4.2. Specification of the uolatili~ model and the associated persistence measures

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Numerous recent studies have relied on more formal time series techniques in

the analysis of high frequency return dynamics both within and across different

markets. The most commonly employed formulation is the GARCH(1, 1) model

proposed independently by Bollerslev (1986) and Taylor (1986). Thus, in order to

evaluate the potential impact of the strong intraday periodicity in this context we

Notes to Table l:

(a) The percentage returns are based on interpolated 5-minute logarithmic average bid-ask quotes for

the Deutschemark-U.S. dollar spot exchange rate from October 1, 1992 through September 29, 1993.

Quotes from Friday 21.00 Greenwich mean time (GMT) through Sunday 21.00 GMT have been

excluded, resulting in a total of 74,880 return observations. The length of the different intraday return

sampling intervals equals 5-k minutes. Each time series has a total of T/k non-overlapping return

observations. The sample means have been multiplied by one hundred. The columns indicated by Pl

and pA give the first order autocorrelations for the returns and the absolute returns. The Ljung and Box

(1978) portmanteau test for up to tenth order serial correlation in the returns and the absolute returns

are denoted by Q(10) and QA(I0), respectively. The variance ratio's for the different sampling

frequencies versus the daily return variance are denoted by VR. The corresponding variance ratio

statistics for the absolute returns are given in the VR A column.

(b) The returns are based on 79,280 interpolated 5-minute futures transactions prices for the Standard

and Poor's 500 composite index. The sample period ranges from January 2, 1986 through December

31, 1989, excluding the period from October 15, 1987 through November 13, 1987. Overnight five

minute returns have also been deleted, resulting in a total of 80 intraday return observations from 08.35

through 15.15 for each of the 991 days in the sample. The length of the different intraday return

sampling intervals equals 5.k minutes. Each time series has a total of T/k non-overlapping return

observations. The sample means have been multiplied by one hundred. The columns indicated by p~

and pA give the first order autocorrelations for the returns and the absolute returns. The Ljung and Box

(1978) portmanteau test for up to tenth order serial correlation in the returns and the absolute returns

are denoted by Q(10) and QA(10), respectively. The variance ratio's for the different sampling

frequencies versus the daily return variance are denoted by VR. The corresponding variance ratio

statistics for the absolute returns are given in the VR A column.

2s The negative skewness may be interpreted as evidence of the so-called 'leverage' and/or

'volatility feed-back' effects discussed by Black (1976), Christie (1982) and Nelson (1991), and

Campbell and Hentschel (1992), respectively.

present MA(1)-GARCH(1, 1) estimation results for each of the intradaily sampling

frequencies in Table 2a and b. Formally, the model is defined by

R~,~ = tx(h ) + O(k)etk,,_ 1 + ekt , n~

and

where E,,~_ l(e,~..)= 0 and Et, ~_ l[(e~.) 2 ] = (o-~k.) 2 denotes the conditional return

variance over the subsequent intraday period, with the subscript (t, 0) defined to

equal (t - 1, K). The reported parameter estimates for a(k ) and fl(k) are obtained

Table 2

k T/k o~(k ~ tick) a~k) + tick) Half life Mean lag Median lag

(a) Persistence of MA(1)-GARCH(I, 1) models for intraday DM-$ exchange rate

k _

Rt,n = 100"~i=(n 1)k+l,nkRt,i = I't'(k) + OIk)Etk, n I

+ , k (o.,~o)2 = ,o~k~ +a(k~(et.k, ,_ j) 2 +fl~k)(o'tk,n _l ) 2 t=l , 2 , . . . , 2 6 0 , n=1 , 2 , . . . , 2 8 8 / k

1 74,880 0.193 (0.0l 1) 0.822 (0.009) 1.015 ~ ~ w

2 37,440 0.229 (0.012) 0.774 (0.008) 1.003 ~ co

3 24,960 0.273 (0.018) 0.708 (0.014) 0.981 533 725 488

4 18,720 0.287 (0.019) 0.677 (0.016) 0.964 375 488 320

6 12,480 0.322 (0.035) 0.579 (0.033) 0.901 200 233 138

8 9,360 0.286 (0.028) 0.581 (0.037) 0.868 195 207 108

9 8,320 0.306 (0.035) 0.521 (0.042) 0.828 165 167 81

12 6,240 0.311 (0.047) 0.395 (0.069) 0.706 119 105 35

16 4,680 0.261 (0.039) 0.456 (0.074) 0.718 167 136 < 40

18 4,160 0.270 (0.061) 0.246 (0.124) 0.516 94 67 < 45

24 3,120 0.018 (0.015) 0.969 (0.026) 0.988 6,771 5,919 1,878

32 2,340 0.016 (0.008) 0.975 (0.013) 0.991 12,159 11,219 4,318

36 2,080 0.011 (0.004) 0.978 (0.005) 0.989 11,311 8,293 266

48 1,560 0.011 (0.004) 0.979 (0.005) 0.990 17,084 13,229 1,748

72 1,040 0.007 (0.005) 0.987 (0.004) 0.987 19,585 10,153 < 180

96 780 0.014 (0.008) 0.969 (0.007) 0.983 19,637 13,202 < 240

144 520 0.010 (0.010) 0.960 (0.007) 0.970 16,329 5,988 < 360

(b) Persistence of MA(1)-GARCH(I, 1) models for intraday S&P 500 returns

k __ k

Rt,n = 100"~i=(n I)k+ 1,nkRt,i = t~(k) 4- O(k)~'t,n 1

4-e,k. , (°'tk, n)2- -w(k)+°l(k)(et.nk 1) 2 4- /3~k~(o'kt ,,_l)2, t=l , 2 . . . . . 991, n=1 , 2 . . . . . 80/ k

