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

308 V. Voev
autocorrelated. Indeed, for optimal l-steps ahead forecasts, the sequence of forecast
errors follows a moving average process of order (l − 1). This result can be
expected to hold approximately for any reasonably well-conceived set of forecasts.’
Consequently, it can be shown that the variance of ¯d is, asymptotically,
Var � ¯d � ≈ T −1
�
�l−1
γ0 + 2
k=1
γk
�
, (26)
where γk is the kth autocovariance of dt. The Diebold–Mariano test statistic is
S1 = � �Var � ¯d �� −1/2 ¯d, (27)
where �Var � ¯d � is obtained from Eq. (26) by substituting for γ0 and γk the sample
variance and autocovariances of dt, respectively. Tests are then based on the
asymptotic normality of the test statistic. Noting that we only consider 1-step ahead
forecasts in this paper, the series dt should not be autocorrelated. As already noted
above, this is expected to hold for any reasonably constructed forecasts. Actually,
however, the sample based forecasts are not really reasonable in the sense that
they do not account for the serial dependence of the process they are supposed
to forecast. Thus, the degree of autocorrelation in the dt series, when either h1 or
h2 is a sample based forecast, will correspond to the degree of dependence in the
series to be forecast. For this reason, ignoring autocovariances in the construction
of the Diebold–Mariano tests will lead to an error in the test statistic. To correct
for this we include in �Var � ¯d � the first k significant autocorrelations for each of the
120 series.
Table 1 summarizes the results of the Diebold–Mariano tests carried out
pairwise between all models for all 120 series.
The first entry in each cell of the table shows the number of series (out of
120) for which the model in the corresponding column outperforms the model in
the corresponding row. The second entry corresponds to the number of significant
Table 1 Results from the Diebold–Mariano tests
s ss rm rc src drc dsrc drc− dsrc−
Chol Chol
s – 85/28 106/50 14/1 16/1 47/20 89/37 93/49 100/55
ss 20/0 – 106/47 14/1 16/1 47/20 89/37 92/49 100/55
rm 14/0 14/0 – 7/1 11/1 37/7 73/29 85/33 89/37
rc 106/60 106/61 113/69 – 105/86 119/59 120/88 115/80 117/88
src 104/55 104/56 109/69 0/0 – 119/50 120/86 114/77 117/85
drc 73/12 73/12 83/26 1/0 1/0 – 104/31 98/47 103/58
dscr 31/3 31/3 47/8 0/0 0/0 1/0 – 69/28 83/35
drc (Chol) 27/8 28/8 35/10 5/1 6/1 22/7 51/12 – 91/19
dsrc (Chol) 20/7 20/7 31/8 3/1 3/1 17/6 37/11 29/3 –
Note: Due to the definition of the shrinkage target, the first numbers in the pairs highlighted in
bold do not sum up to 120, since the variance series are unchanged in their respective ‘shrunk’
versions. Thus, in these cases there are only 105 series forecasts to be compared.