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Consider a p-dimensional vector AR-process Xt that is given by
K
Xt = ∑ Гk Xt – k + εt ,
k=1
Where matrices Г𝐤 provide autoregression coefficients and εt ~ N(0, ∑) iid. In
this case all component time series are stationary.
Assume that maximum lag length K0 can be derived either from an economic
background or from a certain rule of the thumb based on number of observations.
Therefore, the model selection problem for a given realization of the process includes
identifying K and non-zero elements of Гk in the search space described by
2 ×K
Ω = {0, 1}p
0
where 0 indicates parameter constraint to be zero, and 1 to the freely
estimated parameter. It can also be mentioned, that for this problem different objective
functions can be considered. For this problem the Bayesian or Schwarz information
̂ | +l lnT/T (where ∑
̂ denotes determinant of the fitted residuals
criterion BIC = ln|∑
covariance matrix and l denotes number of non-zero parameters) is used, but using any
other information criterion (e.g. Akaike information criterion) is possible. It could be
implemented just by replacing the objective function.
We assume that Ω is finite, which allows solving the problem by simple
enumeration of all elements of the set and choosing lag structure that corresponds to
the lowest value of the chosen information criterion. However, this method is not
applicable to the real problem, since even modest values of p2 and K0 returns a big
amount of the lag structures, which makes enumeration practically impossible. Another
way is to consider only a small subspace of the Ω, which could be enumerated.
However, there is no guarantee that optimal lag structure will lie within this subset.
Therefore, using heuristics is reasonable at least as a benchmark.
As mentioned in the previous section, neighborhood for the TA heuristics could
be defined by the means of ε-spheres defined by Hamming distance. In this case,
generally, ε should be chosen to be large enough for not being stuck in a bad local
minima and small enough to result in a guided local search. For the considered problem
with a 5000 iterations per restart Hamming distance of 4 appears to be the most
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