The bootstrapped procedure introduced in this paper is motivated by the loss of size and power from the nuisance parameters that appear in the distribution in Chapter 1. In this section, we test if the parametric bootstrap helps to obtain more accurate test sizes in practice.
For the simulation exercise, we use the exponential smoothing model as in Cheng (2015). The model specification is defined as follows.
t(θn) =yt−ζ0x1,t−βnyt−1(1−exp(−c(x2,t−π0)2)) (2.40)
For this model, there is not need to use drifting sequences of true parameters for ζ and π, the speed at which βn → ∞ will determine the identification category. We assume the following true values in the simulations : ζ0 = 1, π0 = 0 and b = 1. In this exercise we
compare the identification categories as follows, under strong identification we set βn = b, under weak identification we set βn = b/
√
n and under non-identification (limiting case of weak identification) we set βn= 0.
The sample size takes valuen = 100,250 and 500. We set arex1,t, x2,t d
∼N(0,1),c=−1. We let the true errors to be either Normal(0,1) or GARCH(1,1) with ω = 0.1, α = 0.3 and β = 0.6. The wild bootstrap of Liu et al. (1988) using the two point distribution multiplier as in Mammen (1993) to generate the bootstrapped residuals in Step 2. The number of simulations is 1,000. For each simulation, we construct bootstrapped samples using 500 draws. We also refer Chapter 1 for a complete reference of gradient, Hessian and
other expressions required for estimation and construction of the bootstrap as well as other simulation details. The remaining tables are presented in the Appendix.
We consider three null hypotheses in this simulation exercise. The first null hypothesis sets the parameters equal to their true values. To evaluate the power of the test, the second and third hypothesis consider the false null hypotheses of parameters within one and three standard deviations of the true value.
H0β,1 :β =βn H0π,1 :π =π0
H0β,2 :β =βn+σβ H0π,2 :π=π0+σπ (2.41) H0β,3 :β =βn+ 3σβ H0π,3 :π=π0+ 3σπ
The robust sample t-statistic is constructed by
Tn =
√
n(r(ˆθn)−v)
[rθ(ˆθn)B−1( ˆβn) ˆΣn(ˆθn)B−1( ˆβn)rθ(ˆθn)0]1/2
(2.42)
while the standard t-statistic takes the form,
Tns=
√
n(r(ˆθn)−v) [rθ(ˆθn) ˆΣn(ˆθn)rθ(ˆθn)0]1/2
(2.43)
The critical values of the bootstrapped t-statistic are computed using order statistics. Let {Tm
a,n(π)}mj=1 be a sequence of independent draws of the t-statistic with a = ψ, π, θ.
Denote the order statistics by Ta,n,m [1] ≤Ta,n,m [2], ..., etc. The LF and ICS0 critical values are
computed using ca,mn,1−α/2 = inf{c ≥ 0 : P(Tm
n ≤ c) ≥ 1−α/2} and c a,m
n,α/2 = inf{c ≥ 0 : P(Tnm≤c)≥α/2}, as we use two tailed critical values. To construct theICS0 critical value,
we use κn = (ln(n))1/2, as suggested by Andrews and Cheng (2012).
Tables 2.3 and 2.4 compares the results of the parametric bootstrap introduced in this paper and the (unfeasible) asymptotic approximation of Andrews and Cheng (2012). The LF AC and ICS0 AC critical values are unfeasible because it is assumed that the nuisance
parameters are known while the identification category is still unknown. The bootstrapped critical values are feasible as they do not assume that the identification category or the nui- sance parameters are known. Even though the testing ofβ is not valid using the parametric bootstrap, the tables illustrate its performance and its irregular behavior in the weakly identified case.
The results in Tables 2.3 and 2.4 indicate that the parametric bootstrap works excep- tionally well for the cases of strong and non-identification. The critical values that are constructed in these cases are numerically close to the infeasible critical values of Andrews and Cheng (2012). When the model is weakly identified, the critical values work well, but not as well as the infeasible case. The difference in accuracy hinges on the inability of the ˆAn statistic to recognize if the parameters are weakly or strongly identified. When the ˆAn leads to the incorrect conclusion, theICS0 selects the incorrect critical value, and in consequence
rejection rates are usually higher than the correct test size. The least favorable critical values perform better as the simulation exercise shows that the distributions under weak identifi- cation have larger critical values that the strong identification case. The critical values forπ work particularly well compared to the asymptotic approximations of Andrews and Cheng (2012). In the case of the asymptotic approximations, test sizes close to zero when we test with respect to π. As the t-test is centered at π0 which is a nuisance parameter, generating
a grid and taking the supremum of the critical values does not perform well in practice because the critical values are too wide. Using the parametric bootstrap presented in this paper, we can test π without nuisance parameters, which performs as well as the infeasible critical values of Andrews and Cheng (2012). In summary, the parametric bootstrap works very well as long as the ICS0 critical values are able to recognize if the parameters of the
model are weakly or strongly identified.