3.2. Interpretation in terms of a suggestive intraday return model

The pronounced systematic fluctuations in the return correlogram provide an

initial indication that direct ARCH modeling of the intraday return volatility would

be hazardous. Standard ARCH models imply a geometric decay in the return

autocorrelation structure and simply cannot accommodate strong regular cyclical

patterns of the sort displayed in Fig. 4. Instead, it seems intuitively clear that the

combination of recurring cycles at the daily frequency and a slow decay in the

average autocorrelations may be explained by the joint presence of the pronounced

intraday periodicity documented above coupled with the strong daily conditional

heteroskedasticity 20. The following stylized model provides a simple specification

of the interaction between these two components,

N 1 N

R,--- E R,,n = E soZ,,n. (1)

n=l n=l

Here, R t denotes the daily continuously compounded return calculated from the N

uncorrelated intraday return components, Rt, .. The conditional volatility factor for

day t is denoted by ~r t, while s n refers to a deterministic intraday periodic

component and Z,, n is an i.i.d, mean zero, unit variance error term assumed to be

independent of the daily volatility process, {~rt}. Both volatility components must

be non-negative i.e. % > 0 a.s. for all t and s, > 0 for all n. The following

terminology for the normalized, deterministic sample means and covariances for

the periodic structure will prove convenient:

1 N 1 N 1 N

n=~ =- M( s ) = 1, -~ E SnSn_i -= M( ssi), -~ E S2 =- M( s2),

=1 n=l n=l

where s,,+j N =- s n for any integer j and 0 < n < N.

In the absence of intraday periodicity (s, = 1 for all n) the daily returns may be

represented in the form R t = o',N 1/2Y',,=1,NZt, ., where the return component

N-1/2F,,= LNZ,,n is i.i.d, with mean zero and unit variance. Thus, Eq. (1) extends

the standard volatility model for daily returns to an intraday setting with independent

return innovations and deterministic volatility patterns. Of course, this type of

periodicity is annihilated when the returns are measured at the daily frequency. In

particular, letting Z, denote an i.i.d, random variable with E(Z t) = 0 and Var(Z t)

= 1, we have

R t = MI/2( S 2 ) o',Z,, (2)

so that the expected absolute return equals M1/2(s2)~tEIZ, I. Since M1/2($2) > 1,

the expected daily absolute return is an increasing function of the fluctuations in

the intraday periodic pattern. However, this effect is limited to a scale factor.

Letting c = (EIZ, I) -2 - 1 > 0, it follows that for t 4= ~',

Cov( o-,, o;_)

Corr(lRtl, ]RTI) = Var(o.t) + cE(~rt2) " (3)

Hence, the presence of periodic components reduces the overall level of the

interdaily return autocorrelations, without affecting the autocorrelation pattern 21

In contrast, the periodicity may have a strong impact on the autocorrelation pattern

for the absolute intraday returns. Straightforward calculations reveal

Corr(IR,,,,I, IR .... I)

m(ssn_m)COv(o't, ~-) q- CoV(Sn, Sm)E2(o't)

M(s2)Var(%) + CNE( Crt2)M( s 2) + E2( crr)Var( s) '

(4)

where Var(s) ~ m(s 2) - M2(s), Cov(sn, s,,) = M(ss,,_,n) - M2(s) and C N

E-2IZt,,{- 1. Eq. (4) illustrates the interaction between the periodicity in absolute

returns at the intradaily level and the conditional heteroskedasticity at the daily

level. For adjacent trading days the impact of the positive correlation in the daily

return volatility, captured by Cov(~,, o-T), is strong and induces positive dependence

in the absolute returns, but as the distance between t and ~- grows this effect

becomes less important which is consistent with the slow decay in the correlograms

in the bottom panels of Fig. 4. At the same time, the correlograms are

affected by the strong intraday periodicity. For example, consider the display for

the absolute S&P 500 returns in Fig. 4b. The correlations attain their lowest

values around lag forty, or half a trading day. This corresponds to the bottom of

the U-shape for the average absolute returns depicted in Fig. 2b. Clearly, the

population covariance, Cov(s n, Sm), is minimized and significantly negative, at

this frequency. Eq. (4) verifies that the negative correlation between the 5-minute

absolute returns, realized about half-a-day apart, translates into a negative contribution

to the corresponding correlogram at the 3-4 hour frequencies. Likewise,

Fig. 2a indicates that there is strong negative correlation between the absolute

foreign exchange returns in the intervals 80-225 (covering about half-a-day) and

all the remaining 5-minute returns. Not surprisingly, the lower panel of Fig. 4a

verifies that this again results in highly significant troughs in the correlogram

around the 12 hour frequency (and its harmonics). Indeed, the impact is now

sufficiently strong that the absolute return autocorrelations turn negative. This is

truly remarkable given the very large positive autocorrelations found at the daily

frequency and it is testimony to the profound impact of the periodic structure on

the intraday return dynamics. In terms of the specification in Eq. (4), the size of

the second, negative, term of the numerator exceeds the first, positive, term around

the 12 hour frequency.

