Mathematical and Statistical Methods for Actuarial Sciences and Finance

242 Marcella Niglio and Cira Perna

Ta b le 4 .LR linearity test ofrt

pdStat (p-value)

rt 4130. 5996 (1.9343e-05)

The results of the LR test show that a linear structure does not seem to be capabel
of catching the structure of the generating process (this explains the poor performance
of the autoregressive forecasts in [10]). The threshold autoregressive model fitted to
the data is clearly based on a strict parametric structure from which the forecasts are
generated.
Here, we alternatively propose the nonparametric predictor (3), which is more
flexible than that generated from SETAR models, and whose Markov order is selected
following Procedure 1.
For both approaches, we have generated one-step-ahead, out-of-sample forecasts
following an expanding window algorithm over the forecast horizonL=26, which
corresponds to the last six months of the time interval under analysis.
Further, a null threshold value has been fixed for the SETAR model (with threshold
delay given in Table 4) and at each iteration the model has been estimated following
[22].
SETAR and nonparametric least-squares forecasts have been evaluated using the
mean square error and the mean absolute error,MSE(L)=L−^1

∑L

i= 1 (XˆT+i−

XT+i)^2 andMAE(L)=L−^1

∑L

i= 1 |XˆT+i−XT+i|, whose values are compared in
Table 5 where the MSE (and the MAE) of (3) over the MSE (and MAE) of the SETAR
predictions are shown.

Ta b le 5 .MSE (and MAE) of the nonparametric forecasts over the MSE (and MAE) of the
SETAR forecasts

MSE(L)np[MSE(L)thr]−^1 MAE(L)np[MAE(L)thr]−^1

rt 0.839081 0.944058

The better forecast accuracy, in terms of MSE and MAE, of predictor (3) can be
appreciated. It further confirms the good performance of the proposed procedure in the
presence of one-step-ahead forecasts. Moreover, the forecast accuracy seems not to
be affected when different values, of moderate size, are assigned tomin Procedure 2.

5 Conclusions

We have proposed a procedure to select the orderpin the presence of strictly stationary
Markov processes (1). It is based on the use of one-step-ahead predictors generated
from nonparametric Nadaraya-Watson kernel smoothers.