Forecastability In Addition To Economical Modelling

When most people recollect most macroeconomics, what they want is the powerfulness to forecast economical outcomes. However, economists' (of all stripes) reputation every bit forecasters is non especially high. My sentiment is that this is non likewise surprising: what nosotros want forecasters to accomplish is in all likelihood impossible. (I am hardly the foremost mortal to greenback this, every bit variants of this thought cash inward one's chips dorsum at to the lowest degree to Keynes; I could non promise to offering a history of this idea.) However, I recollect if nosotros want to approach macro theory formally, nosotros ask to formalise the notion that outcomes cannot hold upwards forecast, which way nosotros ask to define non-forecastability formally.

This article gives 1 potential definition of forecastability, in addition to and then applies the concept to a uncomplicated stock-flow consistent (SFC) model. It should hold upwards noted that these are my preliminary thoughts, in addition to I believe that the definition volition ask to hold upwards refined.

(Note: I am piece of cake going through the editing of my breakeven inflation book; nonetheless cause got a few other projects going on. Since at that topographic point is no novel content to hold upwards added to the breakeven inflation book, I tin instantly start thinking most my adjacent writing projects. It is possible I mightiness do some other "greatest hits" compilation of weblog articles on banking, but otherwise it would hold upwards on concern wheel analysis. One possible angle for the concern wheel analysis is to organise the word some the notion of forecastability. If the analysis holds up, the championship of the volume mightiness hold upwards something similar "Forecastability of the Business Cycle." The analysis would hold upwards using stock-flow consistent models, in addition to that volition hold upwards the trigger for me to mass out my sfc_models Python framework.)

Analogy: Controllability

The thought of forecastability is past times analogy to controllability (and observability) from command systems theory. Once nosotros write downwards the mathematical specification of a nation infinite model of the organisation nosotros want to develop a command organisation for, nosotros tin in addition to then work a criterion (which depends upon the degree of system) to testify whether the organisation is controllable. If it is controllable, at that topographic point exists a command constabulary that tin force the nation of the organisation from whatever initial nation to a target nation inward finite time.

If it is non controllable, nosotros cause got the province of affairs where at that topographic point are target states which cannot hold upwards achieved nether whatever possible command law. As a uncomplicated example, bring whatever criterion (non-flying!) car. We to a greater extent than oftentimes than non tin movement the auto to whatever signal on a apartment stretch of terrain, but at that topographic point is no machinery for us to position the auto at whatever arbitrary altitude. If nosotros define the nation infinite every bit including the altitude, the auto is uncontrollable.

Controllability is a useful characteristic for dealing with pointy-haired bosses. One tin imagine such a mortal insisting that since the command engineer has a Ph.D. from a fancy-pants university, they should hold upwards able to develop a command organisation that allows the auto to wing at whatever height desired past times the driver. The suffering command engineer tin merely signal to the mathematical definition of controllability, in addition to demo that such a chore is theoretically impossible.

Many economists are inward the career seat of the junior command engineer: they cause got superiors that want them to do the impossible (generate accurate signal forecasts). This is extremely awkward, but mainstream economist profession managed to divulge a way to dodge the question. Instead of giving a raw forecast, they prefer to limited the effects of policies every bit causing deviations from an (unknown) baseline forecast: "if x happens, gross domestic product volition grow y% faster over 1 year." This sounds scientific, in addition to keeps everyone happy. However, such a formulation is past times its nature non-falsifiable, in addition to it does non dispel cynicism most economist forecasts alongside non-economists.*

One in conclusion signal most the analogy. In command systems theory (particularly linear systems theory), it is possible to divulge a number of weather that are mathematically equivalent to controllability. That is, at that topographic point are a laid of properties (of the organisation inward question) x, y, z,... such that the organisation is controllable if in addition to only if ("iff" inward blackboard-speak) belongings x (or y, z) holds. This way that nosotros could work whatever of the properties x,y,z,... instead our master definition of controllability. This is despite the fact that the definitions seem different. When extending the analogy to forecastability, nosotros could run into the job that at that topographic point are a number of potential definitions 1 could use, in addition to although they mightiness hold upwards equivalent for certainly classes of models, they mightiness differ for other classes.

