Palgrave Handbook of Econometrics: Applied Econometrics

Tommaso Proietti 399

1960 1980 2000

–5.0

–2.5

0.0

2.5

5.0

Real-time cyclical component

1960 1980 2000

–5.0

–2.5

0.0

2.5

5.0

Smoothed cyclical component

–20 –10 0 10 20

–0.2

0.0

0.2

Real-time cycle weights

–20 –10 0 10 20

–0.25

0.00

0.25

0.50

Smoothed cycle weights

Figure 9.2 Trend-cycle decomposition with correlated disturbances. Real-time and smoothed
estimates of the cyclical components

trend as a random walk driven by the innovationsξt=yt−E(yt|Yt− 1 ). Writing
the ARIMA representation forytasyt=β+ψ(L)ξt,ψ(L)=θ(L)/φ(L), whereφ(L)
is a stationary AR polynomial of orderpandθ(L)an invertible MA polynomial of
orderq, the BN decomposition can be written as:yt=mt+ct,t=1,...,n, where
mtis the BN trend, andctis the cyclical component.
The trend is defined as liml→∞[y ̃t+l|t−lβ], withy ̃t+l|t=E(yt+l|Yt). Writing
yt+l=yt+l− 1 +β+ψ(L)ξt, taking the conditional expectation and rearranging,
it is easily shown to givemt=mt− 1 +β+ψ( 1 )ξt, whereψ( 1 )=θ( 1 )/φ( 1 )is the
persistence parameter, as it measures the fraction of the innovation at timetthat
is retained in the trend. In terms of the observations,mt=wm(L)yt, where wm(L)
is the one-sided filter:

wm(L)=

ψ( 1 )

ψ(L)

=

θ( 1 )

φ( 1 )

φ(L)

θ(L)

.

The sum of the weights is one, i.e.,wm( 1 )=1.
Thetransitory componentis defined residually asct=yt−mt=ψ∗(L)ξt, where
ψ∗(L)=ψ(L)−ψ( 1 ). Alternative representations in terms of the observationsyt
and of the innovationsξtare, respectively:

ct=

φ( 1 )θ(L)−θ( 1 )φ(L)

φ( 1 )θ(L)

yt, ct=

φ( 1 )θ(L)−θ( 1 )φ(L)

φ( 1 )φ(L)

ξt. (9.13)