Nonparametric prediction in time series analysis 239Ta b le 1 .Distribution of the 100 series of lengthT=70 andT=100 respectively, simulated
fromModel 1
bˆ(T=70)
p [9, 21] [22, 34] [35, 43] Tot.
114 23 5188
21 4 2 7
34 0 0 4
40 1 0 1bˆ (T=100)
p [9,29] [30,44] [45,64] Tot.
119215090
21 3 1 5
32 2 1 5
40 0 0 0Ta b le 2 .Distribution of the 100 series of lengthT=70 andT=100 respectively, simulated
fromModel 2
bˆ(T=70)
p [17, 26] [27, 35] [36, 44] Tot.
115115985
21 0 4 5
33 6 1 10
40 0 0 0bˆ (T=100)
p [15, 31] [32, 47] [48, 64] Tot.
121145489
22 2 3 7
31 0 3 4
40 0 0 0In both cases, the proposed procedure gives satisfactory results on the selection
of the autoregressive order in the presence of a Markov process of order 1 that, as
expected, improves asTgrows.
Note that the good performance is a guarantee for time series of moderate length
T, that rises the interest on the procedure.
As expected, most “well selected” models belong to the last class ofbˆ. It should
not be surprising because the results used in the proposed procedure are mainly given
in asymptotic context.
4 Empirical results on 90-day US T-bill rate
Starting from the theoretical results described in Section 2, the model selection pro-
cedure has been applied to generate forecasts from the weekly 90-day US T-bill
secondary market rates covering the period 4 January 1957–17 December1993. The
time series,Xt, of lengthT=1929, has been extracted from the H.15 release of the
Federal Reserve System (http://www.federalreserve.gov/releases/h15/data.htm).
The 90-day US T-bill has been widely investigated in nonlinear and nonpara-
metric literature (among others [1] and [15]) and, in particular, the data set under
study, plotted in Figure 1, has been analysed in [10] to compare three kernel-based
multi-step predictors. The authors, after computing proper unit-root tests, show the