Промышленный лизинг
Методички
n - 1, 2, ..., К where К = 288/& refers to the number of returns per day. Note that while the 5-minute return series consists of 74,880 observations, the hourly series contains only 6,240 observations and the 1 /2-day return series has a mere 520 observations. These differences should be kept in mind when interpreting the evidence. The standard deviations in Table la grow at a rate almost proportional to the square root of the sampling frequency. This is consistent with the 5-minute returns being approximately uncorrected, although there is a small, but highly significant, negative first order autocorrelation coefficients at the higher frequencies. As mentioned, the weak negative correlation may be the result of spread positioning by dealers causing mean reversion in the quote midpoints; an effect similar to a bid-ask bounce in transactions data 24. In line with this explanation, the p, coefficients generally turn insignificant at the 40-minute and lower frequencies. Further corroborating evidence along these lines is provided by the variance ratio statistics, VR=-*K , (6) where VarT(Rkn) and VarT(Ln=] kRkn) denote the sample variances for the intraday and daily returns, respectively. Expanding the daily variance estimate in the denominator demonstrates that a value of the VR-statistic below unity will result from positive autocorrelation between adjacent return components, while a statistic above one is indicative of predominantly negatively correlated intraday returns 25. Finally, it is worth noting from Table la, that the kurtosis of the DM-$ returns increases almost monotonically with the sampling frequency. The first order autocorrelations of the absolute returns, pA are, not surprisingly, all highly significant for the shorter intervals. However, beyond the 2-hour sampling frequency the autocorrelations drop off very sharply and in fact turn negative at the 8 and 12 hourly frequencies (£ = 96, 144). This is, of course, consistent with the negative region of the 5-minute absolute return correlogram in Fig. 4a. The VRA-statistic reported in the final column of Table la is calculated by replacing Rkn with \RkJ in the definition of VR in Eq. (6) 26. The statistic starts out at 0.05 for the 5-minute returns and rises almost monotonically to 0.69 for the Note that the standard deviation of the 5-minute returns is less than the average quoted bid-ask spread. According to Bollerslev and Melvin (1994), more than half of the DM-$ quotes are posted with a spread of 0.10%, while the second most common and lowest regularly posted spread of 0.05% accounts for about a quarter of the quotations. 25 Formal tests for serial correlation based on the VR-statistic may be calculated as outlined in Lo and MacKinlay (1989). 26 Note that the denominator in this VRA-statistic involves the variance of the sum of the absolute returns rather than the absolute value of the sum of the returns. The expected value of the VR-statistic would not equal unity under the latter definition. Table 1
(b) Summary statistics for intraday S&P 500 returns
12 hourly returns. The results for the multiple day returns reported in Andersen and Bollerslev (1994) continue this near monotone ascent, reaching 1.94 for the biweekly sampling interval. The smooth increase suggests that a common component accounts for a substantial part of the positive higher order dependence in all of the return series. The corresponding p statistics of 0.123 and 0.118 for the weekly and biweekly sampling frequencies also testify to the importance of the interday heteroskedasticity 27. 27 Hence, the VRA-statistics convey a coherent message about the degree of conditional heteroskedasticity in the series. As a set of simple diagnostics, these statistics may therefore be more informative about the nature of the volatility process than the standard Ljung-Box statistics for tenth order serial correlation in the absolute returns, QA(10). which appear both erratic and highly dependent on the sample size. The summary statistics for the S&P 500 index futures returns in Table lb largely parallel those for the DM-$ returns. However, in contrast to the results for the exchange rates, the first order autocorrelations and the VR-statistics in Table lb all indicate a slight positive intraday dependence. Moreover, the equity returns are negatively skewed and display very significant excess kurtosis 28. Finally, the intraday return periodicity, here depicted in Fig. 2b, again have a strong effect on the correlations for the absolute intraday returns, although the decay in the pA coefficients for the lower frequencies is less pronounced than for the exchange rates. 4.2. Specification of the volatility model and the associated persistence measures 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 1: (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-к 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 ratios 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 VRA column. (b) The returns are based on 79,280 interpolated 5-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 equals 5-к 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 ratios 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 VRA column. 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. 1 2 3 4 5 [ 6 ] 7 8 9 10 11 12 13 14 15 |