Estimating the Codifference Function of Linear Time Series Models ...

with
{ f ij = e σα (|s i| α +|s j| α −|s i−s j| α ) 1 (|s i| α +|s j| α −|s i−s j| α) }
2 eσα − 1
+ e σα (|s i| α +|s j| α −|s i+s j| α ) { 1
2 eσα (|s i| α +|s j| α −|s i+s j| α) − 1
}
+ 1
{
h ij = e σα (|s i| α +|s j| α −|s i−s j| α ) 1 (|s i| α +|s j| α −|s i−s j| α) }
2 eσα − 1
{ }
+ e σα (|s i| α +|s j| α −|s i+s j| α )
1 − 1 (|s i| α +|s j| α −|s i+s j| α )
2 eσα
and
g ij = 4σ 2α |s i | α |s j | α
3 Simulation evidence
In this section, we present a simulation study for investigating the small sample properties of the
sample normalized codifference function.
3.1 Practical considerations
Before we proceed, we make a remark about the sample and the population codifference function.
From (6) and the fact that all c j ’s are real, we have that the codifference function of the models
we consider here is a real-valued function but the estimator (10) is complex. Therefore, one
possibility is to use only the real part of the estimator. Because in practice we are working with a
finite sample, the imaginary part of the estimator is still present, but will vanish asymptotically.
In what follows, we are only working with the estimator of normalized codifference Î(k). Hence,
we note that unlike the true normalized codifference (5), from (10) and Corollary 2.3, one can
see that the sample normalized codifference function and its limiting variance depend on s =
{s 1 , . . .,s r }. Apparently Î(·) is defined for all s > 0, and from Theorem 2.1, we know that it
is a consistent estimator. However, in a finite sample, the convergence of the estimator to the
population values depends on the choice of s. Therefore, for estimation, s is a design parameter
which has to be chosen appropriately. In other words, Î(·) should be calculated from those values
of s which gives the most accurate estimates of the true function I(·).
To be more precise, in practice the number of grid points r and more importantly, the location
of s 1 , . . . , s r , have to be chosen. As the normalized codifference function is defined based on the
ecf, we can apply here the known results for ecf. For a fixed r, Koutrouvelis (1980) and Kogon and
Williams (1998) showed that for calculating ecf, the location of the grid points s 1 , . . . , s r , should
be chosen close, but not equal to zero. It has been shown in this case, the ecf will most accurately
estimate the characteristic function. This particular choice of grid points seems to be reasonable
for calculating Î(·), as for instance, can be shown in Figure 1. To determine the location of s i’s,
we suggest to plot Re Î(k) within the interval 0.01 ≤ s ≤ 2, for some values of lag k > 0. These
graphs will show the interval of s around zero which has relatively small bias, as the best location
for evaluating the estimator. The best choice for the interval of s clearly depends on the data
itself and in general also on the lag k. However, we suggest to use s a = 0.01 as the left bound of
the interval, where the best choice for the right bound can be determined from the graphs, i.e.,
as the threshold of s = s b where the graphs Re Î(k), for some lag k, are still relatively flat. The
individual choices for s i ’s can be chosen in one of two ways:
1. If we wish to use equal spacing of s i ’s, we can set s = {s 1 = 0.01, 0.01 + i s b−s a
r−1 , s b}, i =
1, . . .,r − 2. To obtain r, we can minimize the determinant of covariance matrix in (13).
Unfortunately, the covariance matrix in (13) depends on the unknown parameters c j ’s, α, σ
and the distance between s i ’s. One possibility is to replace (13) with its consistent estimate,
however here we consider a different approach for choosing the s i ’s, i.e., with the help of
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