Generate 200 observations from the process

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Reference no: EM131068416

Question- Consider the sinusoidal time series X_t = β₀ + A cos(2πωt + ψ) + W_t, where W_t is a WN(0, σ²). Here, A and ψ are unknown and you may take 0 < ω < 1/2.

(i) Using the identity cos(a ± b) = cos(a) cos(b) sin(a)sin(b) one can re-write this model as X_t = β₀ + β₁ cos(2πωt) + β₂ cos(2πωt) + W_t, t = 1,...,n. Show that for large n the least squares estimates of β₁ and β₂ are approximately

Β^{^}₁≈ (2/n)_t=1Σⁿ(X_t - X^-) cos(2πωt) and Β^{^}₂≈ (2/n)_t=1Σⁿ(X_t - X^-) sin(2πωt)

(ii) For the ease of notation, put α = 2πω and β₀ = 0. Show that X_tsatisfies the AR(2) process X_t = Φ₁X_t-1+ Φ₂X_t-2+W_t, where Φ₁ = 2 cos(a) and Φ₂ = -1.

(iii) Generate 200 observations from the process X_t = 10 cos(πt/6) + W_t, where W_t is a Gaussian (normal) white noise with mean zero and variance 1. Plot the PACF and comment.

Reference no: EM131068416

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