Models Versus Reality

My word of controllability higher upwards deliberately skated over 1 issue: nosotros ask to pull a occupation betwixt mathematical models in addition to reality. The illustration I gave appears intuitively obvious, but at that topographic point are technical details to consider.

The definition of controllability, similar all good posed applied mathematics, a tilt most sets. Once nosotros write downwards a formal organisation that covers the model of interest, controllability is a tilt most the existence of a command input u given an initial nation x in addition to a terminal nation y. It tin only hold upwards tested against a specific mathematical model.

If nosotros expression at the query of driving a auto inward the existent world, at that topographic point are an infinite number of models 1 could use. The simplest (and which ignores a adept bargain of physics) is to assume that whatever given time, nosotros tin laid the administration of move in addition to speed of move (both frontward in addition to backward) independently. This is equivalent to us existence able to laid the x in addition to y velocity to arbitrary levels on a 2-D surface. (The nation vector is the x and y position, every bit good every bit the x in addition to y velocities.) This would in all likelihood hold upwards only adequate if nosotros were discussing a path-planning job where nosotros ask to divulge the optimal trajectory to follow given some routing constraints (including speed in addition to acceleration limits). We tin plug this model into the controllability definition, in addition to reckon that the organisation is indeed controllable on a 2-dimensional surface. However, if nosotros augment the nation infinite to include the height (z axis) every bit good every bit the z velocity, the model fails the controllability test.

This model breaks downwards if our desired trajectory pushes limits. We cannot alter the administration of move in addition to speed arbitrarily quickly, which is rattling apparent if nosotros are driving on H2O ice or a gravel road. We would ask a to a greater extent than complex model to capture to a greater extent than realistic driving dynamics, which takes into concern human relationship angular momentum, tire facing in addition to traction, etc. That said, nosotros tin work intuition from physics to hold upwards able to reckon that fifty-fifty with these added dynamics, nosotros cannot displace the auto to arbitrary altitudes (outside of Hollywood physics...).

This way that inward the driving illustration nosotros gave, nosotros are somewhat lucky. The mathematical concept of controllability is robust across a multifariousness of models for the same physical system. However, this volition non e'er hold upwards the case. I cause got non kept upwards with the developments inward robust command theory since 1998, but every bit of that era, the notion of controllability was nonetheless tied to special models. This way that it was non tied to the notion of model robustness.**

This limits our powerfulness to brand assertions most the forecastability of economies (like I did before inward this article...). Technically, nosotros tin only characterise the non-forecastability of special models (or classes of models). One could in addition to then divulge "similar" models that are inward fact forecastable. If nosotros want to brand statements most the existent world, nosotros would ask to contrast the 2 models, in addition to reckon which improve fits observed data.

Influenza A virus subtype H5N1 Potential definition of Forecastability

Although I am emphasising the formal mathematical nature of the definition, I volition instead beak over the qualities that nosotros hope to capture with a formal definition. In companionship to give a formal definition, nosotros ask to come upwards up with a rattling full general definition of what is an economical model, in addition to that do could easily plough into a self-indulgent moving ridge of definitions. Until this concept is improve pinned down, that do would hold upwards a waste matter of time.

One odd characteristic of the notion of forecastability is that nosotros ask some way to specify the thought of limitations of our noesis of the nation of the economy, or fifty-fifty the model dynamics. It makes trivial sense to beak over forecastability from the perspective of an omniscient observer.

To illustrate the ask for information limitations, allow us imagine a video game which is a conflict betwixt 2 estimator players ("artificial intelligence") inward which the game features "fog of war": non all information for the game nation is available to the players. (A game similar chess has no fog of war.) The fog of state of war could outcome from limited observations of enemy troops inward a state of war game, or it could hold upwards lack of noesis of the opponent's cards (and the deck(s)) inward a carte du jour game. Each estimator musician attempts to win, using its available information set. Unless the game is rigged (or reached a "no win" state), neither musician could predict the outcome of the game. However, from the perspective of the mortal who has laid upwards the estimator game, all of the information is presumably available. The game merely needs to hold upwards run to create upwards one's hear the outcome, in addition to hence the outcome could hold upwards "forecast." If nosotros allow for randomness inward demeanour or inside the game rules, nosotros tin merely run the game a large number of times to teach an empirical justice of the probability distribution of outcomes.

