Mathematical and Statistical Methods for Actuarial Sciences and Finance

110 P. Coretto and M.L. Parrella

Ta b le 1 .Percentage of testing periods for each specified model and window when cases 1–4
occur. Cases 1–4 are as follows. Case 1: the estimated linear coefficients in the parametric
step are jointly equal to zero at levelα=5%, and we do not reject theH 0 hypothesis in
the nonparametric stage at the same level; Case 2: the estimated linear coefficients in the
parametric step are jointly equal to zero at levelα=5%, and we reject theH 0 hypothesis
in the nonparametric stage at the same level; Case 3: the estimated linear coefficients in the
parametric step are jointly different from zero at levelα=5%, and we do not reject theH 0
hypothesis in the nonparametric stage at the same level; Case 4: the estimated linear coefficients
in the parametric step are jointly different from zero at levelα=5%, and we reject theH 0
hypothesis in the nonparametric stage at the same level

Window Regressors (model)

ββ,σβ^2 ,σ

(Model A) (Model B) (Model C)

Case 1

w= 22 49.42 45.595 57.78

w= 66 43.87 44.144 51.195

w= 264 43.449 43.213 52.099

Case 2

w= 22 31.141 13.114 23.925

w= 66 38.41 16.078 28.373

w= 264 38.41 12.53 28.148

Case 3

w= 22 10.155 26.564 9.695

w= 66 8.238 23.909 10.409

w= 264 9.183 25.141 9.465

Case 4

w= 22 9.284 14.728 8.6

w= 66 9.483 15.87 10.023

w= 264 8.959 19.116 10.288

As Case 1 we label the percentage of testing periods where the estimated linear
coefficients in the parametric step are jointly equal to zero at the testing levelα=5%,
and we do not reject theH 0 hypothesis (e.g., the linear relation statistically holds) in
the nonparametric stage at the same level. This case is of particular interest because
the percentage of testing periods when it occurs is not smaller than 43% for all rolling
windows and all sets of regressors. If we combine the results of the two stages (both
the parametric and the nonparametric) this means that in almost half of the testing