### Table 4: Parameters of the Continuous-Time Model

"... In PAGE 33: ...for assembling the various parts are Gamma random variables. Table4 contains the parameters of... ..."

### Table 3: Parameters of the Continuous-Time Model

2003

### Table 4: Numerical results of the Continuous-Time formulation with GAMS 2.50/Cplex 7.0.0 Size Continuous-time model

2002

"... In PAGE 12: ...P7.CP.CL for 10 instances. The results are summarized in Table4 . Final demands vary as combinations of 0s and 20s.... In PAGE 12: ...08 seconds. The numerical results in Table4 are very encouraging. We can conclude that by applying the new continuous-time model, with a small number of events, it is not difficult to get a feasible solution; the CPU time for getting an optimal solution is acceptable.... ..."

### Table 8. Continuous Time Models Estimated from Interest Rate Data: Quantiles of Unconditional Distributions

1995

"... In PAGE 28: ...Table8 about here ||||||||||||{ Table 8 provides insight as to what goes wrong. The Table shows quantiles of the uncon- ditional distributions of the observed three interest rates along with those implied by the estimated models.... In PAGE 28: ...||||||||||||{ Table8 provides insight as to what goes wrong. The Table shows quantiles of the uncon- ditional distributions of the observed three interest rates along with those implied by the estimated models.... In PAGE 28: ... Treating 1; 2; and 3 as three free parameters (YF-Power-Free), provides little improvement, suggesting that the separate exponents are not sharply estimated. Table8 indicates that the YF-Power speci cations do better than the basic Yield Factor models in terms of capturing the right skewness of the unconditional distribution of the interest rates. For exponents of 0.... ..."

### Table 7. Continuous Time Models Estimated from Interest Rate Data: Optimized Value of the Criterion for the Semiparametric ARCH Score.

1995

"... In PAGE 27: ... To compute m( ; ) : = 1 N N Xt=0 @ @ log[f(^ ytj^ yt?L; : : : ; ^ yt?1; )] we use an explicit order 2 weak as described in Section 4 above. For this work, time t is scaled so the interval [t; t + 1] is one week and n0 = 10; which implies the simulation step size is = 0:10: ||||||||||||{ Table7 about here ||||||||||||{ Table 7 summarizes the main results. As can seen be seen from the table, YF-Diagonal and YF-Premium models fare poorly as does the YF-General.... In PAGE 27: ... To compute m( ; ) : = 1 N N Xt=0 @ @ log[f(^ ytj^ yt?L; : : : ; ^ yt?1; )] we use an explicit order 2 weak as described in Section 4 above. For this work, time t is scaled so the interval [t; t + 1] is one week and n0 = 10; which implies the simulation step size is = 0:10: ||||||||||||{Table 7 about here ||||||||||||{ Table7 summarizes the main results. As can seen be seen from the table, YF-Diagonal and YF-Premium models fare poorly as does the YF-General.... In PAGE 28: ...50 for federal funds, but not Trea- sury Bills. The middle part of Table7 reports the concentrated objective function for YF- Power speci cations with the exponents restricted to a common value, 31 = 32 = 33 = ; = 0:60; 0:70; 0:80; 0:90: These speci cations come closer to tting the Semiparametric ARCH score, with the best t at = 0:70; though the model is still rejected at conventional signi cance levels. If the common-value restriction on the exponents is maintained but treated as a free parameter, then the objective function is quite at in the region 0.... In PAGE 28: ...reated as a free parameter, then the objective function is quite at in the region 0.70{0.75, with the point estimate being 0.706 and the objective value hardly improves (YF-Power- Equal in Table7 ). Treating 1; 2; and 3 as three free parameters (YF-Power-Free), provides little improvement, suggesting that the separate exponents are not sharply estimated.... ..."

### Table 5. Continuous Time Models Estimated from Stock Prices: Estimates of Di usion Parameters.Model Score N

1995

"... In PAGE 21: ... All FD speci cations are sharply rejected. ||||||||||||{Table 3 about here ||||||||||||{ ||||||||||||{Figure 1 about here ||||||||||||{ ||||||||||||{Table 4 about here ||||||||||||{ ||||||||||||{ Table5 about here ||||||||||||{ The quasi-t-ratios displayed as a bar chart in the upper panel of Figure 1 suggest why rst order drift class of models are rejected. The large quasi-t-ratio b2 in the location function implies that rst order drift is not adequate: matching the moments of a discrete-time second order autoregression will require second order drift in continuous time.... ..."

