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Методички
through December 31, 1989. The data specify the time, to the nearest 10 seconds and the exact price of the S&P 500 futures transaction whenever the price differs from the previously recorded price 46. The calculation of the returns is based on the last recorded logarithmic prices for the nearby futures contract over consecutive five minute intervals. The price record covers the full trading day in the futures market from 8.30 a.m. (central standard time) to 3.15 p.m. Although, the New York Stock Exchange closes at 3.00 p.m., we retain the last three 5-minute returns from the futures market in the analysis reported on below. The first return for the trading day, i.e. from 8:30 to 8:35 a.m., constitutes another unusual time interval. This period incorporates adjustments to the information accumulated overnight, and consequently displays a much higher average return variability than any other 5-minute interval. In effect, this is not a 5-minute return, and we therefore delete it in the subsequent analysis. Alternatively, it would be possible to account for this special return interval using dummy variables. However, any such procedure is invariably ad hoc in nature. Furthermore, informal investigations reveal little sensitivity to the exact treatment of the overnight returns. We thus elect to work exclusively with the 5-minute returns. Following Chan et al. (1991), we also exclude the October 15 through November 13, 1987 time period around the stock market crash due to the frequent trading suspensions. Outside these four weeks trading suspensions were rare, but did occur. In these instances the missing prices were determined by linear interpolation, leading to identical returns over each of the intermediate intervals. This obviously smoothes the series over the missing data points which will mitigate the effect of sharp price changes subsequent to a trading suspension. Experimentation with exclusion of trading days with missing observations indicate that the findings pertaining to the degree of volatility persistence reported on here are virtually unaffected by this interpolation. All in all, these corrections result in a sample of 991 days, each consisting of 80 intraday 5-minute returns, for a total of 79,280 observations i.e. Rtn, n = 1, 2, ...,80, t = 1, 2, 991. Appendix B. Flexible Fourier form modeling of intraday periodic volatility components From Eq. (7), define, * = 2\og[\R,n - E(R, )\] - log a,1 + log N = log jf, + log Z2 . (A.l) 46 We are grateful to G. Andrew Karolyi for providing us with this 5-minute price series. The same set of data has also been analyzed from a different perspective in Chan et al. (1991). Our modeling approach is then based on a non-linear regression in the intraday time interval, n, and the daily volatility factor, cr x =/(0;cr )+ , (A.2) where the error, u, = log Z2n - £(log Z2 ), is i.i.d. mean zero. In the actual implementation the non-linear regression function is approximated by the following parametric expression, -2 D 0J + j-r 2j-+bKjJn . рп2тг рп2тг + Ц I Уп, cos ----h S i sin N PJ N (A.3) where Nt = N~1 £,= ]Ni = (N + l)/2 and N2 = N1 £,= UNf = (N + 1)(7V + 2)/6 are normalizing constants. For / = 0 and D = 0, Eq. (A.3) reduces to the standard flexible Fourier functional form proposed by Gallant (1981, 1982). Allowing for J > 1 and thus a possible interaction effect between a/ and the shape of the periodic pattern might be important in some markets, however. Each of the corresponding J flexible Fourier forms are parameterized by a quadratic component (terms with /-coefficients) and a number of sinusoids (the y- and -coefficients). Moreover, it may be advantageous to also include time specific dummies for applications in which some intraday intervals do not fit well within the overall regular periodic pattern (the A-coefficients). Practical estimation is most easily accomplished using a two-step procedure. Firstly, a generated x, n series, x, , is obtained by replacing E(R, ) with the sample mean of the returns, Rt and a, with the estimates from a daily volatility model, say &r Substituting <r, for cr, and treating xt n as the dependent variable in the regression defined by Eqs. (A.2) and (A.3) allow the parameters to be estimated by ordinary least squares (OLS). Note that from Eq. (3), 5,2 represents an estimate of М(л2)<тД so that after substitution for cr, in Eq. (A.2), the term - log M(s2) is implicitly included in the constant term in Eq. (А.З), /лт. Let /, =/(§;& n) denote the resulting estimate for the right hand side of Eq. (A.3) 47. The normalization TE = ,i,[T/N]st,n = Ь where [T/N] denotes the number of trading days in the sample, then suggests the following estimator of the intraday periodic component for interval n on day t, E]E,exp(/;, /2)- Note that although the periodic modeling procedure is designed for fitting the Given consistent estimates for <x the resulting parameter estimates will generally be consistent. However, the use of generated regressors may result in a downward bias in the conventional OLS standard errors for the parameter estimates (see Pagan, 1984). average volatility pattern across the N intraday intervals, the second-stage estimation of Eq. (A.3) is based on a time series regression that include all T intraday returns. Utilizing this additional information in the data rather than simply fitting the average intraday pattern, enhances the efficiency of the estimation. The first step of our procedure involves the determination of the daily volatility factor estimates i.e. &t. Given the relative success of the daily MA(1)-GARCH(1, 1) models in explaining the aggregation results for the interdaily frequencies in both markets, this appears to be a natural choice. Next, the number of interaction terms, J and the truncation lag for the Fourier expansion, P, must be determined. This is done primarily on the basis of parsimony i.e. for each of the return series we choose the model that best matches the basic shape of the periodic pattern with the fewest number of parameters. The resulting estimates for the DM-$ returns with J = 0 and P = 6 are, ft, n 0.72 (1.06) 27ГП - 2.51 cos- (-6.15) N 2ттЪп + 0.42 cos- (8.79) N lirbn - 0.12 cos- (-5.38) N (-4.14) 2ттп - 0.40 sin- (-10.44) N 2-7тЗи - 0.09 sin- (-4.89) N 0.22 2тг5 (13.35) - sin - + 5.59 - (4.14) W2 2-jr2n - 0.38 cos- (-3.71) N 2тт4п - 0.02 cos- (-0.53) N 2ттЬп - 0.23 cos- (-12.67) TV 27г2и 4-0.06 sin- (2.70) N 27г4л + 0.35 sin- (20.48) N 2n6n + 0.01 sin- (0.45) N where the numbers in parentheses indicate heteroskedastic robust f-statistics. It is evident from the corresponding fit in Fig. 6a, that this representation provides an excellent overall characterization of the average intradaily periodicity in the DM-$ market. Consistent with Fig. 2a, the basic shape of the periodic pattern appears invariant to the daily volatility level i.e. 7 = 0. In contrast, our preferred model, the S&P 500, returns sets J = 1 and P = 2, fi,n~ - 1.85 (-3.03) - 0.16 / (-0.53) + 1.18 cos - (З.П) Л -3.07 - (1.62)/V, - °-62 h-(- 1.83) - 0.59 sin-(-6.14) 2ттп - 2.68 - (-2.05) TV, (2.99) 2ir2n + 0.28 cos- (2.94) N - 0.14 sin- (-2.31) 2тт2п -0.54 (0.95) - 0.11 / (-0.39) - 0.37 cos- (- 1.06) 2ттп - 1.73 - (-0.98) N\ - 0.30 / (-0.97) -0.12sin- (1.30) + 1.57 - (1.29) №> - 0.69 / , (-2.02) - 0.17 cos- (- 1.97) 2тт2п 2тт2п - 0.03 sin- (-0.50) TV Although few of the coefficients in the expansion corresponding to 7=1 are individually significant, leaving out the interaction effect results in a seemingly 1 2 3 4 5 6 7 8 9 10 11 12 [ 13 ] 14 15 |