I doubt that there is a whole lot of academic research on this topic within economics. This is entirely reasonable, since this is a relatively unimportant effect. The only case where it should have been heeded is in the previously mentioned case of OLG models.
Even in systems theory, I cannot recall much in the way of treatments of this topic. This is the despite the fact that real world engineering systems are now largely all developed using discrete time digital controls. Once again, my feeling is that the observations I am making here are viewed as relatively obvious -- we know that we are losing information when we sample a continuous time system (which converts it into a discrete time system). The only reason that this subject has come up was a response to the writings of Jason Smith, a physicist, which I first responded to in the article "Discrete Time Models And The Sampling Frequency." I hope that this article answers some questions that were raised in readers' comments.
Terminology and AssumptionsWithin systems theory, we refer to the concept of a state variable: a vector of time series which capture all of the dynamics of the system. (Other time series can be constructed as functions of the state variable.)
For simplicity, we will assume that the model is time-invariant, and the state variable is finite dimension. Imposing these assumptions is not just laziness; otherwise, we are rapidly in a position that we can say very little about the properties of the system we are talking about. Additionally, I will just refer to conversions between monthly and quarterly frequencies; the reader is free to generalise the discussion to other frequency conversions.
Basic Example - Compound InterestWe can convert frequencies for systems that do not feature inputs that are external to the system.
The simplest example to look at is the case of a compound interest model. There state variable consists of a single time series -- the bank balance b(t), which starts out at $100.
The monthly model is generated by having the bank balance grow by 1% per month. Written out:
b(t+1) = (1.01) b(t); b(0) = 100.
We can then create an equivalent quarterly model. It is a similar model, except that the growth rate per period is larger. The only trick is that we cannot use the approximate relationship that the quarterly interest rate is triple that of the monthly; we need to take into compounding over three periods. The correct period interest rate is (1.01)^3 = 1.030301.
The model is:
b(τ+1) = (1.030301) b(τ), b(0) = 100.
Note the two time variables are not on the same calendar time scale. The output at t=3 of the monthly model corresponds to the τ=1 period for the quarterly model.
More General ModelsUnder the assumptions, we can describe any monthly model with the dynamics:
x(t+1) = f(x(t)), x(0) = x_0.
(Note: I have not figured out how to embed mathematical notation into these posts, since I want to avoid confusing readers with unnecessary mathematics. The "x_0" should be x with a subscript 0. Remember that x(t) is a vector of time series, and not just a single time series,)
If the system is linear, we can write:
x(t+1) = A x(t), x(0) = x_0,where A is a N x N matrix.
We can generate a quarterly model from the monthly by creating a new dynamic system:
y(τ+1) = g(y(τ)) = f(f(f(y(τ))), y(0) = x_0.
In the linear case, we we have a new dynamics matrix, which is the original matrix A to the third power. It is straightforward to verify that every third value of y(t) will equal x(t) from the monthly model.
A conversion going from quarterly to monthly is more complicated, but presumably can be done. In the linear model case, we need to be able to take the cube root of the matrix A (take the matrix to the power 1/3). There is no guarantee that we will be able to do that operation, in which case we cannot generate a time invariant linear model.
Although we may be able to find a new model which matches the output of the original, there is no guarantee that it is in the same class of models. For example, take a class of stock-flow consistent models in the text by Godley and Lavoie: they specified by a set of behavioural parameters. There is no guarantee that a frequency converted model can be generated by another choice of those parameters. Moreover, care has to be taken when doing the conversion; if we use approximations, the results would presumably be mismatched. (Using my example compound interest system, that would correspond to using a quarterly interest compounding of 1.03 instead of 1.030301).
External Inputs Means That We Cannot Do Frequency ConversionsUnfortunately, as soon as we allow for external inputs into the system (exogenous variables in economist jargon), we can no longer do frequency conversions.
Imagine that we modify our compounding interest model to allow for deposits or withdrawals, which is a time series that is external to the original system that is denoted as u(t). In this case, we start with an initial balance of $0. The new equation is:
b(t+1) = 1.01(b(t) + u(t)), b(0) = 0,
(This equation is saying that deposits or withdrawals take effect at the beginning of the period, and so it affects the balance on the next time point.)
We now will look at what happens if we set u(t) to be a $100 deposit at one time point, and 0 elsewhere.
- If we deposit $100 in the first month (t=1), the bank balance at t=3 will be $100*(1.01)^2 = $102.01.
- If we instead make the deposit in the second month, the bank balance will be $101 at t=3.
- If the deposit is made at t=3, the balance will equal $100.
When we switch to a quarterly frequency, we cannot distinguish between these cases: the monthly cash flows would be aggregated to the same quarterly time series, with an inflow of $100 at τ=1. When simulating the quarterly model, the balance at τ=1 would have to equal $100, since we have no way of knowing whether the true cash flow arrived slightly earlier, allowing for interest to accumulate.
We have lost information as a result of moving to a lower frequency, and there is no way of distinguishing the two different inputs. There is no way of defining the quarterly model to match the output of the monthly model. (This loss of information was the subject of my original article.) The only way to avoid this loss of information is to hide monthly data within the "quarterly" model, which means that it is no longer a true quarterly model.
So What?Models with different frequencies cannot reproduce each other's outputs exactly; this means that the choice of sampling frequency affects model outputs. This is true for any model, and is not an artefact of SFC models, which was Jason Smith's claim.
However, the magnitude of the error is driven by the "compounding" that occurs within the high frequency model (when compared to the lower frequency model). So long as there is not a spectacular time scale difference (for example, a sample time period of monthly versus twenty years), these mismatches are going to be smaller than other sources of model error.
What About Continuous Time?Since there are no continuous time economic series, it makes little sense to insist upon the accuracy of moving from a continuous time model to discrete time model. A continuous time model is already an approximation of true economic data.
However, if one insists upon starting from continuous time, one needs to consult the literature on the numerical approximation of differential equations. The message of that literature is straightforward: unless we have a closed form solution of the differential equation, any discretisation of the differential equation is only approximately correct. The quality of the fit depends upon the frequency composition of the continuous time system versus the sampling frequency, and the quality of the method of approximation.
In other words, small errors are inevitable if we create a model starting from a continuous time approximation of an economic system.
(c) Brian Romanchuk 2015