The ask to specify a limited information laid creates a contrast betwixt a definition of forecastability in addition to the existing concepts of controllability in addition to observability.

We volition instantly plough to steps needed to define forecastability.

The foremost stair is defining what nosotros hateful past times an economical model. Rather than coming upwards with an abstract definition, nosotros merely assume that it captures criterion classes of economical models, such every bit stock-flow consistent (SFC) models every bit good every bit DSGE models. Within this definition, at that topographic point volition cause got to hold upwards a notion of all the variables inside the model. I volition refer to this hither every bit the state vector out of habit, but technically the criterion definition of the nation vector inward dynamic systems theory is a subset of all the variables.

We in addition to then ask to decompose the nation vector into world in addition to individual information, which is a stair that presumably does non seem inward most treatments of economical models.

• Public information includes information similar economical aggregates in addition to prices, which inward regulation tin hold upwards determined past times statistical agencies.
• The individual information would hold upwards mainly the information that describes demeanour of economical agents (sectors), which includes fixed behavioural parameters. (This way that nosotros ask to augment the traditional nation concept from organisation theory with these behavioural parameters; inward engineering systems, those parameters would hold upwards share of the organisation dynamics.)

There is a lot of hidden details inward the sectionalization betwixt world in addition to individual (or hidden) information.

• The dynamics of the model could either hold upwards treated every bit world or private. For example, a production functions could hold upwards either individual or world information. For example, 1 could imagine models where competing firms work dissimilar technologies, in addition to they are non only certainly what the technical capabilities of their competitors are. It is non obvious how to contain doubt some the model construction into the definition.
• Things similar accounting identities are world information, which sensible agents would bring into concern human relationship when making decisions. 
• Government policy could hold upwards either world or hidden. We could cause got models inward which the primal banking company commits to a monetary policy rule, in addition to the parameters of this dominion are assumed to hold upwards known.
• The handling of non straight measured variables could hold upwards complicated. We tin imagine models inward which all actors believe that at that topographic point is some notion of an output gap that is an of import driver of inflation trends, in addition to that an output gap model is truly correct. However, based on real-world experience, it is unreasonable to believe that all agents concur what the floor of the output gap truly is. As a result, nosotros could cause got a model inward which at that topographic point is a hidden "true" output gap, every bit good every bit variables that are output gap estimates -- which is what forecasts would cause got to use. This is rattling dissimilar from an omniscient observer, which would cause got access to the model's truthful output gap every bit a mathematical entity.

Finally, nosotros in addition to then define exact forecastability for a deterministic organisation as: tin an observer "predict the hereafter trajectory" of Earth nation variables on some horizon, when it only has access to world data?  The definition would hold upwards with reference for some fourth dimension t, in addition to hence the only information to hold upwards used inward the forecast is the history before fourth dimension t, along with the hereafter information of world exogenous variables. (That is, nosotros assume that nosotros cause got access to the hereafter path of those exogenous variables. We tin translate the forecast provided every bit existence conditional on the path of the exogenous variable; if the exogenous variable does non evolve every bit specified, the forecast is patently incorrect.)

The "predict the hereafter trajectory" is non-mathematical. I recollect that nosotros could convert this to a mathematical tilt every bit follows. It is equivalent to the existence of an operator that maps the past times history to forecasts, but the private/public information distinction mightiness create some difficulties.

My suspicion is that it volition hold upwards necessary to inject some notion of mensuration mistake into the definition, in addition to and then nosotros switch to a novel version where the resulting forecast errors are "contained" relative to the mensuration errors. I recollect it volition hold upwards necessary to expression at to a greater extent than interesting examples, in addition to reckon how nosotros tin come upwards up with a generalisation of exact forecastability to capture the desired behaviour.

I render to the query of models with random variables at the halt of this article.

Example: Model SIM

I volition work Model SIM -- the simplest non-trivial stock-flow consistent (SFC) model. I cause got written most this model extensively inward the past, every bit it is easiest to explicate concepts with the simplest model.

