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Khi đăng nhập cá nhân hàng tháng, trả lại được phân phối AR (1) với các lỗi thông thường, các yếu tố tích lũy đăng nhập cũng được phát hành bình thường. Sử dụng mô hình AR (1) với các thông số, một, đăng nhậpNày giả định một vị trí bắt đầu trung lập cho quá trình này, | Analytic Calibration of Other Models 73 AR 1 When the individual monthly log-returns are distributed AR 1 with normal errors the log-accumulation factors are also normally distributed. Using the AR 1 model with parameters fl a 7 logSn N np a h a n 2 4-18 where h a n 1 T VẺ 1 - a 2 This assumes a neutral starting position for the process that is Yo fl so that the first value of the series is Y1 fry. To prove equation 4.18 it is simpler to work with the detrended process Zt Yt - fl so that Zt aZt-1 aet. logSn - n Z1 Z2 Zn 4-19 7S1 a 7S1 a.ei a a 7S1 a.ei 7 e 3 4-20 an 17S1 an 2772 aaen-1 7en 4-21 7 a 1 - an1 1 4-22 The et are independent identically distributed N 0 1 giving the result in equation 4.18 so it is possible to calculate probabilities analytically for the accumulation factors. Once again we use as one of the defining equations the mean one-year accumulation E S12 exp M a 2 2 1-1161 where 12 and 7 ah a 12 . Use as a second the 2.5 percentile for the one-year accumulation factor for -09233 and 7 0-18714 as before. Hence we might use 0-007694 as before. This also gives crh a 12 0-18714. It is possible to use one of the other quantiles in the table to solve for a and therefore for 7. However no combination of table values gives a value of a close to the MLE. A reasonable solution is to keep the MLE estimate of a which was 0.082 and solve for 7 0-05224. Checking the other quantiles shows that these parameters satisfy all nine calibration points as well as the mean and variance criteria. 74 THE LEFT-TAIL CALIBRATION METHOD TABLE 4.4 Distribution for R12. r Pr Ri2 r r Pr Ri2 r 0 0.011172 7 0.041055 1 0.007386 8 0.051291 2 0.010378 9 0.063082 3 0.014218 10 0.076379 4 0.019057 11 0.091925 5 0.025047 12 0.557573 6 0.032338 The RSLN Model The distribution function of the accumulation factor for the RSLN-2 model was derived in equation 2.30 in the section on RSLN in Chapter 2. In that section we showed how to derive a probability distribution for the total number of .