Palgrave Handbook of Econometrics: Applied Econometrics

Tommaso Proietti 389

known as a stationarity test and is discussed in Nyblom and Mäkeläinen (1983).
The nonparametric extension to the case whenψtis any indeterministic stationary
process is provided by Kwiatkowski, Phillips, Schmidt and Shin (KPSS) (1992) (see
also Harvey, 2001, for a review and extensions).
The reduced-form representation of (9.1) is an integrated moving average model
of orders (1,1), or IMA(1,1):yt=β+ξt+θξt− 1 ,ξt∼NID(0,σ^2 ), whereyt=
yt−yt− 1. The difference operator can be defined in terms of the lag operatorL,

such thatLdyt=yt−d, for an integerd,as=( 1 −L).
The moving average (MA) parameter is subject to the restriction− 1 ≤θ≤0.
Equating the autocovariance generating functions ofytimplied by the IMA(1,1)

and by the structural representation (9.1), it is possible to establish thatση^2 =

( 1 +θ)^2 σ^2 andσψ^2 =−θσ^2. Hence it is required thatθ≤0, so that persistence,

( 1 +θ), cannot be greater than unity. The variance ratioλ= σψ^2 /ση^2 depends

uniquely onθ,asλ=−θ/( 1 +θ)^2. The ratio provides a measure of relative smooth-
ness of the trend: ifλis large, then the trend varies little with respect to the noise
component, and thus it can be regarded as “smooth.”
The RWpN model has a long tradition and a well-established role in the analysis
of economic time series, since it provides the model-based interpretation for the
popular forecasting technique known asexponential smoothing, which is widely used
in applied economic forecasting and fares remarkably well in forecast competitions
(see Muth, 1960, and the comprehensive reviews by Gardner, 1985, 2006).
Assuming a doubly infinite sample, the one-step-ahead predictions,μ ̃t+ 1 |t, and
the filtered and smoothed estimates of the trend component, denotedμ ̃t|∞, are
given, respectively, by:

μ ̃t+ 1 |t= ̃μt|t=( 1 +θ)

∑∞

j= 0

(−θ)jyt−j, μ ̃t|∞=

1 +θ

1 −θ

∑∞

j=−∞

(−θ)|j|yt−j.

Here,μ ̃t+ 1 |tdenotes the expectation ofμt+ 1 based on the information available
at timet, whereasμ ̃t|∞is the expectation based on all of the information in the

doubly infinite data set. The filter w(L)=( 1 +θ)( 1 +θL)−^1 =( 1 +θ)

∑∞

j= 0 (−θ)

jLj

is known as a one-sided exponentially weighted moving average (EWMA). These
expressions follow from applying the Wiener–Kolmogorov prediction and signal
extraction formulae (see Appendix B). In terms of the structural form parameters,

μ ̃t|∞=
ση^2
ση^2 +σψ^2 | 1 −L|^2
yt, where| 1 −L|^2 =( 1 −L)( 1 −L−^1 ). The filter:

wμ(L)=

ση^2

ση^2 +σψ^2 | 1 −L|^2

=

1 +θ

1 −θ

∑∞

j=−∞

(−θ)|j|Lj,

is known as a two-sided EWMA filter. In finite samples, the computations are
performed by the Kalman filter and smoother (see Appendix C).
The parameterθ(or, equivalently,λ) is essential in determining the weights
that are attached to the observations for signal extraction and prediction. When