1. Structural Approach Potential output - part I
Output Gap detection: all the different approaches Luxembourg, 8-10 June 2016
CONTRACTOR IS ACTING UNDER A FRAMEWORK CONTRACT CONCLUDED WITH THE EUROPEAN COMMISSION

2. Where are we heading to? 𝑌=𝑇𝐹𝑃∙ 𝐿 𝛼 𝐾 1−𝛼
We have provided an application of the structural approach to the estimate of potential output.
In this application, the structural model hinges on a specification of the aggregate production function.
Potential output is determined based on potential labor input, capital stock, and the TFP trend, using a Cobb-Duglas CRTS technology:
Key measurement issues concern potential labor input and TFP.
𝑌=𝑇𝐹𝑃∙ 𝐿 𝛼 𝐾 1−𝛼

3. TFP decomposition
The new TFP method uses a bivariate Kalman Filter (KF), which exploits the link btw. the TFP cycle & the degree of capacity utilization.
This new approach was endorsed by the EU’s Economic Policy Committee (with the Output gap Working Group - OGWG) in December 2009, and adopted in the Autumn 2010 forecasting exercise.
The basic problem with the existing HP filter method is that such univariate techniques tend to produce imprecise estimates at the end of the sample period (especially close to turning points / "boom-bust" episodes).
Compared with HP, the KF leads to less trend TFP revisions, which has important positive gains for policy makers in helping to reduce the degree of uncertainty pertaining to fiscal policy decision making.

4. HP main shortcomings
The “end-point problem” makes the HP filter very sensitive to sharp cyclical turns.
The degree of optimism / pessimism in the last observation can provoke sizeable jumps in the trend, with the danger of producing very misleading signals.
To see the scale of the problem, we display the TFP trend for the EU15 aggregate and its growth rate.

5. Source: European Economy-Economic Papers 420, July 2010
Each curve represents a specific TFP time series estimates of the period , using data available every fifth year from 2000 to Revisions are sizable!

6. KF bi-variate method
KF does not suffer from the end-point problem and can exploit economic information that can improve estimates and predictions.
A particular important variable, revealing information about TFP-trend evolution, is the degree of utilization of production capacity U.
U strongly co-move with the unobserved cyclical component of TFP, hence enabling unbiased extraction of the TFP cycle even at the end of the sample.

7. TFP and capacity utilization composite indicator series for the EU15 are strongly correlated (first-difference), corr.=0.85.

8. Setting up the inference problem
𝑇𝐹𝑃=𝑃×𝐶, and in logs, 𝑡𝑓𝑝=𝑝+𝑐
We want to decompose the tfp in cycle and trend components.
This is done, given the definition of p,c and the assumption that efficiency is a persistent phenomenon (i.e. acyclical). Thus, capacity c captures the TFP cyclical component.
Bear in mind that efficiency p is unobservable, while for c we have some data.
In the essence: we have data on tfp and on 𝑼 𝑲 (a part of c) and want to detect (p,c).

9. In logs, capacity is,
The labor component 𝑢 𝐿 may be correlated with 𝑢 𝐾 if their cyclical component is correlated.
In the production function, cyclicality of labor enters already through L, which depends on the hours of work. However, to keep track of the possibility that some cyclicality is not captured by the latest, it is assumed that,

10. Using the second to substitute out 𝑢 𝐿 in the capacity equation:
The tfp equation becomes:
Rewritten as

11. Bivariate model:
2 observable variables 𝑧 𝑡 ′≡(𝑡𝑓 𝑝 𝑡 , 𝑢 𝐾,𝑡 )′
4 unobservable «states» 𝑥 𝑡 ′≡ 𝑝 𝑡 , 𝜇 𝑡 , 𝑐 𝑡 , 𝑒 𝑢,𝑡 ′
3 process noise terms 𝑤 𝑡 ′≡ 𝑎 𝜇,𝑡 , 𝑎 𝑐,𝑡 , 𝑎 𝑢,𝑡 ′~𝑚𝑁(0,𝑄)

12. In the KF format

13. KF alghoritm At step t, given 𝑃 𝑡|𝑡−1 , 𝑥 𝑡|𝑡−1
Measurement update in t
compute 𝐾 𝑡 using (∎)
observe 𝑧 𝑡 and update estimates to 𝑥 𝑡|𝑡 and 𝑃 𝑡|𝑡 using (∗)
Time update in t
𝑃 𝑡+1|𝑡 = 𝐹 𝑡+1 𝑃 𝑡|𝑡 𝐹 ′ 𝑡+1 + 𝑄 𝑡+1
𝑥 𝑡+1|𝑡 = 𝐹 𝑡+1 𝑥 𝑡|𝑡 + 𝐵 𝑡+1 𝑐 𝑡+1 ⋮
(∗)
𝐾 𝑡 = 𝑃 𝑡|𝑡−1 𝐻 𝑡 𝑇 𝐻 𝑡 𝑃 𝑡|𝑡−1 𝐻 𝑡 𝑇 −1
(∎)

14. Comparison HP vs. KF
EU15 – Actual & trend TFP growth estimates

15. Hodrick-Prescot. HP is very sensitive to the significant drop at the end of the sample: TFP trend estimates are strongly U-shaped btw , with the lowest point around 2009.
Kalman Filter. the trend growth estimate of the bivariate method is much smoother: no sharp fall in ; more moderate rebound after 2009.
KF predictions seem more realistic: there might have been some slow down in the growth rate of the TFP trend, due to the initial shock in 2008, but HP seems over-pessimistic.
Similarly, HP seems over-optimistic in the effect on the trend of the moderate recovery in

16. Where do we head to?
We still have to apply the PF methodology to estimate potential output, and the output gap.
To do so we are left to estimate the NAIRU
This will be done next.