1 79,280 0.137 (0.004) 0.838 (0.005) 0.975 137 168 105

2 39,640 0.180 (0.010) 0.765 (0.011) 0.945 121 138 79

4 19,820 0.223 (0.024) 0.664 (0.036) 0.887 116 118 57

5 15,856 0.230(0.067) 0.630(0.123) 0.861 116 112 49

8 9,910 0.053 (0.027) 0.935 (0.036) 0.988 2,213 2,602 1,559

10 7,928 0.048 (0.018) 0.940 (0.023) 0.988 2,947 3,437 2,043

16 4,955 0.148 (0.333) 0.764 (0.694) 0.912 606 575 240

20 3,964 0.060 (0.049) 0.890 (0.092) 0.951 1,376 1,124 246

40 1,982 0.108 (0.158) 0.798 (0.315) 0.906 1,397 1,128 228

by quasi-maximum likelihood methods assuming the innovations to be conditionally

normally distributed. The corresponding robust standard errors for the estimates

are provided in parentheses (see Bollerslev and Wooldridge, 1992). We note

that, although it usually represents a reasonable approximation, the GARCH(1, 1)

model is not necessarily the preferred specification for the return generating

process in all, or even most, instances. However, estimating the same model across

both asset classes and all return frequencies facilitates meaningful comparisons of

the findings. Moreover, it corresponds to the class of models for which theoretical

aggregation results are available. The MA(1) term is included to account for the

economically minor, but occasionally highly statistically significant, first order

autocorrelation in the returns.

Unfortunately, an unambiguous characterization of the estimated volatility

dynamics and the associated persistence properties is not possible in this non-linear

setting (see Bollerslev and Engle (1993), Bollerslev et al. (1994) and Gallant et

al. (1993) for further discussion of these issues). Hence, we supplement the

parameter estimates for a(k ) and /3~k ) in Table 2a and b with three additional

summary measures for the implied degree of volatility persistence. In particular, if

a(~) +/3(k ) < 1, the j-step ahead prediction for the conditional variance may be

written as

k 2 j ~ 2

where 0 -2 - o~(k)(1 - a(k >-/3(~) -~ equals the unconditional variance of the

Notes to Table 2:

(a) The returns are based on 288 interpolated five minute logarithmic average bid-ask quotes for the

Deutschemark-U.S. dollar spot exchange rate from October 1, 1992 through September 29, 1993.

Quotes from Friday 21.00 Greenwich mean time (GMT) through Sunday 21.00 GMT have been

excluded, resulting in a total of 74,880 return observations. The length of the different intraday retum

sampling intervals equal 5. k minutes. The model estimates are based on T~ k non-overlapping return

observations. The ot(k ) and /3~k ) columns give the Gaussian quasi-maximum likelihood estimates for

the GARCH(1, 1) parameters. Robust standard errors are reported in parentheses. The half life of a

shock to the conditional variance at frequency k is calculated as -log(2)/log(a~k)+/3tk )) and

converted into minutes. The mean lag of a shock to the conditional variance is given by a(k ) +/3ck ) > 1.

The median lag of a shock to the conditional variance is calculated by ½ +[log(l -/3(k~)- 1og)a(k))--

log(2)]/1og(a(k ) + /3(k )) and reported in number of minutes. For 2a~k ) < 1 -/3(k ) the median lag is

less than ½. The median lag is also not defined for a(t) +/3(k ~ > 1.

(b) The returns are based on 79,280 interpolated five minute futures transactions prices for the Standard

and Poor's 500 composite index. The sample period ranges from January 2, 1986 through December

31, 1989, excluding the period from October 15, 1987 through November 13, 1987. Overnight five

minute returns have also been deleted, resulting in a total of 80 intraday return observations from 08.35

through 15.15 for each of the 991 days in the sample. The length of the different intraday return

sampling intervals equal 5. k minutes. The model estimates are based on T/k non-overlapping return

observations. The ~(k) and /3(k ) columns give the Gaussian quasi-maximum likelihood estimates for

the GARCH(I, 1) parameters. See (a) for the definition of the half life, mean lag and median lag

statistics.

return innovations. The 'half-life' of the volatility process is then defined as the

number of time periods it takes for half of the expected reversion back towards or 2

to occur i.e. -log(2).log(a(k ) +/3(k )) i. Alternatively, by defining the conditional

heteroskedastic squared return innovations, v ~ , , - ( ~ n ) 2 - ( ~ , ) 2, the

GARCH(1 1) model may be expressed as an infinite MA model for (E k )2 with , t,n

positive coefficients, 0~ k,

E k = 0"2"~ o/.(k)E(og(k)q-[~(k)) i lp k q_ 1) k ~ 0-2--}- Eokp k

t,n t,n- i t,n t t,n t "

i=1 i=0

This specification suggests the corresponding 'mean lag', a(~)(1 - o~(~- 2/3~k ) +

oe(k )/3(k ) +/3(k)) i and 'median lag', ½ + [log(1 -/3(k ~) - log(oe(k )) - log(2)] •

log(a(k ) +/3(k~) -1 , as additional measures for characterizing the degree of volatility

persistence and the duration of the dynamic adjustment process in squared

returns across the different sampling frequencies 29 Neither the mean nor the

median lag is defined for ce(k ) +/3~k ~ > 1. Also, the median lag is less than 1/2 for

2a(~) +/3(k ) < 1.