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

192
the invertibility condition holds. The proposed modified test procedure works as
follows:
i) Construct u u i as the probability that Yi is less than the actually observed price
change yi:
u u i
bPr½YiyijF i 1; ZiŠ
¼ Xyi
bPr½Yi¼jjF i 1; ZiŠ
(4.1)
j¼ 1
for each i: 1→n, and construct u i l as the probability that Yi is less than the
actually observed price change minus one:
u l i
bPr½Yiyi1jF i 1; ZiŠ
¼ Xyi 1
bPr½Yi¼jjF i 1; ZiŠ:
(4.2)
j¼ 1
ii) Then use this sequence of probabilities to generate a sequence of artificial
random numbers from the conditional uniform distributions on the intervals
[ui l , u i u ], i.e.,
ui U u l i ; uu i ; i : 1 ! n: (4.3)
If the model is correctly specified, the u i’s drawn under the true parameter
values (to calculate u i l and ui u ) are i.i.d. following a uniform distribution on the
interval [0,1] (see the Appendix). Moreover, one can map the u i’s into a standard
normal distribution using u i* ≡ Φ −1 (u i). Under the correct model, the normalized
u i*’s are i.i.d. standard normal distributed.
JBX HAL
R. Liesenfeld et al.
Fig. 10 Autocorrelation function of the density forecast variable ui* for Jack in the Box Inc. (JBX)
and HalliburtonpCompany ffiffi (HAL). The dashed lines mark off the approximate 99% confidence
interval 2:58 n