Models Are Non Frequency Invariant

If nosotros create multiple discrete fourth dimension models of the same mathematical system, in addition to those models are at dissimilar frequencies, those models volition outcome inward dissimilar outputs if those outputs are converted dorsum to a mutual frequency. For example, this agency that the output of an economical model which runs at a monthly frequency volition cause got dissimilar outputs than a quarterly model, if nosotros convert the monthly fourth dimension serial into quarterly. This is unfortunate, but this is to a greater extent than frequently than non going to endure a modest occupation relative to the other issues economical models face. We only require to worry nearly this upshot if nosotros cause got an extremely depression frequency, such every bit seen inward some overlapping generations (OLG) models.

I dubiousness that in that place is a whole lot of academic enquiry on this topic inside economics. This is solely reasonable, since this is a relatively unimportant effect. The only instance where it should cause got been heeded is inward the previously mentioned instance of OLG models.

Even inward systems theory, I cannot shout out upwards much inward the way of treatments of this topic. This is the despite the fact that existent globe technology scientific discipline systems are at i time largely all developed using discrete fourth dimension digital controls. Once again, my feeling is that the observations I am making hither are viewed every bit relatively obvious -- nosotros know that nosotros are losing information when nosotros sample a continuous fourth dimension arrangement (which converts it into a discrete fourth dimension system). The only argue that this champaign of report has come upwards up was a response to the writings of Jason Smith, a physicist, which I initiative of all responded to inward the article "Discrete Time Models And The Sampling Frequency." I promise that this article answers some questions that were raised inward readers' comments.

Terminology in addition to Assumptions

Within systems theory, nosotros refer to the concept of a state variable: a vector of fourth dimension serial which capture all of the dynamics of the system. (Other fourth dimension serial tin hand notice endure constructed every bit functions of the acre variable.)

For simplicity, nosotros volition assume that the model is time-invariant, in addition to the acre variable is finite dimension. Imposing these assumptions is non simply laziness; otherwise, nosotros are speedily inward a seat that nosotros tin hand notice say real picayune nearly the properties of the arrangement nosotros are talking about. Additionally, I volition simply refer to conversions betwixt monthly in addition to quarterly frequencies; the reader is complimentary to generalise the give-and-take to other frequency conversions.

Basic Example - Compound Interest

We tin hand notice convert frequencies for systems that practise non characteristic inputs that are external to the system.

The simplest illustration to await at is the instance of a chemical compound involvement model. There acre variable consists of a unmarried fourth dimension serial -- the banking venture balance b(t), which starts out at $100.

The monthly model is generated yesteryear having the banking venture residuum grow yesteryear 1% per month. Written out:

b(t+1) = (1.01) b(t);  b(0) = 100.

We tin hand notice thus create an equivalent quarterly model. It is a like model, except that the increment charge per unit of measurement per menses is larger. The only fox is that nosotros cannot utilization the gauge human relationship that the quarterly involvement charge per unit of measurement is triple that of the monthly; nosotros require to cause got into compounding over 3 periods. The right menses involvement charge per unit of measurement is (1.01)^3 = 1.030301.

The model is:

b(τ+1) = (1.030301) b(τ), b(0) = 100.

Note the 2 fourth dimension variables are non on the same calendar fourth dimension scale. The output at t=3 of the monthly model corresponds to the τ=1 menses for the quarterly model.

More General Models

Under the assumptions, nosotros tin hand notice depict whatsoever monthly model amongst the dynamics:

x(t+1) = f(x(t)), x(0) = x_0.

(Note: I cause got non figured out how to embed mathematical notation into these posts, since I desire to avoid confusing readers amongst unnecessary mathematics. The "x_0" should endure x amongst a subscript 0. Remember that x(t) is a vector of fourth dimension series, in addition to non simply a unmarried fourth dimension series,)

If the arrangement is linear, nosotros tin hand notice write:

x(t+1) = Influenza A virus subtype H5N1 x(t), x(0) = x_0,

where Influenza A virus subtype H5N1 is a northward x northward matrix.

We tin hand notice generate a quarterly model from the monthly yesteryear creating a novel dynamic system:

y(τ+1) = g(y(τ)) = f(f(f(y(τ))), y(0) = x_0.

In the linear case, nosotros we cause got a novel dynamics matrix, which is the master copy matrix A to the tertiary power. It is straightforward to verify that every tertiary value of y(t) volition equal x(t) from the monthly model.

