Optimal Pair-Trading Decision Rules for a Class of Non-Linear Boundary Crossings by Ornstein-Uhlenbeck Processes Page: 51
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N (Xt-(e-A aXtl_1+(1-e-A t)))2
L(i, A, o-Xt) = __1_e 2( (-e-2^}1)
t=1 272 (1 - e-2at)
We then obtain the log-likelihood as:
N (Xt -(e - A aXt -1+ (1-e- A a1)))2
1 22 ( - -1 1)
l(i, A, u-Xt) = log e 2(d(-e2^^)
t=1 27r 2 (1 - e-2A1t)
(36)
2 N
--log(27r) - log( (1- e-2.At)) A2 -2 Ot E(Xt - e-AAtXt-1 - i(1 - e-AAt))2
2 2 2A (1 - e2A )
The maximum likelihood estimates for ,u, A and - will be the values ft, A\and 8 respectively,
that maximize the above expression. We will solve this numerically from equation 36, using
the maxLik package in R.
5.1.2.2. Method of Least Squares
While we acknowledge that this is a nonlinear least squares problem, from several
numerical examples, we observed that the linear least squares estimate approximate the
nonlinear least squares estimates up to eight decimal places. Thus we will approximate the
optimization solution with linear least squares method throughout this work. This also saves
us from having to deal with complicated hessian matrices that result from the nonlinear least
squares optimization problem, particularly in proving its positive definiteness.
We formulate the least squares regression equation from equation 32 as follows:
(37) Xt = + Xt-1 + Et, Et ~ i.i.d.N(O,o-E)
where
(38) - =P(1 - e-aot)
(39) = e-aot51
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Tamakloe, Emmanuel Edem Kwaku. Optimal Pair-Trading Decision Rules for a Class of Non-Linear Boundary Crossings by Ornstein-Uhlenbeck Processes, dissertation, December 2021; Denton, Texas. (https://digital.library.unt.edu/ark:/67531/metadc1873709/m1/70/: accessed July 17, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .