Palgrave Handbook of Econometrics: Applied Econometrics

Tommaso Proietti 423

predictive validity, as possible bias would emerge. This criterion is adopted by
a number of authors; e.g., Camba-Mendez and Rodriguez-Palenzuela (2003) and
Proietti, Musso and Westermann (2007) assess the reliability of alternative output
gap estimates through their capability to predict future inflation.

9.4.2 Precision

A measurement method is precise if repeated measurements of the same quantity
are in close agreement. Loosely speaking, precision is inversely related to the uncer-
tainty of an estimate. In the measurement of immaterial constructs the sources of
uncertainty would include: (i)parameter uncertainty, due to the fact that the core
parameterscharacterizing modelM, such as the variance of the disturbances
driving the components and the impulse response function, are unknown and
have to be estimated; (ii)estimation error, the latent components are estimated
with a positive variance even if a doubly infinite sample onytis available; (iii)
statistical revision, as new observations become available, the estimate of a signal is
updated so as to incorporate the new information; (iv)data revision.
The first source can be assessed by various methods in the classical approach; it is
automatically incorporated in the interval estimates of the output gap if a Bayesian
approach is adopted, as in section 9.3.2.2. The methods rely on the fundamental
result that, under regularity conditions, the ML estimator ofhas the asymptotic
distribution ̃∼N(,V), whereVis the inverse of the information matrix. Hamil-
ton (1986) proposed a Bayesian marginalization approach, which uses ̃∼N(,V)
as a normal approximation to the posterior distribution of, given the available
data. Then, a measure of the uncertainty of the smoothed estimates of the output
gap, which embodies parameter uncertainty, is:

Var̂(ψt|F)=^1

K

∑K

k= 1

Var(ψt|F, ̃(k))]+

1

K

∑K

k= 1

[

E(ψt|F, ̃(k))−Eˆ(ψt|F)

] 2

, (9.27)

whereEˆ(ψt|F)=^1 K

∑K

k= 1 E(ψt|F, ̃

(k)), and the ̃(k)s are independent draws from

the multivariate normal density N( ̃,V ̃),k=1,...,K, whereV ̃is evaluated at ̃.
According to thedelta methodproposed by Ansley and Kohn (1986), expressing the
output gap estimates as a linear function of the parameter estimation error− ̃
gives:

Var̂(ψt|F)=Var(ψt|F,) ̃ +d() ̃′Vd ̃ () ̃, d() ̃ = ∂

∂

E(ψt|F,) ̃

∣∣

= ̃, (9.28)

where the derivatives ind() ̃ are evaluated numerically using the support of the
Kalman filter and smoother. Similar methods apply for the real-time estimates,
with the ML estimator being based on the information setFt. Quenneville and
Singh (2000) evaluate and compare the two methods, and propose enhancements
in a Bayesian perspective.
In an unobserved component framework the Kalman filter and smoother provide
all the relevant information for assessing (ii) and (iii). For the latter, we can keep
track of revisions by using afixed-point smoothingalgorithm (see Anderson and
Moore, 1979; de Jong, 1989).