Impulse response analysis

3.4.4.1 Theory

The general VAR(p) model has many parameters, and it may be difficult to interpret coef-

ficients due to complex interactions and feedback effects between variables in the model. As

a result, the dynamic properties of a VAR system are often summarised through some struc-

tural analysis. Two main types of structural analysis are impulse response functions and

forecast error variance decompositions. I will employ both analyses to study the dynamics

in the VAR system.

Impulse response analysis traces the effects of a shock to one endogenous variable on

to the other variables in the VAR system. It moreover traces the speed and persistence of

the shocks, and therefore enables the examination of the time structure of the transmis-

sion. I consider in this study generalized impulse response functions17as opposed to tradi-

tional impulse response in that the generalized responses do not require orthogonalization

of shocks and are invariant to the ordering of the variables in the VAR system. Since there

is no economic theory underpinning the causal relationship between country level implied

volatilities, it is reasonable to use generalised impulse response functions to study the inter-

relationship between them without any a priori assumption.

Under the assumption of covariance-stationarity, equation (3.5) can be rewritten as the

infinite moving average representation,

Y

_t

₌ω₊

∞

X

i=0

A

_i

²

_t−i (3.6)

Where thek×kcoefficient matricesAi can be obtained by recursive substitution,

A

i

=Π

1

A

i−1

+Π

2

A

i−2

+ · · · +Π

p

A

i−p

,

i

=1, 2, . . . ,

17_{See Pesaran and Shin (1998).}

withA0=Ik andAj=0when j<0.

An impulse response can be described as the effect of a hypotheticalk×1 vector of shocks

of sizeδ=(δ1, . . . ,δk)0hitting the system at timetcompared with a base-line profile at time

t+n, given the system’s history. Denoting the known history of the system up to timet−1

by the information setΩt−1, the generalised impulse response function ofYt at horizonn, is

defined by

GI

_Y

(n,δ,Ω

_t−1

)=E(Y

t+n

|²

t

=δ,Ω

t−1

)−E(Y

t+n

|Ω

t−1

)

The appropriate choice of hypothesized vector of shocks,δ, is central to the properties of the impulse response function. The generalised impulse response function shocks one element

of²t, say thejth element, and integrates out the effects of other shocks using an assumed or

historically observed distribution of errors. Then we have

GI

_Y

(n,δ

_j

,Ω

_t−1

)=E(Y

t+n

|²

j t

=δ

j

,Ω

t−1

)−E(Y

t+n

|Ω

t−1

)

. (3.7)

Assuming²t has a multivariate normal distribution with mean vector zero and covariance

matrixΣ=(σi j), the conditional expectation is

E(²

_t

_|²

_{j t}

₌δ

_j

)₌(σ

1j

,σ

2j

,· · ·,σ

k j

)

0

σ

j j−1

δ

j

=Σe

j

σ

j j−1

δ

j

whereej is ank×1 selection vector with unity at itsjth element and zeros elsewhere. Hence,

the generalised impulse response of the effect of a shock in the jth equation onYt+nat time

tis given by











A

_n

Σe

_j

p_σ

j j





















δ

j

p_σ

j j











,

n₌0, 1, 2, . . . .

Settingδj=pσj j, the scaled generalised impulse response function is thek×1 vector:

ψ

j

(n)=

p_σ1_{j j}

A

n

Σe

j

,

n=0, 1, 2, . . . .

(3.8)

3.4.4.2 Results for the Global group

Figure 3.5 presents the results of impulse response analysis with 2 standard error bounds.

The results show that given a shock to any variable in the system, all other variables will

have a positive contemporaneous response to that shock, but the magnitude of responses

is different for each variable. A shock in the change of implied volatility in Japan only gives

rise to about a half unit response of change of implied volatility for U.S., U.K., and Eurozone,

while those responses are much stronger towards a shock in the U.S., U.K., or Eurozone. The

responses are more than 1 unit. Japan clearly has a lagged response to the Western markets.

It has a very small response to the U.S. market on day 1 but has an increased response on day

2. This could be explained by the fact that Japan is on the far East of the globe, and there is

about 8.5 hours difference between the close of the Japanese market and the open of the U.S.

market. Any information not captured on day 1 is flowing into day 2 when Japanese markets

opens about 3 hours after the U.S. market closes, which gives Japan a stronger reaction on

day 2 than day 1.

Similarly Japan has a lagged response to the two European markets, but the responses

on day 2 are decreasing possibly because any news in these two markets has been commu-

nicated with the U.S. market during their 2 hours trading overlap, and is captured by the

responses to the U.S. market. While the impulse responses of Japan to the U.S., U.K., and

Eurozone are positive on the next day, those of U.S., U.K., and Eurozone are corrected on

day 2. This indicates that the responses of U.S., U.K., and Eurozone to the shocks of Japan

suggests that the U.S., U.K., and Eurozone markets lead the Japanese market by one day.

