Can Nosotros Falsify Models Alongside Time-Varying Parameters?
Introduction
In previous articles (first inwards series), I discussed the concept of forecastability: is it possible to forecast the futurity outputs of a mathematical model, based on a past times history of world information? (The Definition needs to distinguish betwixt the information available to the forecaster versus all the information inwards the model.) It should survive emphasised that this is a holding of mathematical models; whether it extends to the existent footing depends upon whether nosotros believe a detail model is a adept representation of reality.The normal justification of using mathematical economical models is that nosotros tin purpose them to brand forecasts, as well as thus i mightiness assume that the model existence forecastable is a useful property. My sentiment is that nosotros genuinely wishing the opposite: since existent footing forecasting exercises tend to neglect inwards detail ways, nosotros wishing models that are similarly difficult to forecast. For example, the Quantity Theory of Money provides a straightforward model that is easily forecast. And inwards the existent world, the Quantity Theory of Money is easily rejected empirically (by those of us who do non wishing to come upward up amongst stories to manipulate the information thus that they gibe the desired theory).
What are Parameters?
When doing mathematics, nosotros demand to survive careful of what definitions nosotros use. One tin legitimately purpose the same English linguistic communication discussion inwards dissimilar ways, thus long every bit y'all arrive clear what Definition y'all use. My intuition for parameters comes from engineering, as well as they would unremarkably survive fixed constants. (A ameliorate term for my intuition is coefficient.)If nosotros begin amongst linear systems theory, nosotros are interested inwards the evolution of a laid of state variables, which nosotros stack into a vector denoted $x$ past times tradition. If nosotros confine ourselves to the laid of linear, time-invariant, finite dimensional systems, the solid soil dynamic equation is of the form:
$$
x(t+1) = Influenza A virus subtype H5N1 x(t) + B u(t),
$$
where $x$ is the state, $u$ is a vector of input variables, spell $A$ as well as $B$ are fixed matrices of the appropriate size. (In economics, the models are frequently of the shape $x(t) = Ax(t-1) + Bu(t)$, that is, at that topographic point is no lag from the output to the solid soil variable.) The elements of $A$ as well as $B$ are fixed parameters.
For example, a elementary chemical compound increment arrangement amongst a increment charge per unit of measurement of 1% is given by:
$$
x(t+1) = 1.01 x(t) + u(t)
$$
(with $x,u$ existence single-dimensional fourth dimension series). In this case, the $A$ matrix simply has i chemical ingredient ($a_{1,1}$), which has a fixed value of 1.01.
Importantly, amongst a linear system, nosotros cannot select the dynamic equation incorporate expressions where the solid soil variables multiply each other (or other complex expressions). There is a strict segmentation betwixt fixed parameters (the $A, B$ matrices above) as well as solid soil variables, which vary over time.
In economical models, nosotros select complicated systems where nosotros do select solid soil variables interacting inwards a nonlinear way. For example, if nosotros render to model SIM (which I discussed inwards the previous article; run across the references therein), the consumption constituent depends on multiplying income as well as wealth past times the alpha parameters (which are the propensity to eat out of income as well as wealth). If those parameters are fixed (as I assumed inwards the previous article), nosotros cease upward amongst a linear fourth dimension invariant system. However, i could imagine a specification of the model where the alpha parameters themselves modify every bit a constituent of time. They are elevated to existence solid soil variables, as well as nosotros select a nonlinear system. As a result, the intuition for economists for what constitutes a parameter would survive variables that are non economical fourth dimension series, which are either fixed coefficients or allowed to vary.
(In my Python sfc_models framework, all variables are assumed to survive fourth dimension series; the alone fixed parameters would survive the ones that are hard-coded every bit constants.)
As I noted inwards the previous article, forecasting the output of a linear arrangement amongst fixed parameters is straightforward (assuming that cardinal variables are conduct measured, which is the commons condition). Things are rattling dissimilar if nosotros permit for time-varying parameters.
Simplest Stock Market Model
If nosotros announce the total render index for a stock marketplace every bit $x$, the simplest possible stock marketplace model is:$$
x(t+1) = a(t)x(t),
$$
(where $a(t)$ tin survive thought of every bit $1 + r(t)$, amongst $r$ the one-period return).
So long every bit the total render index stays away from zero, this model tin reproduce whatsoever observed fourth dimension history for returns. (If the total render index goes to zero, it stays there, according to this model.)
This is non a useful model for reality, but the query is: why?
The showtime affair to examine is forecastability. If nosotros assume that the fourth dimension serial $a$ is non world information, the model is non-forecastable. This fits amongst the empirical results regarding marketplace efficiency, amongst the simplest version is that it is difficult to forecast stock marketplace returns.
The side past times side affair to intend most is statistical tests. However, thus long every bit our information laid does non select the stock index going to cypher as well as and thus bouncing back, this model tin easily reproduce whatsoever historical fourth dimension series. (One could imagine a stock index disappearing because of a nation of war or like calamity, as well as and thus a novel laid of equities arising thereafter. From the perspective of the master copy owners, the novel index would survive a dissever entity.) In other words, it volition ever move past times statistical tests of validity. This is i ground I do non run across a blind reliance on statistical tests every bit existence best practices for assessing the usefulness of models.
Instead, the ground why this is a bad model seems to survive that it provides no useful information (other than the supposition that stock markets volition non larn to cypher as well as and thus rise). It is every bit good flexible: it tin gibe whatsoever observed information set.
In communication systems theory, the notion of information is quantifiable: it is related to a current of bytes y'all are sending downwardly about channel. Unfortunately, inwards this context, the declaration that a model does non bring useful information does non seem to survive conduct quantifiable. However, nosotros mightiness survive able to capture the thought past times comparison the model to other models. Is it possible to give away a shape of models coming from a smaller laid of models that also tin survive gibe to historical data? In this case, nosotros could select models amongst stochastic volatility, which also tin survive gibe to historic data, but cannot reproduce the total laid of deportment shown past times assuming no restrictions on menstruum returns.
Alternatively, i mightiness run across that the model is thus flexible that it tin survive gibe to the output of a broad diversity of models that select theoretical content that contradicts that of the model inwards question. In this case, i could struggle that the model cannot falsified, every bit it tin survive gibe to a almost whatsoever plausible laid of data.
Mainstream Macro
The job amongst mainstream macroeconomic theory is that it is every bit good heavily reliant upon time-varying parameters, such every bit the natural charge per unit of measurement of unemployment or the natural involvement charge per unit of measurement (or whatever fad terms select replaced those names for the concepts). The models seem to offering about information most futurity developments inwards the economy, but non a lot: since those cardinal parameters cannot survive conduct measured, nor forecast.The agency to evidence this is run across whether mainstream economical techniques could survive gibe to other classes of models, peculiarly those that rely on dissimilar underlying assumptions (e.g., stock-flow consistent models). Given the flexibility of the natural involvement charge per unit of measurement inwards adapting to real-world data, I run across such a gibe existence achieved.
In other words, the statistical tests used to "prove" that the policy charge per unit of measurement drives action inwards the existent footing could survive applied to models inwards which nosotros know the policy charge per unit of measurement has no such effect. This is theoretically awkward, to say the least.
Concluding Remarks
Allowing for drift inwards cardinal parameters is i agency to brand model deportment to a greater extent than interesting. The terms is that the model may cease upward offering no theoretical content, as well as existence unable to survive rejected empirically.(c) Brian Romanchuk 2018
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