Potential Output: Concepts and Measurement - Department of Labour

Darren Gibbs 87
The HP filter is derived by minimising the sum of the squared deviations of
a variable (in this case, output), Q t
, from its trend τ, subject to a smoothness
constraint, λ, that penalises squared variations in the growth of the trend series.
That is, the HP filter is calculated to minimise:
T
T−1
QHP =
∑ t − τ t + λ ∑
τ t − τ − τ t − τ t
t=
1
t=
2
[ + 1 1 −1
]
2
( Q ) ( ) ( )
Thus, the calculated HP trend series is a function of the smoothness constraint,
and of both past and future values of output. The user can determine the
smoothness in the trend series by choosing an appropriate value for the
smoothness parameter. Higher values of λ imply a larger weight on the
smoothness of the series, so that in the limit, as λ becomes arbitrarily large, the
trend series will converge on a linear time trend. As Laxton and Tetlow note, this
would be consistent with an extreme Keynesian model in which supply shocks
are deterministic, and variations in output come almost entirely from demand
shocks. Conversely, a very small value of λ will effectively eliminate the penalty
function, and thus the HP trend series will be equal to the actual series. This
would be consistent with an extreme real-business-cycle model in which most
variations in output are also variations in potential or trend output, and hence
are driven by supply shocks.
King and Rebelo (1989) show that the optimal value of λ is a function of the
ratio of the variances of these shocks and that, for some data generating process
of permanent and temporary shocks, the HP filter is an optimal (minimumvariance)
linear inverse filter. However, there is considerable debate concerning
the relative variance of supply and demand shocks. In practice, users of the HP
filter have tended to set λ = 1600 when smoothing quarterly data following the
initial approach of Hodrick and Prescott. An alternative approach is to choose λ
so as to generate an output gap series which is consistent with the results of
other business cycle dating techniques.
Relative to the trends through peaks or linear time trend approach, the HP
filter approach has the advantage that it allows for structural change to occur at
any point in the series (although the rate of change is constrained by the choice
of λ). However, this approach still has a number of weaknesses.
First, the choice of λ can have a large effect on the conclusion reached, as can
the choice of sample period chosen. If an ‘average’ level of λ is chosen, any
dramatic change in the actual series will tend to be attributed to both demand
and supply disturbances. Even if this attribution is correct on average, in the case
of pure positive demand shocks the technique will tend to understate the
amount of excess demand in the economy, while understating the amount of
excess supply in the case of pure positive supply shocks. For example, during a
period of disinflation when monetary conditions are being tightened, the HP
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