Controlling the Transfer Function in the Analytic DFA: an Alternative Weighting Approach

1. Introduction

In order to illustrate the topic I'll rely on the structure of a practically important estimation problem, namely the real-time estimation of the business-cycle. For that purpose I'll assume that a series is filtered by a bandpass filter with bandpass [1,8] years, see the following R-code analytical_dfa_bp_restriction.r (newest version from 13.08.2010). Note that I apply this filter to a random-walk process in the R-code which is unlikely to be very appealing to economists but this is a toy example and I'm treating a statistical estimation problem here. Economists often complain that real-time estimates of cycles aren't in time. I'll propose a filter design which extracts a particular component `exactly': the amplitude function equals one and the time-shift vanishes. Moreover, the quality of this approximation in the vicinity of the targetted frequency can be arbitrarily improved (at costs of poorer performances in the stop band). Finally, I'll show that the optimization is still quadratic i.e. the problem is tractable analytically.  

2. The Criterion

The minimum revision error criterion can be seen in expression 5 in Wildi10:

sum over k of |Gamma(omega_k)-\hat{\Gamma}(omega_k)|^2 I{NX}(omega_k)

Now assume that we add a weighting function W(omega_k):

sum over k of W(omega_k)*|Gamma(omega_k)-\hat{\Gamma}(omega_k)|^2 I_{NX}(omega_k)

If this weighting function is growing unboundedly towards a prespecified frequency omega_0 different from zero, then we expect that :

\hat{\Gamma}(omega_0) converges to Gamma(omega_0)

Assuming that the periodogram doesn't vanish in omega_0 (otherwise the component would not exist and the whole extraction operation would be meaningless, of course). But since Gamma(omega_0) is a real number, \hat{\Gamma}(omega_0) must be real too and therefore

• The time-shift vanishes in omega_0
• The amplitude match is perfect

Note that from a philosophical perspective I prefer the following additive weighting scheme which is implemented in my R-code (but you could straightforwardly the above multiplicative scheme):

sum over k of Gamma(omega_k)-\hat{\Gamma}(omega_k)|^2 * (W(omega_k)+I_{NX}(omega_k))

i.e. the weighting supplied by the user is additive. This manner the user, through his particular weighting, acts like an independent process whose spectral density - W(omega) - adds to the original spectrum of the process as measured by the periodogram, I_{NX}(omega_k). I like this idea of the `independent parallel' user much more  than the `serial' user and it seems much more natural, voilà.

3 The Weighting Function: ZPC-Filters

In order to obtain the intended perfect fit in omega_0 we need a weighting function which behaves like the pseudo-spectral density of an I(1)-process with unit-root in frequency omega_0, see chapter 6 in Wildi2008. The corresponding operator would look like this

W'(omega)=1/|P e(i omega)-e(i omega_0)|^2 

where P=1. But there is a much smarter way to proceed by adjoining a MA(1) in the numerator to obtain

W''(omega)=|Z e(i omega)- e(i omega_0)|^2 / |P e(i omega)-e(i omega_0)|^2 

with P=1 and Z>=P real numbers. For Z=P an identity is obtained and for Z>P a unit-root weighting results. This kind of filters, called ZPC-filter is used in the traditional DFA and it is described at length in section 3.2 in Wildi2008. The trick is that the real number Z>=1 controls for the width of the `spectral peak' in the weighting function W(omega) and therefore the local match of Gamma by \hat{Gamma} around omega_0 can be controlled. The following graph illustrates the `tightening' effect obtained by varying Z for omega_0=pi/24 (amplitude is truncated above 10...).

Please note that 1000 corresponds to pi and 500 to pi/2 i.e. x-axis are frequencies. Added 13.08.2010: Before proceeding further it is useful to account for undesirable nuisances by normalizing this weighting function:

W'''(omega):=W''(omega) / (|Z e(i pi)- e(i omega_0)|^2 / |P e(i pi)-e(i omega_0)|^2 )

Thus W'''(pi)=1. Finally, in an additive setting (my preferred) one I'm using

W(omega):=W'''(omega)-1

i.e. W(omega) is scaled and satisfies W(pi)=0. The following graph is generated in my code:

We have now the essential ingredients to control amplitude and time-shift of the real-time filter towards omega_0 through a very flexible weighting function W relying on a zpc-filter design.

4. Examples

4.1 Varying P

The example at the end of my code entitled "# Example: P is varied" proposes a bandpass filter whose weighting function gradually converges to a unit-root with frequency pi/24 i.e. P converges towards 1 (it's not equal to one due tu numerical issues...).

The following graph illustrates how amplitude and time-shift gradually converge towards 1 and 0 respectively as P approaches 1 (up-dated 13.08.2010):

One can see that the amplitude function is approaching 1 in pi/24. And here we got the detail around pi/24 (which corresponds to the x-value 3) up-dated 13.08.2010:

The convergence of both amplitude and time shift in pi/24 as P approaches 1 (the pole approaches the unit circle) are clearly visible. Interesting effects could be obtained by varying the filter length L but I'll let you experiment with the code (imposing constraints requires sufficiently flexible designs i.e. a `large' L).

 4.2 Varying Z: First Order Restriction

The following is based on the example called "# Example : first order restriction, Z is varied" in my code. P is now fixed to 1 (unit-root weighting: for numerical reasons I'm using P slightly larger than one...) and Z is varied. Let me just briefly reemphasize what I would like to do here:

• I'd like to impose a unit-root constraint at frequency pi/24 such that amplitude and time-shift functions of the real-time bandpass filter match exactly the target signal (in pi/24).
• The unit-root weighting function W(omega) relies on a ZPC design which enables to tighten or to widen the constraint around pi/24.
• A larger Z implies a broader weighting i.e. the `quality' of the approximation towards omega_0 will improve at costs of poorer filter characteristics in the stopband, for example.
• In order to show these features, however, the real-time filter must be sufficiently flexible to allow for `flatter' approximations towards omega_0: I need to address a large filter order.

For my purpose I selected an MA of length 300 (overfitting will be unavoidable but that's not my point here). The following graph plots amplitude and time-shift functions when Z is varied according to my code (click on the graph to magnify) up-dated 13.08.2010.  

The `outlier' (red line in the graph) corresponds to Z=P i.e. W=1: no emphasize of frequency omega_0. All other filters satisfy A(pi/24)=1 and Phi(pi/24)=0 - up to rounding errors which are due to the fact that P is not exactly one - but the quality of this approximation depends on Z. The following graph magnifies the above plot around pi/24 (which corresponds to x=3 on the abscissa) up-dated 13.08.2010:

(I removed the Z=P filter, red line in the previous graph) One can see that the slope of the time-shift (recall that the time shift is the phase divided by the frequency) in the lower graph decreases such that the shift is closer to 0 in a wider range of frequencies around pi/24 for increasing Z. Similarly, the amplitude is closer to one in wider intervall around pi/24. Interestingly, the better/wider approximation around pi/24 translates into poorer stopband properties as can be seen in the previous graph: the amplitude is increasing towards pi which means that more noise will alter the output signal (no free lunch!).

You can run my code and try this example and you could try smaller filter orders too:

• the smaller the filter order the more constrained the design and the more difficult it will be to control for the quality of the local fit around pi/24 (ultimately the identity will result i.e. no filtering at all).
• This is one of the main advantages of the traditional DFA - which is unfortunately intractable analytically -: it is much more flexible and parsimonious so that much less parameters are needed to obtain similar or better approximations.