La violencia contra los trabajadores humanitarios y los cooperantes

such, we would like to explore other functional forms that could better capture the dynamics of the these parameters. The necessity for a functional form is clear when it comes to potential extrapolations and interpolations of the work, but we conjec- ture that this may lead to significant errors in the South African market with its ’developing’ designation in the work done inNUS-RMI (2016) and the relatively few firms listed on the JSE when compared to the United States economy or other larger, developed economies.

We would additionally like to review the treatment of missing and extreme val- ues within our sample. We have greatly oversimplified the treatment of these vari- ables, owing to the scope of this study. Different levels of winsorization are ex- plored in the work done inNUS-RMI(2016), with this not being necessary for our work, owing to the possibly erroneous assumption that the descriptive variables are multivariate-normally distributed. The normality assumption in conjunction with the imposed upper limit on the sample size, greatly reduces the probability of exceptionally large outliers that would need to be winsorized. Additionally, the treatment of missing variables and their classification has been greatly simplified due to the relatively small simulated data set. The work done inDuan et al.(2012) andNUS-RMI(2016) use different frequencies of data when calculating the SIGMA and DTD variables, but both have clear definitions for classifying variables as miss- ing. The classification and treatment of these missing variables would undoubtedly add an additional level of complexity to this study.

Finally, as already mentioned in the previous Chapter 6, we have greatly sim- plified the Sequential Monte Carlo algorithm from its presentation and use inNUS- RMI(2016). We would like to include the tempering sequence in our algorithm that was initially introduced inDel Moral et al.(2006). We suspect that this will assist with our weighting of particles and would serve the same purpose as quasi-random numbers in better distributing weights and particle significance. We would addi- tionally like to run the resampling step and move step multiple times within a sin- gle iteration until the efficient sample size has moved above its critical value. This would create a more diverse particle set with better weighting much earlier in the algorithm. Finally, we would like to vary the standard deviation of the Metropolis- Hastings kernel distribution more throughout the execution of the algorithm. This would increase the range of particles that are given an opportunity to be selected as potential candidates to represent the underlying distribution. These are all in- clusions that we believe would further improve our results.

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Appendix

Appendix A

By applying It ¯o’s Lemma to Equation (2.1) we obtain that ln(Vt) is normally dis-

tributed with meanµ − σ₂2tand variance σ2t. We therefore have the density for ln(Vt): fVt(v) = 1 √ 2πσ2_texp − v − µ − σ₂2
t
2 2σ2_t ! . We now use the method of transformations to find the density of:

Et= h(Vt) = VtN (dt+) − Le−r(T −t)N (dt−).

We can do this as h(v) is increasing for all v such that fVt(v) > 0, which a deriva-

tion similar to the calculation of the Black-Scholes delta will verify. We therefore know that h(·) is one-to-one and therefore an inverse exists such that v = h−1(u). We can now calculate the density of Etusing the method of transformations and

Jacobians. It must be noted that h−1(·)is analogous to the inverse of Equation (2.2). We therefore have: fEt(u) = fVt[h −1 (u)]     dh−1(u) du     . From which it follows that:

fEt(u) = 1 √ 2πσ2_texp − v − µ − σ₂2
t
2 2σ2_t ! 1 N (dt+) .

Appendix B

Taking the partial derivative of Equation (2.4) with respect to µ we get: ∂ ∂µL(µ, σ, δ) = − 1 2σ2 n X t=2 1 ht  − 2htln _ˆ Vt(σ, δ) At × At−1 ˆ Vt−1(σ, δ)  + 2  µ −σ 2 2  h2_t  . If (ˆµ, ˆσ, ˆδ)is an optimal solution to the maximization problem, then by setting the above equation equal to zero we can determine that:

ˆ µ = ˆ σ2 2 + 1 Pn t=2ht ln ˆ Vn(ˆσ, ˆδ) An × A1 ˆ V1(ˆσ, ˆδ)  .

We can substitute this into Equation (2.4) to arrive at our new maximization prob- lem in Equation (2.5) of two variables.