Palgrave Handbook of Econometrics: Applied Econometrics

Tommaso Proietti 403

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00

0.2

0.4

0.6

0.8

1.0 BK

Ideal

Bandpass m = r = 6

HP bandpass (m = 2, r = 0)

Figure 9.3 Gain function of the ideal business cycle bandpass filter, the BK filter and two
model-based filters

usingm=r=6 and the two cut-off frequenciesωc 1 = 2 π/32 (corresponding to
a period of eight years for quarterly data) andωc 2 = 2 π/6 (one and a half years);
such large values of the parameters yield a gain which is close to the ideal boxcar
function. The HP bandpass curve is the gain of the Wiener–Kolmogorov filter for
extracting the componentψtin (9.20) withm=2,r=0, andωc 1 ,ωc 2 given above.
In this case the leakage is larger but, as shown in Proietti (2004), taking large values
ofmandris detrimental to the reliability of the end of sample estimates.

9.2.7 Applications of model-based filtering: bandpass cycles and the
estimation of recession probabilities

We present two applications of the model-based filtering approach outlined in the
previous section. Our first illustration deals with the estimation and the assessment
of the reliability of the deviation cycle in US GDP. The cycle is defined as the
highpass component extracting the fluctuations in the level of log GDP that have
a periodicity smaller than ten years (40 quarters). To evaluate model uncertainty,
we fit three models to the logarithm of GDP, namely a simple random walk, or

ARIMA(0,1,0) model (σˆ^2 =0.8570), an ARIMA(1,1,0) model (the estimated first-
order autoregressive coefficient isφˆ=0.33 andσˆ^2 =0.8652), and finally, we
consider the ARIMA(2,1,2) model fitted in section 9.2.5, whose parameter estimates
were reported in Table 9.1.
The estimates of the lowpass component corresponding to the three models
obtained by settingm=2,r=0 (and thusλ=1600 andωc=0.158279) are
displayed in the top right-hand panel of Figure 9.4, along with the Leser–HP cycle.