Question- Consider the sinusoidal time series Xt = β0 + A cos(2πωt + ψ) + Wt, where Wt 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 Xt = β0 + β1 cos(2πωt) + β2 cos(2πωt) + Wt, t = 1,...,n. Show that for large n the least squares estimates of β1 and β2 are approximately
Β^1 ≈ (2/n)t=1Σn(Xt - X-) cos(2πωt) and Β^2 ≈ (2/n)t=1Σn(Xt - X-) sin(2πωt)
(ii) For the ease of notation, put α = 2πω and β0 = 0. Show that Xt satisfies the AR(2) process Xt = Φ1Xt-1+ Φ2Xt-2+Wt, where Φ1 = 2 cos(a) and Φ2 = -1.
(iii) Generate 200 observations from the process Xt = 10 cos(πt/6) + Wt, where Wt is a Gaussian (normal) white noise with mean zero and variance 1. Plot the PACF and comment.