Palgrave Handbook of Econometrics: Applied Econometrics

Tommaso Proietti 395

roots of the AR polynomial are a pair of complex conjugates with modulusρ−^1
and phase&; correspondingly, the spectral density displays a peak around&.
Harvey and Trimbur (2003) further extend the model specification by proposing
a general class of model-based filters for extracting trend and cycles in macro-
economic time series, showing that the design of lowpass and bandpass filters
can be considered as a signal extraction problem in an unobserved components

framework. In particular, they consider the decompositionyt=μmt+ψkt+ (^) t,
where (^) t∼NID(0,σ
2 ). The trend is specified as anmth-order stochastic trend:
μ 1 t = μ1,t− 1 + ζt
μit = μi,t− 1 + μi−1,t, i=2,...,m.
(9.9)
This is the recursive definition of anm−1-fold integrated random walk, such that
mμmt=ζt. The componentψktis akth-order stochastic cycle, defined as:
[
ψ 1 t
ψ 1 ∗t
]
=ρ
[
cos& sin&
−sin& cos&
][
ψ1,t− 1
ψ1,∗t− 1
]

• [
  κt
  0
  ]
  ,
  [
  ψit
  ψit∗
  ]
  =ρ
  [
  cos& sin&
  −sin& cos&
  ][
  ψi,t− 1
  ψi∗,t− 1
  ]


• 
  

  [
  ψi−1,t
  0
  ]

  
  

. (9.10)

The reduced form representation for the cycle is:

( 1 − 2 ρcos&L+ρ^2 L^2 )kψkt=( 1 −ρcos&L)kκt.

Harvey and Trimbur show that, asmandkincrease, the optimal estimators of the
trend and the cycle approach the ideal lowpass and bandpass filter, respectively.

9.2.5 Models with correlated components

Morley, Nelson and Zivot (2003; henceforth, MNZ) consider the following
unobserved components model for US quarterly GDP:

yt = μt+ψt t=1, 2,...,n,

μt = μt− 1 +β+ηt,

ψt = φ 1 ψt− 1 +φ 2 ψt− 2 +κt,

(

ηt

κt

)

∼ NID

[(

0

0

)

,

(

ση^2 σηκ

σηκ σκ^2

)]

, σηκ=rσησκ.

(9.11)

It should be noticed that the trend and cycle disturbances are allowed to be contem-
poraneously correlated, withrbeing the correlation coefficient. The reduced form

of model (9.11) is the ARIMA(2,1,2) model:yt=β+φ(θ(LL))ξt, ξt∼NID(0,σ^2 ),

whereθ(L)= 1 +θ 1 L+θ 2 L^2 andφ(L)= 1 −φ 1 L−φ 2 L^2. The structural form is exactly
identified, both it and the reduced form have six parameters. The orthogonal
trend-cycle decomposition considered by Clark (1987) imposes the overidentifying
restrictionr=0.