5. The dynamics of filtered and standardized intraday returns

This section proposes a general framework for modeling of high frequency

return volatility that explicitly incorporates the effect of the intraday periodicity.

The preceding section suggests that this is a prerequisite for meaningful time

series analysis. Our approach is motivated by the stylized model in Section 3.

While the model almost certainly is overly simplistic, the previous analysis

suggests that the representation does capture the dominant features of our foreign

exchange and equity return series and thus may serve as a reasonable first

approximation.

Specifically, consider the following decomposition for the intraday returns,

o"l st,rtZt, n

R,. n=E(Rt,n) + U,/2 , (7)

where E(Rt, n) denotes the unconditional mean, and N refers to the number of

return intervals per day. Notice that this represents a generalization of the model in

Section 3, in that the periodic component for the nth intraday interval, st,,,, is

allowed to depend on the characteristics of trading day, t 35. Given the absence of

any economic theory for stipulating a particular parametric form for the intraday

periodic structure, a flexible nonparametric procedure seems natural. Although no

one procedure is clearly superior, the smooth cyclical patterns documented in Fig.

2a and b naturally lend themselves to estimation by the Fourier flexible functional

form introduced by Gallant, 1981, 1982 36. In a related context, Dacorogna et al.

(1993) have proposed estimating the periodicity in the activity in the foreign

exchange market as the sum of three polynomials corresponding to the distinct

geographical locations of the market. Returns measured on their resulting theta-time

scale correspond closely to our filtered returns defined and analyzed below.

However, one advantage of the approach advocated here is that it allows the shape

of the periodic pattern in the market to also depend on the overall level of the

volatility; a feature which turns out to be important for the equity market. Also,

the combination of trigonometric functions and polynomial terms are likely to

result in better approximation properties when estimating regularly recurring

patterns. Furthermore, our approach for estimating st. n utilizes the full time series

dimension of the returns data, as opposed to simply estimating the average pattern

across the trading day. Full details of the approach are provided in Appendix B.

Meanwhile, it is clear from the estimated average intraday periodic patterns

depicted in Fig. 6a and b, that the fitted values, ~t.n, provide a close approximation

to the overall volatility patterns in both markets. Of course, the usefulness of the

procedure will ultimately depend upon the degree to which it is successful in

identifying the periodic components in a temporal dimension as well. If so, the

approach may serve as the basis for a nonlinear filtering procedure that could

eliminate the periodic components prior to the analysis of any intraday return

volatility dynamics 37

This section proposes a general framework for modeling of high frequency

return volatility that explicitly incorporates the effect of the intraday periodicity.

The preceding section suggests that this is a prerequisite for meaningful time

series analysis. Our approach is motivated by the stylized model in Section 3.

While the model almost certainly is overly simplistic, the previous analysis

suggests that the representation does capture the dominant features of our foreign

exchange and equity return series and thus may serve as a reasonable first

approximation.

Specifically, consider the following decomposition for the intraday returns,

o"l st,rtZt, n

R,. n=E(Rt,n) + U,/2 , (7)

where E(Rt, n) denotes the unconditional mean, and N refers to the number of

return intervals per day. Notice that this represents a generalization of the model in

Section 3, in that the periodic component for the nth intraday interval, st,,,, is

allowed to depend on the characteristics of trading day, t 35. Given the absence of

any economic theory for stipulating a particular parametric form for the intraday

periodic structure, a flexible nonparametric procedure seems natural. Although no

one procedure is clearly superior, the smooth cyclical patterns documented in Fig.

2a and b naturally lend themselves to estimation by the Fourier flexible functional

form introduced by Gallant, 1981, 1982 36. In a related context, Dacorogna et al.

(1993) have proposed estimating the periodicity in the activity in the foreign

exchange market as the sum of three polynomials corresponding to the distinct

geographical locations of the market. Returns measured on their resulting theta-time

scale correspond closely to our filtered returns defined and analyzed below.

However, one advantage of the approach advocated here is that it allows the shape

of the periodic pattern in the market to also depend on the overall level of the

volatility; a feature which turns out to be important for the equity market. Also,

the combination of trigonometric functions and polynomial terms are likely to

result in better approximation properties when estimating regularly recurring

patterns. Furthermore, our approach for estimating st. n utilizes the full time series

dimension of the returns data, as opposed to simply estimating the average pattern

across the trading day. Full details of the approach are provided in Appendix B.

Meanwhile, it is clear from the estimated average intraday periodic patterns

depicted in Fig. 6a and b, that the fitted values, ~t.n, provide a close approximation

to the overall volatility patterns in both markets. Of course, the usefulness of the

procedure will ultimately depend upon the degree to which it is successful in

identifying the periodic components in a temporal dimension as well. If so, the

approach may serve as the basis for a nonlinear filtering procedure that could

eliminate the periodic components prior to the analysis of any intraday return

volatility dynamics 37