390 Structural Time Series Models
θ=0,ytis a pure random walk, and then the current observation provides the best
estimate of the trend:μ ̃t+ 1 |t= ̃μt|t= ̃μt|∞=yt. Whenθ=−1, the trend estimate,
which is as smooth as possible, is a straight line passing through the observations.
The RWpN model provides a stripped to the bone separation of the transitory and
permanent dynamics that depends on a single smoothness parameter, which deter-
mines the weights that are assigned to the available observations for forecasting
and trend estimation. Its use as a misspecified model of economic fluctuations for
out-of-sample forecasting, using multi-step (or adaptive) estimation, rather than
maximum likelihood (ML) estimation, has been considered in the seminal paper
by Cox (1961) and by Tiao and Xu (1993). Proietti (2005) discusses multi-step
estimation of the RWpN model for the extraction of trends and cycles.
9.2.2 The local linear model and the Leser–HP filter
In the local linear trend model (LLTM) the trendμtis an integrated random walk:
yt = μt + ψt, ψt∼NID(0,σψ^2 ), t=1, 2,...,n,
μt = μt− 1 + βt− 1 + ηt, ηt∼NID(0,ση^2 ),
βt = βt− 1 + ζt, ζt∼NID(0,σζ^2 ).(9.2)It is assumed thatψt,ηtandζtare mutually and serially uncorrelated. Forσζ^2 = 0
the trend reduces to a random walk with constant drift, whereas forση^2 =0 the
trend is an integrated random walk (^2 μt=ζt− 1 ).
The above representation encompasses a deterministic linear trend, arising when
bothση^2 andσζ^2 are zero. Second, it is consistent with the notion that the real time
estimate of the trend is coincident with the value of the eventual forecast function
at the same time (see section 9.2.5 on the Beveridge–Nelson decomposition).
The LLTM is the model for which the Leser filter is optimal (see Leser, 1961). The
latter is derived as the minimizer, with respect toμt,t=1,...,n, of the penalized
least squares (PLS) criterion:
PLS =∑n
t= 1 (yt−μt)(^2) +λ∑n
t= 3 (
(^2) μ
t)
(^2).
The parameterλgoverns the trade-off between fidelity and smoothness and it is
referred to as thesmoothnessorroughness penaltyparameter. The first addend of
PLSmeasures the goodness-of-fit, whereas the second penalizes the departure from
zero of the variance of the second differences (i.e., a measure of roughness). In
matrix notation, ify=(y 1 ,...,yn),μ=(μ 1 ,...,μn), andD={dij}is then×n
matrix corresponding to a first difference filter, withdii =1,di,i− 1 =−1 and
zero otherwise, so thatDμ=(μ 2 −μ 1 ,...,μn−μn− 1 )′, we can write the criterion
function asPLS=(y−μ)′(y−μ)+λμ′D^2
′
D^2 μ. Differentiating with respect to
μ, the first-order conditions yield:μ ̃=(In+λD^2
′
D^2 )−^1 y. The rows of the matrix
(In+λD^2
′
D^2 )−^1 contain the filter weights for estimating the trend at a particular
point in time. The solution arising forλ=1600 is widely known in the analysis
of quarterly macroeconomic time series as the Hodrick–Prescott filter (henceforth,
HP; see Hodrick and Prescott, 1997); the choice of the smoothness parameter for