A Desirable but Intriguing By-Product (in the Presence of Disruptive Turning Points)

In (1) I argued: "-- as a by-product -- performances (MSE-contributions by the time-shift) are implicitly emphasized at the practically relevant turning-points (the extrema) of the original series. In many applications, these are the time-points towards which a decision-maker is supposed to (well...) make a decision...", see (1) for the simple geometric argument. Isn't that feature, namely the implicit emphasis of turning points, both desirable and intriguing? 

What's About?

Let's illustrate the topic by relying on a short piece of code posted in (1): I apply a simple equally-weighted filter (lowpass) to a cosine; the code computes the cosine, the filtered cosine as well as the squared filter error and plots the series together with the squared filter error:


# Input series
omega<-pi/61
len<-120
x<-cos(1:len*omega)

# Filter weights: equally-weighted filter of length 12

L<-12
bkh<-rep(1/L,L)
# Normalization (so that scale of output is equal to scale of input)
norm_c<-abs(sum(bkh*exp(1.i*(0:(L-1))*omega)))
bk<-bkh/norm_c

# Filter series
yhat<-rep(NA,len)
for (i in L:len)
  yhat[i]<-sum(bk*x[i:(i-L+1)])

# Plot series, filtered series and squared error
par(mfrow=c(2,1))
ts.plot(x,col="blue",main="Series(blue), filtered series (red) and squared error (black)")
lines(yhat,col="red")
abline(h=0)
ts.plot((x-yhat)^2)

The plot should look like this:

One observes

• a shift of the filter output (red) 
• the (squared) error introduced by the time-shift (black-line): it is maximal at the zero-crossings (of the series) and minimal at the extrema
• notice that the error is entirely due to the time-shift because we normalized the filter (the scale of the output is the same as the scale of the input)

Emphasizing the time-shift (trying to reduce the shift of the above filter) would decrease MSE mostly (more specifically) at the zero-crossings. If the input series (blue line) corresponds to differences of a series (I'm generally working with log-returns instead of prices) then the zero-crossings would correspond to local extrema or turning-points of the data in levels. We now assume differenced data.

Back to the Topic

Emphasizing the time-shift (i.e. trying to reduce it) results automatically (by a simple geometric argument) in smaller MSE specifically at the turning points, at costs of all other time-points. Quite intriguing... because in practice (in a real-time perspective) one generally does not know whether a turning point is occurring or just occurred: as an example, declaring the end of the great recession in the US (June 2009) took the business-cycle dating committee more than a year to effectively commit (Sept. 2010). In spite of this 'ignorance' (about the occurrence or not of a turning point), controlling the time-shift (of the concurrent filter) would automatically -- by construction -- emphasize MSE (contributions by the shift) at the turning points in real-time. Is this desirable feature unconditional or is there a hidden assumption behind the geometric argument?

Assumption

In general, time series are neither smooth nor deterministic. So let's assume that our target is a smooth trend and that we work in (first) differences of a 'typical' economic time series ('typical here means: close to random-walk).

• Besides the trivial fact that the (differenced) series should cross the zero-line (otherwise the original series is just drifting away without turning points) the 'slope' of the (differenced) trend should be maximal at the zero-crossings, as was the case for our cosine above. In that case, reducing the shift would have a maximal effect on MSE (at the turning point).
• This means that the curvature (second derivative) of the trend in the original series (for example an asset-price or a macro-aggregate like GDP) should be 'large' (ideally maximal) in absolute value at the turning points, as was the case for the cosine.

I might be inclined to call such turning points 'disruptive' because the rate of change (acceleration/deceleration) is supposed to be large/maximal (in absolute value) in their vicinity.

Are turning points of typical economic series disruptive in the above sense? Maybe. But even if a majority of turning points aren't disruptive, the argument would still be valid for the few potentially most harmful ones.

• Not only would a control of the time-shift (of the filter) emphasize MSE-contributions at the turning-points, typically (at costs of all other time points); it would furthermore emphasize performances specifically at the potentially most harmful ones.

A way for controlling the time-shift is proposed later on (customization based on ATS-trilemma).

One (at least) small interjection is allowed with regards to macro-data:

• maybe the curvature is strong at the historical turning-points (in the revised data)
• but is it also strong in the real-time data (subject to revisions)?

Maybe.