5.3. Filtered equity returns

We now turn to the corresponding findings for the S&P 500 returns. In

interpreting the results, it is important to recognize that the estimated intraday

periodicity now involves interaction terms between the daily volatility level and

 (a) See Table 2a for construction of the raw return series. The method for standardizing the returns,

/~t i is described in the main text.

(b) See Table 2b for construction of the raw return series. The standardized returns, Rt.i, are generated

as described in the main text.

the Fourier functional form, so that not only the level but also the shape of the

volatility pattern varies with o-,. Thus, our stylized deterministic periodic model

discussed in Section 3 is not strictly valid in this context i.e. generally st, n 4: s ....

for t 4= r. Counter to the results for the DM-$ returns, this time-varying volatility

component may weaken the autocorrelations for the raw absolute returns as,

e f f e c t i v e l y , additional noise is injected into the returns process. Fig. 7b seem to

indicate that this is indeed the case, as the correlogram for the filtered absolute returns, I Rt,n[, lies substantially above that for the raw series, [R,.,,I. Interestingly,

this does not just occur at the 5-minute sampling frequency; the absolute return

autocorrelation across the nine different intraday frequencies, as measured by pA,

QA(10) and VR A, are all markedly higher than the corresponding statistics for the

raw returns in Table lb 45. This is also in line with the MA(I)-GARCH(1, 1)

estimation results reported in Table 5b. The parameter estimates obtained as we

I move from the ~ day to the 40-minute return horizon are again consistent with the

theory. Thereafter, the sum ~(k) and /3(k) decay slightly, but more importantly ~(k)

starts to increase. However, all the intraday estimates now consistently p~nt

towards a very high degree of volatility persistence and in all instances c~(~) +/3(k )

are higher than the estimates for the raw return series in Table 2b. Note also, that

in line with the findings for the DM-$ returns, the 5 day correlogram in Fig, 7b

for the 5-minute filtered S&P 500 returns still retains a distinct periodic pattern,

indicating the presence of even more complicated stochastic volatility components.

Nonetheless, the simple filtering procedure again succeeds in eliminating a large

proportion of the systematic intraday variation in the absolute returns and in so

doing has unveiled a cleaner and starkly different picture of the volatility

dynamics.

We now turn to the corresponding findings for the S&P 500 returns. In

interpreting the results, it is important to recognize that the estimated intraday

periodicity now involves interaction terms between the daily volatility level and

 (a) See Table 2a for construction of the raw return series. The method for standardizing the returns,

/~t i is described in the main text.

(b) See Table 2b for construction of the raw return series. The standardized returns, Rt.i, are generated

as described in the main text.

the Fourier functional form, so that not only the level but also the shape of the

volatility pattern varies with o-,. Thus, our stylized deterministic periodic model

discussed in Section 3 is not strictly valid in this context i.e. generally st, n 4: s ....

for t 4= r. Counter to the results for the DM-$ returns, this time-varying volatility

component may weaken the autocorrelations for the raw absolute returns as,

e f f e c t i v e l y , additional noise is injected into the returns process. Fig. 7b seem to

indicate that this is indeed the case, as the correlogram for the filtered absolute returns, I Rt,n[, lies substantially above that for the raw series, [R,.,,I. Interestingly,

this does not just occur at the 5-minute sampling frequency; the absolute return

autocorrelation across the nine different intraday frequencies, as measured by pA,

QA(10) and VR A, are all markedly higher than the corresponding statistics for the

raw returns in Table lb 45. This is also in line with the MA(I)-GARCH(1, 1)

estimation results reported in Table 5b. The parameter estimates obtained as we

I move from the ~ day to the 40-minute return horizon are again consistent with the

theory. Thereafter, the sum ~(k) and /3(k) decay slightly, but more importantly ~(k)

starts to increase. However, all the intraday estimates now consistently p~nt

towards a very high degree of volatility persistence and in all instances c~(~) +/3(k )

are higher than the estimates for the raw return series in Table 2b. Note also, that

in line with the findings for the DM-$ returns, the 5 day correlogram in Fig, 7b

for the 5-minute filtered S&P 500 returns still retains a distinct periodic pattern,

indicating the presence of even more complicated stochastic volatility components.

Nonetheless, the simple filtering procedure again succeeds in eliminating a large

proportion of the systematic intraday variation in the absolute returns and in so

doing has unveiled a cleaner and starkly different picture of the volatility

dynamics.