### Table 7. Estimates of Unobserved Components Models: Discrete Time and Continuous Time

"... In PAGE 28: ... The results are contained in Tables 7 and 8. Table7 contains the results of estimating the discrete time and continuous time trend- plus-cycle models. The discrete time estimates are taken directly from Harvey (1989, p.... In PAGE 30: ...2843 Figures in parentheses are standard errors. misspeci ed in some way, and Harvey (1989) does indeed nd that the discrete time model in Table7 is inferior to a cyclical trend model in which t also depends on t 1 and yt = t+ t. Further investigations with continuous time cyclical trend models may be fruitful, but are beyond the scope of this simple illustration.... ..."

### Table 1 Convergence of Percentage Errors in Option Prices The table compares the percentage errors in option prices as the partition is refined. The true option prices correspond to the continuous time model under pricing measure Q.

2004

"... In PAGE 16: ... Of course, as the time partition becomes very fine, both the GARCH-Jump and Euler approximations converge to the same theoretical value. Table1 provides more details of percentage errors of the GARCH-Jump approximate prices 7In our model, the same jump affects both return and volatility. If one wants to switch off just one of them, two separate jump sources need to be built into the approximating model.... In PAGE 17: ...Jump approximate prices are generally within 2:5% of their continuous-time limits, regardless of moneyness. Table1 Here Even if one prefers to begin with modeling prices and volatilities by a bivariate process in continuous time, as above, there are significant advantages in estimating the parameters of the process from the approximating discrete-time NGARCH-Jump model. In fact, we have also shown that the GARCH approximation is a more efficient way of computing option prices as compared to the use of the Euler approximation scheme.... ..."

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### Table 8. Estimates of Continuous Time Unobserved Components Model with Di erential-Di erence Equation Cycle

"... In PAGE 28: ...3 years compared to 7 years.15 The estimates in Table8 , for the di erential-di erence equation cycle, are based on three choices of truncation parameter for the spectrum. Using the same method as in the simulations in the previous section, the truncation parameter M is determined by M = [T ] + 1(T 62 N ) for = f0:25; 0:50; 0:75g (the corresponding values of M are 3, 8 and 22 respectively).... ..."

### Table 6. Continuous Time Models Estimated from Stock Prices: Con dence Intervals on the Parameters of the Preferred Model. Criterion Di erence 95% Con dence Limits

1995

"... In PAGE 22: ... Our preferred model is, therefore, Preferred Model dHt = (ah + ahhHt)dt + (bh1 + bhh1Ht)dW1t + (bh2)dW2t 0 t lt; 1 dXt = (ax + Vt)dt + (bx2 + bxx2jXtj + eHt)dW2t dVt = (avvVt + avxXt)dt yt?L = T (Xt; b2; b3) t = 0; 1; : : : where T (z; b2; b3) = b0 + b1z + b2z3:4Iz 0(z) + b3(?z)3:4Iz lt;0(z) with b0 and b1 determined by R T (z; b2; b3) (z) dz = 0 and R T 2(z; b2; b3) (z) dz = 1: Upon application of Ito apos;s lemma, we obtain a representation of our preferred model as a stochastic di erential equation in four state variables with one component observed. dHt = (ah + ahhHt)dt + (bh1 + bhh1Ht)dW1t + (bh2)dW2t 0 t lt; 1 dXt = (ax + Vt)dt + (bx2 + bxx2jXtj + eHt)dW2t dVt = (avvVt + avxXt)dt dZt = [(ax + Vt)T 0(Xt; b2; b3) + 1 2(bx2 + bxx2jXtj + eHt)T 00(Xt; b2; b3)]dt + (bx2 + bxx2jXtj + eHt)T 0(Xt; b2; b3)dW2t yt?L = Zt t = 0; 1; : : : Table6 presents Wald and criterion di erence 95% con dence intervals for the parameters of the preferred t.... In PAGE 23: ...Table6 about here ||||||||||||{ A reader has remarked that the quasi-t-ratios shown in Figure 1 have not been orthog- onalized so that, due to multicolinearity, there is no guarantee that a modi cation to the model suggested by the largest quasi-t-ratio is necessarily the best possible modi cation one might make. The reader is correct in this, some other modi cation might be better and perhaps orthogonalization, which is numerically feasible, might be bene cial.... ..."