Palgrave Handbook of Econometrics: Applied Econometrics

D.S.G. Pollock 269

are respectively the trend component and the transitory component. This is the
so-called Beveridge–Nelson decomposition.
The trend component of the Beveridge–Nelson decomposition is a first-order
random walk with drift, whereas the transitory component is an ARMA process.
The distinguishing feature of the decomposition is that both components have the
same forcing function. It is easy to see that:

τ(z)=

θ( 1 )

α( 1 )

α(z)

θ(z)

y(z), (6.78)

which is to say that the estimate of the trend is derived by applying an ordinary
linear filter to the data sequence. The effect of the filter is to eliminate the ARMA
factor from the data so as to deliver a pure random walk.
A common objection to the Beveridge–Nelson decomposition is that the result-
ing trend is liable to be too rough. This is a consequence of the fact that a random
walk that is an accumulation of independently and identically distributed random
variables comprises elements at all frequencies up to the limiting Nyquist frequency
ofπradians per sample period. Also, the decomposition makes no provision for
the presence of seasonal fluctuations in the data. A more elaborate model can be
proposed with the aim of overcoming these objections.
Consider the multiplicative seasonal ARIMA model of Box and Jenkins (1976),
which can be represented by the equation:

∇d(z)∇sD(z)y(z)=μ(z)+

θ(z)(zs)

α(z)A(zs)

ε(z). (6.79)

Here,α(z)andθ(z)are the autoregressive and moving-average polynomials that
have appeared in equation (6.74), whereasA(z)and(z)are seasonal operators.
Whereas∇(z)continues to represent the ordinary difference operator, there is now
a seasonal difference operator∇s(z)= 1 −zs=( 1 −z)S(z), which forms the differ-
ences between the data from the same season (or month) of two successive years.
The factors of this operator are the ordinary difference operator and a seasonal

summation operatorS(z)= 1 +z+z^2 +···+zs−^1. A decomposition can now be
found of the formy(z)=τ(z)+σ(z)+ζ(z), where:

τ(z)=

1

∇d+D

{μ(z)+α(z)ε(z)}, (6.80)

σ(z)=

β(z)

SD(z)

ε(z), (6.81)

ζ(z)=

γ(z)

α(z)A(zs)

ε(z), (6.82)

are, respectively, the trend, the seasonal component and the transitory component.
If the degreed+Dof the (ordinary) difference operator exceeds unity, then the
trend is liable to be smoother than one generated by a first-order random walk.
Also, the effect ofα(z)might be further to attenuate the high-frequency elements
of the forcing function, thereby enhancing the smoothness of the trend.