recent developments in high frequency financial ... - Index of

The performance analysis of chart patterns 215
function in order to approximate the time series of prices Pj,k, we set the state
variable equal to time, XPj;k
m(x) may be expressed as:
¼ t: For any arbitrary x, a smoothing estimator of
^mðxÞ ¼ 1
l
X l
j¼1
!jðxÞPj;k; (10)
where the weight ωj(x) is large for the prices Pj,k with XPj;k near x and small for
those with XPj;k far from x. For the kernel regression estimator, the weight function
ωj(x) is built from a probability density function K(x), also called a kernel:
KðxÞ 0;
Z þ1
1
KðuÞdu ¼ 1: (11)
By rescaling the kernel with respect to a parameter h > 0, we can change its
spread:
1
KhðuÞ
h Kðu=hÞ;
Zþ1 KhðuÞdu ¼ 1 (12)
and define the weight function to be used in the weighted average (10) as:
1
!j;h Kh x XPj;k =ghðxÞ (13)
ghðxÞ
1
l
X l
j¼l
Kh x XPj;k : (14)
Substituting (14) into (10) yields the Nadaraya–Watson kernel estimator ^mhðxÞ
of m(x):
^mhðxÞ ¼ 1
l
X l
j¼l
!j;hðxÞPj;k ¼
Pl j¼1 Kh x XPj;k Pj;k
Pl j¼1 Kh
: (15)
x XPj;k
If h is very small, the averaging will be done with respect to a rather small
neighborhood around each of the XPj;k ’s. If h is very large, the averaging will be over
larger neighborhoods of the XPj;k’s. Therefore, controlling the degree of averaging
amounts to adjusting the smoothing parameter h, also known as the bandwidth.
Choosing the appropriate bandwidth is an important aspect of any local-averaging
technique. In our case we select a Gaussian kernel with a bandwidth, hopt,j ,
computed by Silverman (1986):
KhðxÞ ¼ 1
h ffiffiffiffiffi p e
2
x2
hopt;k ¼ 4
3
1=5
2h 2 (16)
kl 1=5 ; (17)