Tommaso Proietti 407leakage from the non-stationary component. Moreover, the filter seriously distorts
the evidence for the comovements among detrended series.
The issue of spuriousness is problematic, at least, if not tautological. The main
difficulty stems from the fact that it ties in with a more fundamental question
concerning what is indeed the cycle in economic time series. If we adhere to the
bandpass paradigm of viewing the cycle as consisting of those fluctuations within
a give range of periodicity, than the case for spuriousness is much less compelling.
Another source of concern among practitioners, especially for the conduct of
monetary policy, relates to the end-of-sample behavior of the Leser–HP filter: the
real-time estimates would be subject to “end-of-sample bias,” since they result from
the application of a one-sided filter and will suffer from both phase shifts and ampli-
tude distortions. One has to separate two issues: as we hinted before, the IMA(2,2),
for which the Leser–HP filter is optimal, is usually misspecified for macroeconomic
time series. As a result, the cycle estimates have no optimality properties. Model-
based bandpass filtering is aimed at overcoming this limitation. Having said that,
it is a fact of life that, for a correctly specified model, the optimal real time signal
extraction filter will be one-sided and thus will produce phase shifts and amplitude
distortions.
9.3 Multivariate models
Information on the output gap is contained in macroeconomic variables other
than aggregate output, either because those variables provide alternative measures
of production, or because they are functionally related to the output gap. In this
section we start from the consideration of a bivariate model that, along with an
output decomposition, includes an inflation equation. We then extend the model
to include other variables, such as the unemployment rate and industrial produc-
tion, and consider the estimation of a monthly model using quarterly observations
on real GDP.
9.3.1 Bivariate models of real output and inflation
Price inflation carries relevant information for the output gap. The definition of
the latter as an indicator of inflationary pressure and, correspondingly, of potential
output as the level of output consistent with stable inflation, makes clear that a rig-
orous measurement can be made within a bivariate model of output and inflation,
embodying a Phillips curve relationship. The Phillips curve establishes a relation
between the nominal price, or wage, inflation rate,pt, where, e.g.,ptis the log-
arithm of the consumer price index (CPI), and an indicator of excess demand,
typically the output gap (ψt).
A general specification is the following:
δ(L)pt=c+θψ(L)ψt+γ(L)′xt+ξpt, (9.21)wherecis a constant,xtdenotes a set of exogenous supply shocks, such as changes
in energy prices and terms of trade, andξptis WN. Often the restriction is imposed