Influenza A virus subtype H5N1 conversion going from quarterly to monthly is to a greater extent than complicated, but presumably tin hand notice endure done. In the linear model case, nosotros require to endure able to cause got the cube root of the matrix A (take the matrix to the ability 1/3). There is no guarantee that nosotros volition endure able to practise that operation, inward which instance nosotros cannot generate a fourth dimension invariant linear model.

Although nosotros may endure able to detect a novel model which matches the output of the original, in that place is no guarantee that it is inward the same cast of models. For example, cause got a cast of stock-flow consistent models inward the text yesteryear Godley in addition to Lavoie: they specified yesteryear a laid of behavioural parameters. There is no guarantee that a frequency converted model tin hand notice endure generated yesteryear some other alternative of those parameters. Moreover, tending has to endure taken when doing the conversion; if nosotros utilization approximations, the results would presumably endure mismatched. (Using my illustration chemical compound involvement system, that would stand upwards for to using a quarterly involvement compounding of 1.03 instead of 1.030301).

External Inputs Means That We Cannot Do Frequency Conversions

Unfortunately, every bit shortly every bit nosotros let for external inputs into the arrangement (exogenous variables inward economist jargon), nosotros tin hand notice no longer practise frequency conversions.

Imagine that nosotros alter our compounding involvement model to let for deposits or withdrawals, which is a fourth dimension serial that is external to the master copy arrangement that is denoted every bit u(t). In this case, nosotros start amongst an initial residuum of $0. The novel equation is:

b(t+1) = 1.01(b(t) + u(t)), b(0) = 0,

(This equation is maxim that deposits or withdrawals cause got upshot at the offset of the period, in addition to thus it affects the residuum on the adjacent fourth dimension point.)

We at i time volition await at what happens if nosotros laid u(t) to endure a $100 deposit at i fourth dimension point, in addition to 0 elsewhere.

• If nosotros deposit $100 inward the initiative of all calendar month (t=1), the banking venture residuum at t=3 volition endure $100*(1.01)^2  =  $102.01. 
• If nosotros instead brand the deposit inward the instant month, the banking venture residuum volition endure $101 at t=3.
• If the deposit is made at t=3, the residuum volition equal $100.

When nosotros switch to a quarterly frequency, nosotros cannot distinguish betwixt these cases: the monthly cash flows would endure aggregated to the same quarterly fourth dimension series, amongst an inflow of $100 at τ=1. When simulating the quarterly model, the residuum at  τ=1 would cause got to equal $100, since nosotros cause got no way of knowing whether the truthful cash current arrived slightly earlier, allowing for involvement to accumulate.

We cause got lost information every bit a outcome of moving to a lower frequency, in addition to in that place is no way of distinguishing the 2 dissimilar inputs. There is no way of defining the quarterly model to jibe the output of the monthly model. (This loss of information was the champaign of report of my master copy article.) The only way to avoid this loss of information is to enshroud monthly information inside the "quarterly" model, which agency that it is no longer a truthful quarterly model.

So What?

Models amongst dissimilar frequencies cannot reproduce each other's outputs exactly; this agency that the alternative of sampling frequency affects model outputs. This is truthful for any model, in addition to is non an artefact of SFC models, which was Jason Smith's claim.

However, the magnitude of the fault is driven yesteryear the "compounding" that occurs inside the high frequency model (when compared to the lower frequency model). So long every bit in that place is non a spectacular fourth dimension scale deviation (for example, a sample fourth dimension menses of monthly versus xx years), these mismatches are going to endure smaller than other sources of model error.

What About Continuous Time?

Since in that place are no continuous fourth dimension economical series, it makes picayune sense to insist upon the accuracy of moving from a continuous fourth dimension model to discrete fourth dimension model. A continuous fourth dimension model is already an approximation of truthful economical data.

However, if i insists upon starting from continuous time, i needs to consult the literature on the numerical approximation of differential equations. The message of that literature is straightforward: unless nosotros cause got a unopen cast solution of the differential equation, whatsoever discretisation of the differential equation is only simply about correct. The character of the fit depends upon the frequency composition of the continuous fourth dimension arrangement versus the sampling frequency, in addition to the character of the method of approximation.

In other words, modest errors are inevitable if nosotros create a model starting from a continuous fourth dimension approximation of an economical system.

(c) Brian Romanchuk 2015