5. Variance decomposition
As we know, a vector autoregression VAR can be written as a vector moving average VMA. Thus, Eq. 3 can be iterated backward infinite times to obtain
X = m +
j = 0
A
1 j
o
t − j
5 where m = I + A
1
+ A
2
+ … A
is the unconditional mean of X
t
. The fact that Eq. 5 is the VMA representation of Eq. 3 in that variables x
it
are expressed in terms of the current and past values of the various types of shocks o
it
. The VMA representation of Eq. 3 is an essential feature of Sims 1980 methodology in that it allows a tracing out of the time path of the various shocks
on the variables contained in the VAR system. Since there are 11 variables excluding Thailand in our system, there will be totally 11 × 11 = 121 impulse
response functions. It is extremely complicated to plot the impulse response functions to represent the behavior of the X
it
series in response to the various shocks e
it
. Instead, we choose to use ‘forecast error variance decomposition’ to show the
proportion of the movements in a sequence say country 1, x
1t
due to its own shocks o
it
versus shocks from other countries e
2t
– o
11t
. If we use Eq. 5 to conditionally forecast X
t
+
n
the n-period forecast error is X
t + n
− E
t
X
t + n
=
n − 1 j = 0
A
1 j
o
t + n − j
Focusing solely on the x
1t
, sequence, we see that the variance of the n-step ahead forecast error variance of x
1t + n
is s
1
n
2
= s
1 2
n − 1 j = 1
a
1,2
j
2
+ ... + s
11 2
n − 1 j = 1
a
1,11
j
2
where A
1 j
= [a
pq
j]
11 × 11
and Varo
it
= s
i 2
Thus, the ratio of W
1
i = s
i 2
n − 1 j = 1
a
1,i
j
2
s
1
n
2
represents the proportion of movements in country 1, x
1t
, due to shocks from country i, o
it
. If o
2t
, – o
11t
shocks explain none of the forecast error variance of x
1t
at all forecast horizons, we can say that the x
lt
, sequence is ‘fully exogenous’. In such a circumstance, the x
1t
sequence would evolve independently of the o
2t
– o
11t
, and x
2t
– x
11t
, sequence. At the other extreme, o
2t
– e
11t
shocks could explain all the forecast error variance in the x
1t
sequence at all forecast horizons, so that x
1t
sequence would be ‘fully endogenous’. The ratio W
i
i , for i = 1, 2,…11 that is the proportion of movements in country i, x,t, which can be explained by its own shock
Table 8 Variance decomposition before the crisis
ait, can be used to represent the ‘degree of exogeneity’ of country i in response to the financial crisis. We also compare the ratio for each country before and during
the period of the crisis, in order to understand whether the ‘degree of exogeneity’ was reduced as a response to the financial crisis.
Tables 8 and 9 present, respectively, the results of variance decomposition for the period before, and during, the crisis. Each entry in the tables denotes the percentage
of forecast error variance of markets on the left-hand side explained by the markets at the top. These entries are all convergent values in 24-day horizon. To facilitate
the understanding of Tables 8 and 9, we also plot the results in Fig. 2.
Table 10 shows the proportion of market movements that can be explained by its own shocks, or the ‘degree of exogeneity’, before and during the period of the crisis.
This indicates that the ‘degree of exogeneity’ for all countries has been significantly reduced, implying no countries are exogenous to the financial crisis. The degree of
exogeneity in three markets, US, Australia and Taiwan, was reduced less 7.68, 4.00 and 7.67, respectively. This also suggests that the markets of Australia and
Taiwan passively responded to other country’s innovations during the period of the Asian financial crisis.
6. Summary
Kategori