On the consistency of bootstrap testing for a parameter on the boundary of the parameter space
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On the consistency of bootstrap testing for a parameter on the boundary of the parameter space. / Cavaliere, Giuseppe; Nielsen, Heino Bohn; Rahbek, Anders.
In: Journal of Time Series Analysis, Vol. 38, No. 4, 07.2017, p. 513–534 .Research output: Contribution to journal › Journal article › Research › peer-review
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TY - JOUR
T1 - On the consistency of bootstrap testing for a parameter on the boundary of the parameter space
AU - Cavaliere, Giuseppe
AU - Nielsen, Heino Bohn
AU - Rahbek, Anders
PY - 2017/7
Y1 - 2017/7
N2 - It is well known that with a parameter on the boundary of the parameter space, such as in the classic cases of testing for a zero location parameter or no autoregressive conditional heteroskedasticity (ARCH) effects, the classic nonparametric bootstrap – based on unrestricted parameter estimates – leads to inconsistent testing. In contrast, we show here that for the two aforementioned cases, a nonparametric bootstrap test based on parameter estimates obtained under the null – referred to as ‘restricted bootstrap’ – is indeed consistent. While the restricted bootstrap is simple to implement in practice, novel theoretical arguments are required in order to establish consistency. In particular, since the bootstrap is analysed both under the null hypothesis and under the alternative, non-standard asymptotic expansions are required to deal with parameters on the boundary. Detailed proofs of the asymptotic validity of the restricted bootstrap are given and, for the leading case of testing for no ARCH, a Monte Carlo study demonstrates that the bootstrap quasi-likelihood ratio statistic performs extremely well in terms of empirical size and power for even remarkably small samples, outperforming the standard and bootstrap Lagrange multiplier tests as well as the asymptotic quasi-likelihood ratio test.
AB - It is well known that with a parameter on the boundary of the parameter space, such as in the classic cases of testing for a zero location parameter or no autoregressive conditional heteroskedasticity (ARCH) effects, the classic nonparametric bootstrap – based on unrestricted parameter estimates – leads to inconsistent testing. In contrast, we show here that for the two aforementioned cases, a nonparametric bootstrap test based on parameter estimates obtained under the null – referred to as ‘restricted bootstrap’ – is indeed consistent. While the restricted bootstrap is simple to implement in practice, novel theoretical arguments are required in order to establish consistency. In particular, since the bootstrap is analysed both under the null hypothesis and under the alternative, non-standard asymptotic expansions are required to deal with parameters on the boundary. Detailed proofs of the asymptotic validity of the restricted bootstrap are given and, for the leading case of testing for no ARCH, a Monte Carlo study demonstrates that the bootstrap quasi-likelihood ratio statistic performs extremely well in terms of empirical size and power for even remarkably small samples, outperforming the standard and bootstrap Lagrange multiplier tests as well as the asymptotic quasi-likelihood ratio test.
KW - Faculty of Social Sciences
KW - bootstrap
KW - boundary
KW - ARCH
KW - location model
KW - C32
U2 - 10.1111/jtsa.12214
DO - 10.1111/jtsa.12214
M3 - Journal article
VL - 38
SP - 513
EP - 534
JO - Journal of Time Series Analysis
JF - Journal of Time Series Analysis
SN - 0143-9782
IS - 4
ER -
ID: 164405276