, . . . , m, ai0 > 0, ai1, . . . , aiq, bi1, . . . , bip 0, andqj=1 aij +

pk=1 bik < 1. 2

Assumption 3.4. Fori = 1, . . . , m, all eigenvalues of E(Ait Ait) lie inside

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the unit circle, where denotes the Kronecker product and

Ait =

ai12it aiq2it bi12it bip2it

Iq1 0(q1)1 0(q1)p

ai1 aiq bi1 bip0(p1)q Ip1 0(p1)1

. 2

Assumption 3.5. t is symmetrically distributed. 2

Assumption 3.6. For n = 1, 2, . . ., | GVnG |= 0. 2

Assumptions 3.1 and 3.2 are, respectively, Assumptions 2.1(b) and 2.1(d) in

Phillips (1998). Instead of assuming i.i.d. {t}, here we adopt Assumptions 3.3-3.4.Assumptions 3.3-3.4 are the necessary and sufficient conditions for E(vec[tt]vec[t

t]) 0,

there exists N = N() such that for all n > N, Prob[min(G) < ] < . Thus, (a)

is shown. Further, we can write:

G1 G1 = G1(G G)G1 = Op(T1/2), (B. 1)

given (a) and Lemma 3.2. Thus (b) is also shown. Finally, from (B.1),

G1 G1 = G1(G G)G1 (G1 G1)(G G)G1

= G1(G G)G1 + Op(T1),

by (b) and Lemma 3.2. Thus, (c) is also shown. 2

Lemma B.2. Suppose Assumptions 3.1-3.5 hold.

vec[V1/2n V1/2n ]=

1

2(1/2 D1n )m(hn hn) + (1/2 Dn)Lm

(2 2) + Op(T1)

= Op(T1/2). 2

Proof. Given Vn = DnDn in Model (3.1), we can write:

V1/2n V1/2n

= (Dn Dn)1/2 + Dn(1/2 1/2) + (Dn Dn)(1/2 1/2) (B. 2)

Given Lemma 3.3, it is not difficult to show that Dn Dn = Op(T1/2) (see, forinstance, Lemma A.2 of Sin, 2004). On the other hand, D1n = Op(1), therefore,

Dn Dn = 12

D1n [(D2n D2n) (Dn Dn)2]

=1

2D1n (D

2n D2n) + Op(T1). (B. 3)

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In spite of the fact that 1/2 is symmetric, it is beneficial to write:

= 1/21/2

1/21/2

= (1/2 1/2)1/2 1/2(1/2 1/2). (B. 4)

Therefore,

[(1/2 Im) + Kmm(1/2 Im)]vec[1/2 1/2] = vec[ ]. (B. 5)

First note that is positive definite. By Theorem A.1(c), for all > 0, there exists

N = N() such that for all n > N, Prob[min() < 2] <

Prob[min(

1/2) 0, there exists N = N() such that for all n > N, Prob[min(GVnG) < 2]

Impulse Response and Variance Decomposition in the Presence of Time-Varying Volatility With Applications to Price Discovery

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