For to a greater extent than details, the reader is directed to:

• My online articles that refer to the model: https://cococalamanco.blogspot.com/search?q=model+SIM; or
• my volume on SFC models.

The model is a straightforward 3 sector model, with a identify sector, concern sector, in addition to government. The concern sector does non do rattling much; it is assumed to e'er intermission even, in addition to hence it has no holdings of fiscal assets (nor are at that topographic point inventories). Government spending is specified every bit an exogenous fourth dimension series, in addition to essentially determines the other variables (once behavioural parameters are fixed).

This illustration shows some of the awkwardness of the definition of information availability. It would presumably hold upwards defined every bit having access to the fourth dimension serial of historical data. I depict the method of the solution determination inward the article "Finding the Solution inward a Simple SFC Model." In summary, each period, the provide in addition to demand functions are lined upwards in addition to simultaneously solved. In particular, inward companionship for the concern sector to e'er intermission even, it has to hold upwards able to predict demand precisely inward the electrical flow catamenia (so that it tin check production to demand). So the solution method implies that the concern sector has "perfect information" most the electrical flow catamenia supply/demand situation. This is shoved nether the rug if nosotros define the information availability inward terms of historic fourth dimension series; nosotros tin sentiment the solution determination to hold upwards some form of equilibrating physical care for that is buried inside the macro specification. (If nosotros had a pure agent-based model, this job would disappear.) In other words, the notion of "information availability" mightiness non check one's intuition.

For now, nosotros assume that all behavioural parameters (including the revenue enhancement rate) are constant. Furthermore, nosotros assume that all entities cause got access to the total model specification.

We tin instantly plough to the query of individual in addition to world information.

• The identify consumption share is defined inward terms of a distich of propensity to eat parameters (out of income, out of wealth). Only the identify sector knows what those parameters are for sure. If at that topographic point is no source of mensuration error, my guess is that it would hold upwards to a greater extent than oftentimes than non possible to create upwards one's hear precisely those parameters with access to a finite history of world variables.*** 
• The concern sector is constrained to intermission even, in addition to presumably all entities know this, in addition to hence it has no hidden variables associated with its behaviour. More complex models would allow for concern sector behavioural variables.
• Government policy is specified inward terms of authorities consumption in addition to a fixed revenue enhancement rate. The revenue enhancement charge per unit of measurement has to hold upwards world information (as otherwise collection is somewhat difficult...); the status of authorities consumption volition hold upwards discussed below.
• All other fourth dimension serial are public.

From the perspective of the government, model SIM is (generally) forecastable (after a catamenia of observation). It knows what it is planning to do with honour to authorities consumption, in addition to it tin dorsum out the identify sector behavioural parameters from observed data. It tin in addition to then infer what the reply to its consumption fourth dimension serial past times plugging inward the identify propensity to eat parameters into its re-create of the model.

For the household, forecastability depends upon how nosotros care for authorities consumption. If it knows the hereafter trajectory of authorities consumption, it knows its ain consumption function, in addition to tin infer the path of all variables. If the identify sector has no thought what authorities consumption volition be, it cannot predict what volition happen. (Finding a solution inward a catamenia requires knowing the electrical flow value of authorities consumption.)

From whatever other perspective, the forecastability status is similar to the cases for the authorities in addition to household. It is possible to justice the identify consumption function, but forecastability depends on knowing the hereafter path of authorities consumption.

The fact that model SIM is forecastable nether the proposed definition is desired. Qualitatively, if nosotros know the path for authorities spending, nosotros tin forecast what volition happen, every bit the dynamics are almost only driven past times a multiplier on authorities spending (with lagging dynamics). We ask to a greater extent than interesting concern wheel models for them to cash inward one's chips non-forecastable.

If nosotros allow for non-public structural changes inward the behavioural parameters (consumption share alpha parameters, revenue enhancement rates), forecastability in addition to then depends on time. That is, imagine that nosotros allow the possibility that some parameter changes from 1 value to some other at some fourth dimension t. Forecasts generated before fourth dimension t will hold upwards invalidated, but after observing the information after fourth dimension t, it volition hold upwards possible to pivot downwards the novel value for the parameter. Once again, it would hold upwards possible to brand accurate forecasts.

