International macroe.. - Free

2.7. FILTERING 69Now let ˜λ ∼ U[0, π] 29 . Imagine that we take a draw from this distribu-1.510.5-0.500 0.4 0.8 1.2 1.6 2 2.4 2.8 3.2 3.6 4 4.4 4.8 5.2 5.6 6-1-1.5Figure 2.2: π/2Phase shift. Solid: cos(t), Dashed: cos(t + π/2).tion. Let the realization be λ, and form the time-seriesq t = a cos(ωt + λ). (2.100)Once λ is realized, q t is a deterministic function with periodicity 2π and ωphase shift λ but q t is a random function ex ante. We will need thefollowing two basic trigonometric relations.Two useful trigonometric relations. Let b and c be constants, and i bethe imaginary number where i 2 = −1. Thencos(b + c) = cos(b)cos(c) − sin(b)sin(c) (2.101)e ib = cos(b)+i sin(b) (2.102)(2.102) is known as de Moivre’s theorem. You can rearrange it to getcos(b) = (eib + e −ib )2, and sin(b) = (eib − e −ib ). (2.103)2i29 Youonlyneedtoworryabouttheinterval[0, π] because the cosine function issymmetric about zero—cos(x) =cos(−x) for0≤ x ≤ π