A Very Brief Introduction To Time Domain Sampling
Systems engineers were forced to develop the theory of sampling in the post-war in response to the rise of digital electronics. All electronics are in fact analog (continuous time), but in a digital circuit, we have taken the convention of ignoring the value of signals except during the short periods when a timing circuit is active. We can therefore get away with treating the system as stepping between discrete points in time.
The chart above illustrates the mathematical operation of sampling. We have decided to sample two time series at a frequency of 1 hertz, that is, every second. The two series are two pure cosine waves, with a frequency of 1 and 2 respectively.
The red lines indicate the sample times, (t = ...,-1,0,1,2,...). The sampled discrete time series are defined by the value of the input continuous time series at those time instants.
We immediately see a problem: both sampled series are given by the constant series 1,1,1,... Under sampling, we cannot distinguish these series, nor can we even distinguish them from a pure constant.
A full explanation of this effect requires diving into frequency domain analysis techniques. But to summarise, a sampled series can only reproduce the low frequency components (the "spectrum") of the input signal; the range of reproduction is the set of frequencies [0,f/2), where f is the sampling frequency. In other words, with the sampling frequency of 1 in my example, we can only distinguish sinusoids with frequencies less than 0.5.
Higher frequency components do not disappear, they are "aliased" into a lower frequency. We see this in the example: the two high-frequency sinusoids effectively turn into constant signals, with a frequency of 0.
Effects On ModelsThe effect of sampling on models is a more complex topic. Since you are throwing out high frequency dynamics, you can end up with ghost dynamics that are associated with the sampling frequency (or one-half of the sampling frequency) and these dynamics are not associated with anything in the original continuous time system.
Luckily for control systems engineers, the physical dynamics of engineering systems are at a much lower frequency than the timing frequency of digital electronics. The fastest moving dynamics in an aircraft air frame are on the order of 10 hertz or so, whereas even 1960s era digital electronics ran at kilohertz frequencies (thousands of samples per second). The physical components are effectively "standing still" when compared to the electronics, so engineers can get away with some fairly basic approximations when translating a continuous time model to discrete time model. The high frequency dynamics created by sampling have no interaction with the physical system model, given the large gap in frequency responses.
That said, possession of an engineering degree does not mean that you will get it right. I was an anonymous referee on a paper written by three control systems engineering academics which developed a control strategy for a discrete time model. They took a standard control systems laboratory system (an inverted pendulum), sampled the model using a standard approximation, and then applied their control law to that discrete time system.
It looked good (at least to them) in discrete time. However, when I translated what was happening back to continuous time, I found that the tip of the inverted pendulum was whipping around at a significant fraction of the speed of light. Needless to say, I suggested that this was perhaps not the best way of designing control systems.
Returning to economics, we see some problems created by sampling. Economic models are often developed at a quarterly frequency (4 samples per year), which is a lower frequency than quite a bit of economic dynamics. For example, we see a seasonal ripple in many time series (for example, consumer spending around Christmas), which we cannot hope to capture in at a quarterly frequency. Importantly, American recessions are often dated to be only a several months in length (5-8 months, say), which is hard to translate into a quarterly time series. Within a recession, there is a downward self-reinforcing slide in activity, which is then arrested by the action of the automatic stabilisers. Capturing such differing dynamics in 1-2 time samples seems to be a difficult problem, particularly the action of the automatic stabilisers (which require deviation away from trend growth, presumably for more than one time point).
The inability to capture such fleeting dynamics may be one reason why many mainstream models are biased towards assuming that private sector activity is inherently stable. Since unstable dynamics disappear at the standard quarterly frequency, they cannot affect parameter estimates by definition.
The area of theory which faces the greatest problems are overlapping generations (OLG) models. In some classes of these models, a single time point is a "generation," and an individual lives for 2 or 3 periods. This implies a sample period of at least 20 years, which eliminates practically all business cycle dynamics.
I am not a fan of OLG models, as I discussed in earlier articles. Essentially, they are a way to put forth dubious theories to justify austerity policies; since the business cycle is eliminated, it is impossible to properly analyse the effects of fiscal policy and government debt. This is essentially how the mainstream found a way to ignore Functional Finance (which argues the way to gauge the effect of fiscal policy is to look at how it affects the business cycle): completely change the subject.
Not Just For ModelsMy comments here are not solely aimed at formal "models," even economy-watchers need to keep this in mind.
The NBER in the United States dates recessions using monthly data. This is certainly a better approach than the "two quarters of declining real GDP" that is used in other countries. In a slow growth environment, it may not be that hard to trigger two consecutive declining quarters, even though the slowdown was not material. Quarterly data are often quite rich, but they may be available with only too much of a lag to be useful.
Going to a higher frequency above monthly is also difficult. There are only a handful of series available at higher frequencies (job claims, monetary statistics), and they can easily be disrupted by unusual holidays or weather. We need to wait for monthly data -- which naturally tend to smooth out very short disturbances -- in order to confirm what weekly data are telling us.
Information Transfer Economics?This article is partially a response to claims made by the physicist Jason Smith in the article "More like stock-flow inconsistent."
He summarises his argument as:
TL;DR version: ΔΔ in SFC models [Stock-Flow Consistent models] has units of 1/time and therefore assumes a fundamental time scale on the order of a time step.As discussed above, any discrete time model has a "fundamental time scale" on the order of one time step; this is not a particular property of SFC models.
He complains that SFC model proponents describe their models as "just accounting," which misses how SFC modellers describe their modelling strategy. The initial hope was that the majority of model dynamics would be specified by the accounting relationships, but reality intervened: it was clear a long time ago that "behavioural" relationships matter a lot. Unfortunately, even superficially similar models could end up with quite different dynamics as a result of small changes to behavioural rules. We can use SFC models to gain an understanding of the economy, but we need to be realistic about what can be accomplished with reduced order models.
Smith advances what he calls "information transfer" models. They are largely based on analogies to models in physics. Unfortunately, he buries his discussion in a layer of physics (and his own) jargon, even though there is limited reason to believe that this will help the comprehension of readers who are not physicists. I have attempted to follow his logic, and I have had a hard time discerning the advantages of his suggested methodologies.
He advances continuous time models as being the "true" model for economic dynamics. Since monetary transactions settle at the end of day, the inherent frequency of transactions are truly only at a daily frequency. Although a much higher frequency than quarterly, that is a long way away from a hypothetical continuous time model.
Continuous time models also suffer from other defects.
- We have very little reason to be certain that solutions to the differential equations exist, or are unique, once we start to incorporate nonlinearities.
- Transactions, such as transfer payments, take the form of the transfer of non-zero amounts of money at particular times; continuous time models assume that there is a continuous infinitesimal flow between entities. The only way to replicate the large transfers is to invoke dubious concepts like the "Dirac delta function," which is not actually a function; and once we start using these delta "functions," we no longer can clearly characterise what space time series objects inhabit.
- Very few economic decisions, such as hiring a new worker, can be truly made at a high frequency.
- Continuous time models require an infinite number of state variables to simulate time delays. Time delays appear throughout economic analysis; even forward-looking agents cannot react instantaneously to new information.
Once we start to study economic aggregates, we are dealing with aggregated transactions across accounting time intervals. The behavioural relations that we get will end up as approximations, and those approximations may vary based on the sampling frequency.
Concluding RemarksThe sampling frequency of a model matters, and modellers should spend some time questioning the effect of the chosen sampling frequency. That said, it is unclear whether using a higher frequency beyond monthly will add any value for macroeconomic models. Quarterly frequency models are likely to be acceptable, but they may miss the unstable dynamics around recessions.
(c) Brian Romanchuk 2015