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.P. Blake, G. Kapetanios Economics Letters 69 2000 15 –23
i.i.d. observations. Results for sequences of dependent data exist for some cases such as AR or MA models see Bose, 1988 and Bose, 1990 indicating that the improvement is of order o 1 rather than
p 21 2
O T .
p
In our framework the bootstrap can be applied as follows. Once the original test statistic, denoted ˆ
ˆ by S has been obtained we retrieve the set of residuals,
he , . . . , e j from 1. We then resample
1 T
randomly with replacement from the set of the residuals to obtain a bootstrap sample of residuals ˆ
ˆ ˆ
ˆ ˆ
e , . . . , e where each e has been drawn with replacement from
he , . . . , e j and stars denote
1 T
t 1
T
generic bootstrap quantities. We then carry out the artificial neural network test on the bootstrap ˆ
ˆ sample of residuals,
e , . . . , e . Repeating this process N times where N is the number of bootstrap
1 T
replications we obtain a set of bootstrap test statistics, S , . . . , S . We then use these samples to
1 N
construct the bootstrap distribution of our test statistic. More specifically the estimated P-value of a given test statistic S is given by
N
O
1S S
n 51 n
ˆ ]]]]]
p 5 N
where 1 ? is the indicator function taking the value 1 when its argument is true and zero otherwise. We will refer to the bootstrap test as RBF-B. Under the null hypothesis, the error terms in 1 are i.i.d.
and therefore the resampling scheme described above is justified. Under an ARCH alternative the random resampling should ensure that the dependence between the resampled residuals is negligible
asymptotically thereby providing a consistent testing procedure under certain conditions. Establishing that the error sequence in 1 is either mixing or near epoque dependent see e.g. Davidson, 1994, pp.
261–277 under a generalised ARCH alternative should be sufficient for the testing procedure to be consistent. Nevertheless the sufficiency of mixing or near epoque dependence is conjectured and a
rigorous proof remains to be provided.
4. Monte Carlo study
As we mentioned earlier, both the residuals and the squared residuals of 1, at given lags, are used as possible inputs to the neural network. We refer to the tests using residuals in levels as ANN-L,
RBF-L and RBF-BL tests respectively, and similarly the tests using squared residuals as ANN-S, RBF-S and RBF-BS.
˜ Following Peguin-Feissolle 1999 we set q 5 10 and q 5 3 for ANN-L and ANN-S. We carry out
Monte Carlo experiments for three widely used ARCH-type specifications. The first is a standard ARCH model. The second is a generalised ARCH GARCH model and the third is an exponential
GARCH E-GARCH model. The first case coincides with that of Peguin-Feissolle 1999 for comparative purposes. The other setups are considered because of their popularity in the literature. For
all cases the regression model 1 is given by
y 5 0 ? 25 1 0 ? 5x 1 e
t t
t
x 5 0 ? 7x 1
n , n | NID0,1
t t 21
t t
The data y and x have been normalised to lie between 0 and 1 for the ANN-L and ANN-S tests.
t t
A .P. Blake, G. Kapetanios Economics Letters 69 2000 15 –23
19
The specification of h for the three experiments we will consider together with the lag order of the
t
residuals used as inputs to the neural network are: •
Case 0: p51 e | NID0, 1
t
• Case 1: p53
2 2
2
h 5 0 ? 2 1 0 ? 2 e
1 0 ? 4 e
1 0 ? 3 e
,
t t 21
t 22 t 23
] e 5 h v , v | NID0, 1
t
œ
t t
t
• Case 2: p53
2 2
2
h 5 0 ? 2 1 0 ? 1
e 1 0 ? 3
e 1 0 ? 1
e 1 0 ? 1h
t t 21
t 22 t 23
t 21
1 0 ? 2h 1 0 ? 1h
,
t 22 t 23
] e 5 h v , v | NID0, 1
t
œ
t t
t
Fig. 1. P-discrepancy.
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.P. Blake, G. Kapetanios Economics Letters 69 2000 15 –23
• Case 3: p53
log h 5 0 ? 2 1 0 ? 1
z 1 0 ? 3
z 1 0 ? 1
z 1 0 ? 1 log h
t t 21
t 22 t 23
t 21
1 0 ? 2 log h 1 0 ? 1 log h
,
t 22 t 23
] e 5 h v , v | NID0, 1 z 5 uv u 2 Euv u 1 nv , n 5 2 0 ? 5
t
œ
t t
t t
t t
t
The negative value given to n for the E-GARCH setup reflects the so-called leverage effect
commonly found in stock prices denoting the asymmetric responses to positive and negative shocks. For all cases four different sample sizes were considered to investigate the effect of the number of
observations on the performance of the tests. The sample sizes considered were 50, 100, 150 and 250. For each sample size, T samples of size T 1 200 were constructed with 200 initial observations
discarded to minimise the effect of starting values, which are set to zero. For each experiment 1000 replications were carried out.
Fig. 2. Size-power curve: Case 1.
A .P. Blake, G. Kapetanios Economics Letters 69 2000 15 –23
21
5. Monte Carlo results