Analogously, the U.K. and Eurozone market have a lagged response to the U.S. market while

the U.S. markets responds to the two European markets only on day 1, which indicates that

the U.S. market leads the U.K. and Eurozone market by one day. The figure shows clearly

that the impact of the shocks are positive on day one or day two, and start to decay there-

after, which suggests that these markets are efficient in that international news are processed

within two days.

The time sequence of markets is not arbitrary. On a given date, Japan is at the top, Eu-

rozone and U.K. come in the middle, and U.S. at the bottom; simply because the Japanese

market is the first to open and U.S. is the last to close. But one can argue that which mar-

ket comes first depends on where one cuts into the chain, and we should expect to result

in the same transmission mechanism under different ordering. Hence, I advance U.S. and

European market each to the top of the chain by appropriately lagging the other variables.

Figure 3.6 shows the impulse responses when the change of implied volatilities are grouped

asU St,Eur ozonet+1,U Kt+1and J Pt+1. It can be seen thatU St has a lagged response to

U Kt+1andEur ozonet+1with an increased next day response. This is because the two con-

tinental markets are grouped such that they are at the two far ends of the chain, thatU Kt+1

andEur ozonet+1have an immediate impact onU St+1but inevitably not much onU St. The

pattern of the impulse responses of Japan to shocks in the U.K. and Eurozone is the same as

in the previous discussion meaning that the U.K. and Eurozone markets lead Japanese mar-

ket by one day. But J Pt+1 does not have a lagged response to shocks inU St as J Pt+1 sits

immediately afterU St, of which information is processed contemporaneously. Likewise for

U.K. and Eurozone, the impact of shock to U.S. on daytis fully processed on dayt+1, sug-

gesting that U.S. leads U.K. and Eurozone market by one day.

show thatJ Pt+1has a stronger impact onEur ozonet+1andU Kt+1than onEur ozonet and

U Kt. Putting Japan on the bottom of the chain helps us to see that the shocks to U.S., U.K.,

and Eurozone on daytis fully incorporated by Japan on dayt+1, which suggests that U.S.,

U.K., and Eurozone market leads Japanese market by one day. Similarly, Eur ozonet and

U Kt have a lagged response toU St, which suggests that the U.S. market leads U.K. and Eu-

rozone market by one day. In summary, all three sets of impulse responses tend to agree that

the Japanese market is a follower in this global setup while U.S. is the dominant leader in

implied volatility spillover. The European markets sit in the middle where it follows the U.S.

and leads Japan.

The aforementioned argument can be further supported by a closer look at the VAR re-

gressions. Table 3.5 shows theR2of each component regression equation under a VAR(3)18

specification, as well as the breakdown of lags included in the estimation. The table reports

results of all three groups under different ordering of the variables. Group one shows that

when variables are grouped on the same timeline dayt, lags up to 3 periods are included in

the estimation for each of the variable. The Japan equation has the highestR2=0.28 whereas

U.S. has the lowest at 0.06. When U.S. is advanced to the top of the timeline and JP, Euro-

zone, and U.K. are lagged one day in the second group,R2changed significantly. The lags

included in the U.S. equation aret−1,t−2 andt−3 while those included in JP, Eurozone,

and the U.K. equation are t,t−1 and t−2. R2 increased substantially for the U.S. equa-

tion to 0.44 from previous 0.06 by including Eur ozonet,U Kt, and J Pt in the estimation,

indicating that changes of implied volatility ofEur ozonet,U Kt, andJ Pt have a significant

impact on the changes of implied volatility of U.S. on dayt. Omitting the lag ofU St in the

equation of Eur ozonet+1,U Kt+1, and J Pt+1 significantly reduced the explanatory power.

18_{Different ordering of the variables requires distinct VAR specification. The system is best described by}

a VAR(7), VAR(3), and VAR(3) respectively for each group. But in order to disentangle the marginal effect of inclusion(exclusion) of one specific lag in a particular equation, I employ a VAR(3) model for all groups for consistent comparison.

R2decreased to 0.04, 0.04, and 0.14 respectively, indicating thatU St has a strong influence

onEur ozonet+1,U Kt+1, andJ Pt+1. In the third group when Japan is ordered the last of the

chain, results show that R2of Japan equation reduced to 0.05 whenEur ozonet,U Kt, and

U St are not included in the equation, indicatingEur ozonet,U Kt, andU St have a signifi-

cant impact on J Pt+1. The results show consistent evidence that the U.S. market impacts

other markets on the following day, and the two European markets impact Japanese market

on the following day.

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