The pronounced systematic fluctuations in the return correlogram provide an

initial indication that direct ARCH modeling of the intraday return volatility would

be hazardous. Standard ARCH models imply a geometric decay in the return

autocorrelation structure and simply cannot accommodate strong regular cyclical

patterns of the sort displayed in Fig. 4. Instead, it seems intuitively clear that the

combination of recurring cycles at the daily frequency and a slow decay in the

average autocorrelations may be explained by the joint presence of the pronounced

intraday periodicity documented above coupled with the strong daily conditional

heteroskedasticity 20. The following stylized model provides a simple specification

of the interaction between these two components,

N 1 N

R,--- E R,,n = E soZ,,n. (1)

n=l n=l

Here, R t denotes the daily continuously compounded return calculated from the N

uncorrelated intraday return components, Rt, .. The conditional volatility factor for

day t is denoted by ~r t, while s n refers to a deterministic intraday periodic

component and Z,, n is an i.i.d, mean zero, unit variance error term assumed to be

independent of the daily volatility process, {~rt}. Both volatility components must

be non-negative i.e. % > 0 a.s. for all t and s, > 0 for all n. The following

terminology for the normalized, deterministic sample means and covariances for

the periodic structure will prove convenient:

1 N 1 N 1 N

n=~ =- M( s ) = 1, -~ E SnSn_i -= M( ssi), -~ E S2 =- M( s2),

=1 n=l n=l

where s,,+j N =- s n for any integer j and 0 < n < N.

In the absence of intraday periodicity (s, = 1 for all n) the daily returns may be

represented in the form R t = o',N 1/2Y',,=1,NZt, ., where the return component

N-1/2F,,= LNZ,,n is i.i.d, with mean zero and unit variance. Thus, Eq. (1) extends

the standard volatility model for daily returns to an intraday setting with independent

return innovations and deterministic volatility patterns. Of course, this type of

periodicity is annihilated when the returns are measured at the daily frequency. In

particular, letting Z, denote an i.i.d, random variable with E(Z t) = 0 and Var(Z t)

= 1, we have

R t = MI/2( S 2 ) o',Z,, (2)

so that the expected absolute return equals M1/2(s2)~tEIZ, I. Since M1/2($2) > 1,

the expected daily absolute return is an increasing function of the fluctuations in

the intraday periodic pattern. However, this effect is limited to a scale factor.

Letting c = (EIZ, I) -2 - 1 > 0, it follows that for t 4= ~',

Cov( o-,, o;_)

Corr(lRtl, ]RTI) = Var(o.t) + cE(~rt2) " (3)

Hence, the presence of periodic components reduces the overall level of the

interdaily return autocorrelations, without affecting the autocorrelation pattern 21

In contrast, the periodicity may have a strong impact on the autocorrelation pattern

for the absolute intraday returns. Straightforward calculations reveal

Corr(IR,,,,I, IR .... I)

m(ssn_m)COv(o't, ~-) q- CoV(Sn, Sm)E2(o't)

M(s2)Var(%) + CNE( Crt2)M( s 2) + E2( crr)Var( s) '

(4)

where Var(s) ~ m(s 2) - M2(s), Cov(sn, s,,) = M(ss,,_,n) - M2(s) and C N

E-2IZt,,{- 1. Eq. (4) illustrates the interaction between the periodicity in absolute

returns at the intradaily level and the conditional heteroskedasticity at the daily

level. For adjacent trading days the impact of the positive correlation in the daily

return volatility, captured by Cov(~,, o-T), is strong and induces positive dependence

in the absolute returns, but as the distance between t and ~- grows this effect

becomes less important which is consistent with the slow decay in the correlograms

in the bottom panels of Fig. 4. At the same time, the correlograms are

affected by the strong intraday periodicity. For example, consider the display for

the absolute S&P 500 returns in Fig. 4b. The correlations attain their lowest

values around lag forty, or half a trading day. This corresponds to the bottom of

the U-shape for the average absolute returns depicted in Fig. 2b. Clearly, the

population covariance, Cov(s n, Sm), is minimized and significantly negative, at

this frequency. Eq. (4) verifies that the negative correlation between the 5-minute

absolute returns, realized about half-a-day apart, translates into a negative contribution

to the corresponding correlogram at the 3-4 hour frequencies. Likewise,

Fig. 2a indicates that there is strong negative correlation between the absolute

foreign exchange returns in the intervals 80-225 (covering about half-a-day) and

all the remaining 5-minute returns. Not surprisingly, the lower panel of Fig. 4a

verifies that this again results in highly significant troughs in the correlogram

around the 12 hour frequency (and its harmonics). Indeed, the impact is now

sufficiently strong that the absolute return autocorrelations turn negative. This is

truly remarkable given the very large positive autocorrelations found at the daily

frequency and it is testimony to the profound impact of the periodic structure on

the intraday return dynamics. In terms of the specification in Eq. (4), the size of

the second, negative, term of the numerator exceeds the first, positive, term around

the 12 hour frequency.