International macroe.. - Free

76 CHAPTER 2. SOME USEFUL TIME-SERIES METHODSwhere λ is the penalty attached to the volatility of the trend component.For quarterly data, researchers typically set λ = 1600. 33 Noting that∆ 2 τ t+1 = τ t+1 − 2τ t + τ t−1 ,differentiate (2.114) with respect to τ t andre-arrange the Þrst-order conditions to get the Euler equationsq 1 − τ 1 = λ[τ 3 − 2τ 2 + τ 1 ],q 2 − τ 2 = λ[τ 4 − 4τ 3 +5τ 2 − 2τ 1 ],. .q t − τ t = λ[τ t+2 − 4τ t+1 +6τ t − 4τ t−1 + τ t−2 ], t =3,...,T − 2. .q T −1 − τ T −1 = λ[−2τ T +5τ T −1 − 4τ T −2 + τ T −3 ],q T − τ T = λ[τ T − 2τ T −1 + τ T −2 ].Let c =(c 1 ,...,c T ) 0 ,q=(q 1 ,...,q T ) 0 ,andτ =(τ 1 ,...,τ T ) 0 ,andwritethe Euler equations in matrix formwhere the T × T matrix G is given byG =⎡⎢⎣q =(λG + I T )τ, (2.115)1 −2 1 0 ··· ··· 0−2 5 −4 1 0 ··· ··· 01 −4 6 −4 1 0 ··· ··· 00 1 −4 6 −4 1 0.. .. . .. .0 0 1 −4 6 −4 1 0. 0 1 −4 6 −4 1. 0 1 −4 5 −20 ··· ··· 0 1 −2 1Get the trend component by τ =(λG+I T ) −1 q. The cyclical componentfollows by subtracting the trend from the original observationsc = q − τ =[I T − (λG + I T ) −1 ]q.33 The following derivation of the Þlter follows Pederson [121].⎤⎥⎦.