back start next
[start] [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66] [67] [68] [69] [70] [71] [72] [73] [74] [75] [ 76 ] [77] [78] [79] [80] [81] [82] [83] [84] [85] [86] [87] [88] [89] [90] [91] [92] [93] [94] [95] [96] [97] [98] [99] [100] [101] [102] [103] [104] [105] [106] [107] [108] [109] [110] [111] [112] [113] [114] [115] [116] [117] [118] [119] [120] [121] [122] [123] [124] [125] [126] [127] [128] [129] [130] [131] [132] [133] [134]
76 Figure 7.5 does not exhibit strong maxima for each full business day. Yet the remaining small maxima indicate a certain "heat wave" component. 7.3.2 Short and Long Memory The autocorrelation function of volatility decays at a hyperbolic rate rather than an exponential rate. In studies based on daily FX prices (e.g., Taylor, 1986) or weekly FX prices (e.g., Diebold, 1988), the number of observations is usually too small for outright rejection of either a hyperbolic or an exponential decay of the autocorrelation functions. In studies with longer daily series such as Ding et al. (1993), evidence of long memory is found with the S&P 500 from January 1928 to August 1991 (17,055 observations). To illustrate the presence of the long memory, two curves, one hyperbolic and one exponential, are drawn in Figure 7.5 together with the empirical autocorrelation functions. The hyperbolic curve approximates the autocorrelation function much more closely than the exponential curve. This behavior of volatility is similar to the, fractional noise process of Mandelbrot and Van Ness (1968) and Mandelbrot (1972), which exhibits hyperbolic decay in the autocorrelation function and thus the long memory serial dependence. The hyperbolic ( ,) and exponential (fe) functions used in the analysis above have the following form: where the parameters are k, h, and . determines the lag order of the autocorrelation function. The exponential function cannot simultaneously capture the short and longterm persistence, whereas the hyperbolic function is able to capture both successfully. For the hyperbolic function, values vary from 0.2 to 0.3 depending on the FX rate, whereas h is remarkably stable around 0.28 for all the rates. In Figure 7.4 and the first panel of Figure 7.5, the number of lags are limited to 720 intervals (i.e., 10 days) at the 20min data frequency. In the second panel of Figure 7.5, the number of lags are extended to 4320 (i.e., 60 days) in #scale. The decay in the volatility autocorrelations is more rapid after 10 days. This type of pattern is not specific to USDDEM, but is also found in longer time intervals and other FX rates. To explore this behavior further, we compute the autocorrelation function of daily returns (business days) for up to 200 lags and a sample of 20 years. The result is presented in Figure 7.6 and indicates the persistence of the hyperbolic behavior even at the daily frequency. A process that exhibits a hyperbolic decay in its autocorrelation function is the "fractional noise" of Mandelbrot and Van Ness (1968), which is a purely selfsimilar fractal. We test the empirical significance for this theoretical process. In Mandelbrot (1972), the autocorrelation function of fractional noise processes is given by Mr) = kx~\ and fe(r) = ke  / (7.6) / + 12 2/2 + /1 (7.7)
0.2 Hi o.o  1 1 r~...... i 1 1 0 50 100 150 200 Lag in Working Days FIGURE 7.6 Autocorrelation function of the absolute business day volatilities in the !?time scale. The data are for the USDDEM rate from June I, l973,toJune, I, I993. The hyperbolic (solid curve) and the exponential functions (dotted curve) are superimposed on the empirical autocorrelation function. The 95% confidence intervals are for an identically and independently distributed Gaussian process. where / is the lag parameter and H the Hurst exponent, which lies between 0.5 and 1 for "persistent" fractional noise. For a large number of lags (/), the autocorrelation function converges to a % H (2 H  1) /2("" (7.8) which has a hyperbolic decay. The autocorrelations of absolute returns in Figures 7.5 and 7.6 also follow a hyperbolic decline. The exponent 2(H  1) of Equation 7.8 from the USDDEM volatilities is H = 0.87 in Figure 7.5 and H = 0.86 in Figure 7.6. From the H values, the factor H(2H  1) leads to 0.64 and 0.62, respectively. These values are empirically found to be much lower, which are 0.25 and 0.20, respectively. This indicates that volatility does not follow a pure fractional noise process. Volatility is positive definite and has a skewed and fattailed distribution, whereas the distribution function of pure fractional noise is Gaussian.
In Peters (1989, 1991), the existence of fractional noise in the returns rather than volatility has been investigated similar to Equation 5.10. These findings claim that a drift exponent different from 0.5 necessarily indicates fractional noise. This conclusion holds only if the distribution forms are stable, but Figure 5.6 does not support this claim. We, therefore, conclude that the return process does not support the fractional noise hypothesis. Unlike volatility, the returns themselves exhibit no significant autocorrelation (see the thin curves in Figures 7.4 and 7.5). 7.4 the heterogeneous market hypothesis In the earlier sections, we analyzed the presence of two stylized facts. Namely, a hyperbolic decay of the volatility autocorrelations and the "heat wave" effect. Volatility characterizes the market behavior more deeply than just indicating the size of current or recent price movements. It is the visible "footprint" of less observable variables such as market presence and also market volume (for which information is hardly available in FX markets). The fact is that, contrary to traditional beliefs, volatility is found to be positively correlated to market presence, activity, and volume. Karpoff (1987), Baillie and Bollerslev (1989), and Muller et al. (1990), emphasize the key role of volatility for understanding market structures. The serial correlation studies of LeBaron (1992b,c) show that subsequent returns are correlated in lowvolatility periods and slightly anticorrelated in highvolatility periods. In continuous samples mixed from both lowvolatility and highvolatility periods, this effect indicates that the forecastability of return is conditional to volatility. Thus, volatility is also an indicator for the persistence of trends. These properties of volatility lead us to the hypothesis of a heterogeneous market, as opposed to the assumption of a homogeneous market where all participants interpret news and react to news in the same way. The heterogeneous market hypothesis is characterized by the following: 1. Different agents of the heterogeneous market have different time horizons and dealing frequencies. On the side of high dealing frequencies, there are the FX dealers and market makers (who usually have to close all their open positions before the evening); on the side of low dealing frequencies, there are the central banks, commercial organizations, and, for example, the pension fund investors with their currency hedging. The different dealing frequencies clearly mean different reactions to the same news in the same market. The market is heterogeneous with a "fractal" structure of the participants time horizons as it consists of shortterm, mediumterm, and longterm components. Each such component has its own reaction time to news, related to its time horizon and characteristic dealing frequency. If we assume the memory of volatility of one component to be exponentially declining with a certain time constant, as in a GARCH( 1,1) process, the memory of the whole market is composed of many such exponential declines with different time constants. The superposition of many
[start] [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66] [67] [68] [69] [70] [71] [72] [73] [74] [75] [ 76 ] [77] [78] [79] [80] [81] [82] [83] [84] [85] [86] [87] [88] [89] [90] [91] [92] [93] [94] [95] [96] [97] [98] [99] [100] [101] [102] [103] [104] [105] [106] [107] [108] [109] [110] [111] [112] [113] [114] [115] [116] [117] [118] [119] [120] [121] [122] [123] [124] [125] [126] [127] [128] [129] [130] [131] [132] [133] [134]
