5.4 Two hundred observations of a time series, X1, ..., X200, gave the following sample statistics:
sample mean: x̄200 = 3.82;
sample variance: γ̂(0) = 1.15;
sample ACF: ρ̂(1) = 0.427;
ρ̂(2) = 0.475;
ρ̂(3) = 0.169.
a. Based on these sample statistics, is it reasonable to suppose that {Xt − μ} is white noise?
b. Assuming that {Xt − μ} can be modeled as the AR(2) process Xt − μ − ϕ1(Xt−1 − μ) − ϕ2(Xt−2 − μ) = Zt,
where {Zt} ~ IID(0, σ2), find estimates of μ, ϕ1, ϕ2, and σ2.
c. Would you conclude that μ = 0?
d. Construct 95 % confidence intervals for ϕ1 and ϕ2.
e. Assuming that the data were generated from an AR(2) model, derive estimates of the PACF for all lags h ≥ 1.