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present MA(l)-GARCH(l, 1) estimation results for each of the intradaily sampling frequencies in Table 2a and b. Formally, the model is defined by

Rt,n = + в(к)е1кп-\ + etk,n>

{o-tknf = + <*<*,( e,* -,)2 + /З(4)(ст-Д ,)\

where Et<n l(etkn) = 0 and Etn ,[(e* )2] = (сгД)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 j3,k) are obtained

Table 2

к T/k /3(lt) a(t)+/(*) Half life Mean lag Median lag

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

 = 100- E,. (л-\)k + \,nk Ri.i - м<*) + e(t)e,!n- e (v,kJl = V(k) + a(k)( t = 1, 2, , 260, n = 1, 2, , 288/jfc 74,880 0.193 (0.011) 0.822 (0.009) 1.015 37,440 0.229(0.012) 0.774 (0.008) 1.003 24,960 0.273 (0.018) 0.708 (0.014) 0.981 18,720 0.287 (0.019) 0.677 (0.016) 0.964 12,480 0.322 (0.035) 0.579 (0.033) 0.901 9,360 0.286 (0.028) 0.581 (0.037) 0.868 8,320 0.306 (0.035) 0.521 (0.042) 0.828 6,240 0.311 (0.047) 0.395 (0.069) 0.706 4,680 0.261 (0.039) 0.456 (0.074) 0.718 <40 4,160 0.270 (0.061) 0.246 (0.124) 0.516 <45 3,120 0.018 (0.015) 0.969 (0.026) 0.988 6,771 5,919 1,878 2,340 0.016(0.008) 0.975 (0.013) 0.991 12,159 11,219 4,318 2,080 0.011 (0.004) 0.978 (0.005) 0.989 11,311 8,293 1,560 0.011 (0.004) 0.979 (0.005) 0.990 17,084 13,229 1,748 1,040 0.007 (0.005) 0.987 (0.004) 0.987 19,585 10,153 < 180 0.014(0.008) 0.969 (0.007) 0.983 19,637 13,202 < 240 0.010(0.010) 0.960 (0.007) 0.970 16,329 5,988 <360

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

 100-E,-, a- !)*+ 1. nk Rt,i ,k ( t= 1, 2, ...,991, n = 1, 2, .. .,80/* 79,280 0.137 (0.004) 0.838 (0.005) 0.975 39,640 0.180 (0.010) 0.765(0.011) 0.945 19,820 0.223 (0.024) 0.664 (0.036) 0.887 15,856 0.230 (0.067) 0.630 (0.123) 0.861 9,910 0.053 (0.027) 0.935 (0.036) 0.988 2,213 2,602 1,559 7,928 0.048 (0.018) 0.940 (0.023) 0.988 2,947 3,437 2,043 4,955 0.148 (0.333) 0.764 (0.694) 0.912 3,964 0.060 (0.049) 0.890 (0.092) 0.951 1,376 1,124 1,982 0.108 (0.158) 0.798 (0.315) 0.906 1,397 1,128

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(i) and /3(t) in Table 2a and b with three additional summary measures for the implied degree of volatility persistence. In particular, if a(k) + P(k) < 1 tne J steP ahead prediction for the conditional variance may be written as

where cr2 = w(t)(l - a(Jt) - j3(il)) 1 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 return sampling intervals equal 5 к minutes. The model estimates are based on Т/ к non-overlapping return observations. The a(t) and Bm columns give the Gaussian quasi-maximum likelihood estimates for the GARCHO, 1) parameters. Robust standard errors are reported in parentheses. The half life of a shock to the conditional variance at frequency к is calculated as - log(2)/log(a(i) + /3(t)) and converted into minutes. The mean lag of a shock to the conditional variance is given by a((:) + /3(t) > 1. The median lag of a shock to the conditional variance is calculated by \ +[log(l - Bk)) - log)a(t)) - log(2)]/log(a(t) + /3(J)) and reported in number of minutes. For 2a(i) < 1 - /3((:) the median lag is less than j. The median lag is also not defined for a(t) + /3(t) > 1.

(b) The returns are based on 79,280 interpolated five minute futures transactions prices for the Standard and Poors 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 к minutes. The model estimates are based on T/k non-overlapping return observations. The a(t) and /8(t) columns give the Gaussian quasi-maximum likelihood estimates for the GARCHO, 1) parameters. See (a) for the definition of the half life, mean lag and median lag statistics.

E \{<+i)2

2 + Ы + У[ )2--2\

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 cr2 to occur i.e. - log(2) log(a(/t) + f3{k))~l. Alternatively, by defining the conditional heteroskedastic squared return innovations, vk - (ek )2 - (atkn)2, the GARCH(l, 1) model may be expressed as an infinite MA model for (еД,)2 with positive coefficients, вк,

GO CO

6* = a2 + a(k) £( <*) + pa))- 1 + vl = a2 + £

i = 1 i = 0

This specification suggests the corresponding mean lag, a(t)(l - a((l) - 2j3ik) + a(k) A*) + /W~ and median lag, ~ + [log(l - (3(k)) - log(a(t)) - log(2)] log(a(ll) + (3{k))~], 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 a(k) + ДА) > 1. Also, the median lag is less than 1 /2 for 2 (t) + 1-

4.3. Interpretation of the GARCH results for different return frequencies

This section summarizes the evidence from fitting standard GARCH models to the return series at different frequencies. Particular emphasis is placed on the type of distortions that may be induced by the strong periodic intraday patterns which are ignored in these models. There are a couple of indirect ways to gauge the effect. First, there are theoretical predictions about the relation between the parameters at various frequencies. If these are most obviously violated at the particular frequencies where the intraday periodicity are expected to assert the maximal impact, this is therefore consistent with the periodic pattern being a dominant source of misspecification for these models. Second, to the extent that the periodic pattern is a strictly deterministic intraday phenomenon as suggested in Section 3, the distortions should be absent from models estimated at daily or multiple-day frequencies. Consequently, if the theoretical aggregation results work satisfactorily at the multiple-day frequencies but break down intradaily then this is further evidence of a significant impact of the periodic pattern on the dynamic properties of the intraday volatility process. We also relate our findings to the prior estimates reported from intraday volatility modeling. The comparison shows that our results are fully consistent with the diverse set of estimates reported in the literature once we control for the different return frequencies employed in the studies. Finally, the explicit incorporation of the cyclical pattern in Section 5 verifies that most of the distortions attributable to the intraday volatility cycle may

29 The mean lag is given by E, 0=J6,\ whereas the median lag, m, is implicitly defined by E,-,o,me,* = l/2-Ei n,o=e* (see Harvey, 1981).

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