The Awkwardness of Randomness

If nosotros innovate randomness, nosotros ask to relax the definition. I currently do non cause got a adept feeling how this could hold upwards done. One mightiness hold upwards tempted to follow inward the lines of neoclassical economic science in addition to finance in addition to work a definition along this line: the expected value of forecasts volition equal the expected value of the variables inward the truthful model.

The job is that this is a rattling weak status (although it volition hold upwards rattling congenial to neoclassical academics). It turns out to hold upwards the same affair every bit marketplace seat efficiency -- on average, electrical flow marketplace seat pricing is an unbiased forecast of hereafter pricing (modulo financing costs). The implication is that it is hard to brand super-normal returns inward the market, which is a tilt that appears to fit observed behaviour. That is, 1 could debate that it holds (in the context of fiscal markets).

In fixed income terms, the work of the previous definition using expectations would hold upwards equivalent to proverb that frontward rates are identical to forecast hereafter rates. This is an unreasonable work of the term "forecast": sure, it may hold upwards hard to brand coin trading frontward 10-year rates, but it is abundantly clear that realised 10-year rates only hitting previous forwards on exceedingly rare occasions.

Instead, nosotros would ask to cause got a notion of a confidence interval: at that topographic point is a p% gamble that realised hereafter variables are inside some tolerance of the forecast. So long every bit nosotros impose somewhat strict weather on the tolerance, 10-year U.S.A. Treasury forwards would non reckon this test. However, it mightiness hold upwards truthful for short-rate forecasts: exterior of crises, forwards are pretty adept predictor of curt rates.

(It is extremely probable that I volition render to this theme at greater length. If 1 were to bring the DSGE modellers' description of DSGE models at human face upwards value, the models would seem to imply forecastability using the expectations definition. However, I recollect it is possible to heighten reasonable objections to that characterisation. It is clear that the status of DSGE models would hold upwards the greatest challenge to coming upwards with a formal definition for forecastability.)

Concluding Remarks

This volition probable hold upwards the get-go of a sequence of articles discussing the forecastability of concern wheel models. As model SIM shows, it is non right to say that all macro models volition hold upwards non-forecastable. My conjecture is that if nosotros confine ourselves to "interesting" concern wheel models, non-forecastability volition hold upwards a criterion feature. If the conjecture is correct, in addition to if 1 accepts that these concern wheel models capture reality, nosotros should hold back existent basis forecasting exercises to hold upwards doomed to failure.

Footnotes:

* The sensible approach is to give conditional forecasts: nether a certainly assumption, this is what nosotros recollect volition happen. Many mainstream academic economists (as good every bit heterodox academics) correctly debate that this is what nosotros are supposed to hold upwards doing. However, I spent my entire career inward manufacture interacting with economists, in addition to it is clear to me that this is non what industrial employers of economists want them to do. I do non give forecasts inward my writings since I sentiment expectations most what forecasting tin attain every bit ridiculous. One could attempt to accommodate the definition I outline hither to grip the notion of conditional forecasts, but at the fourth dimension of writing, I reckon some pitfalls with a formal definition of status forecastability. It would hold upwards rattling piece of cake for a prospective definition of conditional forecastability ending upwards existence mathematically equivalent to forecastability, whereas inward my view, it needs to hold upwards a much weaker property.

** For linear time-invariant systems, model robustness was largely a frequency domain concept. In the frequency domain, non-controllable (and non-observable) subspaces of the nation infinite disappear from the mathematical specification. As a result, almost all linear robust command theory only industrial plant with systems that are controllable in addition to observable past times definition. For nonlinear systems, this may no longer hold upwards the case, but the inability to characterise the solutions to nonlinear systems inward a build clean fashion made the number moot.

*** There volition hold upwards trajectories for which this is impossible. For example, if wealth inward the historical information is e'er zero. nosotros cause got no way to calibrate the propensity to eat out of wealth. (More generally, the matrix used for estimation would non hold upwards invertible.) However, the laid of histories for which this is truthful is a laid of mensurate nada relative to the infinite of all possible histories. This is a theme that volition in all likelihood hold upwards revisited inward later on articles.

(c) Brian Romanchuk 2018