Shumway Stoffer Time_Series_Analysis_and_Its_Applications__With_R_Examples 3rd edition (1)

362 6 State-Space Models
Table 6.2. Comparison of Standard Errors
Asymptotic Bootstrap
Parameter MLE Standard Error Standard Error
φ .865 .223 .463
α −.686 .487 .557
b .788 .226 .821
σ w .115 .107 .216
σ v 1.135 .147 .340
in the observation equation, A t = z t , Γ = α, R = σv, 2 and S = 0. The parameter
vector is Θ = (φ, α, b, σ w , σ v ) ′ . The results of the Newton–Raphson
estimation procedure are listed in Table 6.2. Also shown in the Table 6.2
are the corresponding standard errors obtained from B = 500 runs of the
bootstrap. These standard errors are simply the standard deviations of the
bootstrapped estimates, that is, the square root of ∑ B
b=1 (Θ∗ ib − ¯Θ i ∗)2 /(B−1),
where Θ i , represents the ith parameter, i = 1, . . . , 5, and ¯Θ i ∗ = ∑ B
b=1 Θ∗ ib /B.
The asymptotic standard errors listed in Table 6.2 are typically much
smaller than those obtained from the bootstrap. For most of the cases, the
bootstrapped standard errors are at least 50% larger than the corresponding
asymptotic value. Also, asymptotic theory prescribes the use of normal
theory when dealing with the parameter estimates. The bootstrap, however,
allows us to investigate the small sample distribution of the estimators and,
hence, provides more insight into the data analysis.
For example, Figure 6.9 shows the bootstrap distribution of the estimator
of φ in the upper left-hand corner. This distribution is highly skewed with
values concentrated around .8, but with a long tail to the left. Some quantiles
are −.09 (5%), .11 (10%), .34 (25%), .73 (50%), .86 (75%), .96 (90%), .98
(95%), and they can be used to obtain confidence intervals. For example, a
90% confidence interval for φ would be approximated by (−.09, .96). This
interval is ridiculously wide and includes 0 as a plausible value of φ; we will
interpret this after we discuss the results of the estimation of σ w .
Figure 6.9 shows the bootstrap distribution of ̂σ w in the lower right-hand
corner. The distribution is concentrated at two locations, one at approximately
̂σ w = .25 (which is the median of the distribution of values away from
0) and the other at ̂σ w = 0. The cases in which ̂σ w ≈ 0 correspond to deterministic
state dynamics. When σ w = 0 and |φ| < 1, then β t ≈ b for large
t, so the approximately 25% of the cases in which ̂σ w ≈ 0 suggest a fixed
state, or constant coefficient model. The cases in which ̂σ w is away from zero
would suggest a truly stochastic regression parameter. To investigate this
matter further, the off-diagonals of Figure 6.9 show the joint bootstrapped
estimates, (̂φ, ̂σ w ), for positive values of ̂φ ∗ . The joint distribution suggests
̂σ w > 0 corresponds to ̂φ ≈ 0. When φ = 0, the state dynamics are given by
β t = b + w t . If, in addition, σ w is small relative to b, the system is nearly de-