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] 59 arbitrary and depends on the model used. In the context of an efficient market with no arrival of information, Roll (1984) has assumed a similar bias.Now, we are ready to compute the expectation of rf from Equation 5.20, using Equations 5.21 and 5.22 and the independence of r* and e,,= E(V2) = E((r2>) = q*2 + n2(5.23)The squared observed returns are thus biased by the positive amount of rj2.Empirical measures of (r2) are not only biased but also contain a stochastic error, which is defined as the deviation of (r2) from its expectation q2. The variance of this stochastic error can be formulated(V) - Q2)1 = E ((r2)2 - 2 (r2> q2 + qa) (5.24)The last form of this equation has the expanded terms of the square. The first term, (r2)2, can be explicitly written by inserting Equations 5.12 and 5.20; the other two terms can be simplified by inserting Equation 5.23. We obtain£/-l)2(5.25)The first term is somewhat tedious to compute because of the two squares and the sum. We expand the squares to get many terms for which we have to compute the expectation values. All of those terms that contain r* or s to an odd power have a zero expectation due to the symmetry of the normal distribution and the independence of r* and s. The expectations of r*2 and e2 can be taken from Equations 5.21 and 5.22. The expectations of the fourth moments of normal distribution are = 3[E(rf2)]2 = 3(5.26)E(sf) = E (*< ,) = 3[E(e?)]2 = \n* "(5.27)as found in (Kendall etal., 1987, pp. 321 and 338), for example. By inserting this and carefully evaluating all the terms, we obtaina2 = -±1 + 2( +2) Q*2 n2 + »2 + \»+] ,4 4 (528)By inserting Equation 5.23, we can express the resulting stochastic error variance either in terms of q*,a2 =4 *9 7(5.29)or in terms of g,a2 = -64 + r,4 (5.30)n \n nA /Now, we know both the bias 2 of an empirically measured (r2) and the variance of its stochastic error. For reporting the results and using them in the scaling law computation, two alternative approaches are possible:1. We can subtract the bias rj2 from the observed {r2) and take the result with a stochastic error of a variance following Equation 5.30, approximating q2 by (r2). We do not recommend this here because rj1 is only approximately known and thus contains an unknown error. However, the idea of bias modeling and bias elimination is further developed in Chapter 7.2. We can take the originally obtained value of (r2) and regard the bias rj2 as a separate error component in addition to the stochastic error. This is an appropriate way to go, given the uncertainty of rj2.Following the second approach, we formulate a total error with variance CTt2tal, containing the bias and the stochastic error. The stochastic error is independent of the bias by definition, so the total error variance is the sum of the stochastic variance and the squared bias<,,;,„ = + = \e+ (. + i-J).r* (53.)This is the final, resulting variance of the total error of (r2>.For the application in the scaling law, we can use a good approximation for large values of n ~ > 1, which is reasonable even for small values of . By dropping higher order terms from Equation 5.31, we obtain"L. * -Q4 + I4 * ~ (r2)2 + I4 (5-32)In the last form, the theoretical constant q2 has been replaced by its estimator (r2), see Equation 5.23.The mean squared return with error can be formulated as follows:wi.r,error = ( 2) ± Jri4 + ~ ( 2)2 (5.33)V nwhere the second term is the standard deviation of the error according to Equation 5.32.The scaling law is usually formulated for ((r2))1/2 rather than (r2), as in Equation 5.10. Applying the law of error propagation, we obtainv witherror - \r / 31 d{r2) « { - (r2)\/2 . / I4 , {r2}The scaling law fitting is done in the linear form obtained for log((r2))/2 (see Equation 5.17). Again applying the law of error propagation, we obtainlo2(r2>/2 - lo2(r2>1/2 ± d°g/2 / ai + id> ¸\ /withcrror - 6\ / 31 d