2. The (Core of the) K+S Model of the Macroeconomy

The core of the K+S model is the model that Dosi et al. (Reference Dosi, Fagiolo and Roventini2008) develop to model capital- and consumption-goods production and consumption; to that core Dosi et al. (Reference Dosi, Fagiolo and Roventini2010) later add a Keynesian component, Dosi et al. (Reference Dosi, Fagiolo and Napoletano2013) Minskyan credit dynamics, and Dosi et al. (Reference Dosi, Fagiolo, Napoletano and Roventini2015) a monetary policy rule. My presentation of the core will follow the exposition of the “baseline model” that Dosi et al. (Reference Dosi, Napoletano, Roventini and Treibich2017) present in section 2 of their most recent work on the model. But my presentation will skip consumption and instead include one of the extensions of the core: the monetary policy rule.

Dosi et al. (Reference Dosi, Napoletano, Roventini and Treibich2017: 66) model capital-goods production as production of heterogeneous machine tools (“capital goods”) by a firm i that uses labour () to produce the tools (Q _i,t ), and that invests a fraction of its past sales to search for process and product innovation () and imitation (). While successful process innovation/imitation increases labour productivity (), successful product innovation/imitation impacts the productivity of the machines. Whether the firm successfully innovates/imitates is determined by a draw from a probability distribution, the parameters of which (ζ ₁, ζ ₂ ∈ ]0; 1]) define the search capabilities of the firm (i.e. its capabilities to innovate/imitate).

The graph in Figure 1 can be used to illustrate capital-goods production. Here, the arrows stand for relations of causal dependence. Dosi et al. use causal vocabulary only informally. But they conduct simulated policy experiments that involve manipulations of parameters (for instance, of ζ ₁ and ζ ₂) and observations of ensuing changes in other variables (for instance, in Q _i,t ) (cf. end of this section). Therefore, the relations that they believe obtain between the variables in capital-goods production qualify as causal in the sense of the interventionist account of causality. The interventionist account also underlies the inference method that Guerini and Moneta (Reference Guerini and Moneta2017) employ when trying to identify the parameters of the K+S model.Footnote ²

Figure 1. Capital-goods production in the K+S model.

According to the interventionist account, Y causally depends on X relative to a set V of preselected variables iff there is a possible intervention on X that changes Y, while all other variables in V remain unchanged (Woodward Reference Woodward2003: 55, 59). Woodward is not entirely clear about the exact modality of the term ‘possible’ in ‘possible intervention’.Footnote ³ But at one point, he characterizes possible interventions as “hypothetical”, i.e. as turning the antecedents of counterfactuals into true statements (Hitchcock and Woodward Reference Hitchcock and Woodward2003: 183). And in this sense, there is clearly a possible intervention on that changes Q _i,t , while all other variables of the K+S model remain unchanged; there is clearly a possible intervention on B _i,t that changes Q _i,t , while all other variables of the K+S model remain unchanged; and so on. It is quite another question whether there are real interventions: whether the relations of causal dependence that Dosi et al. believe obtain in the target system of the K+S model really do obtain in the target system. In order to show that they obtain in the target system, one will have to identify the parameters of the (core of the) K+S model, and sections 4–6 will present three problems that get in the way of parameter identification (or causal inference) in macroeconomics.

Dosi et al. (Reference Dosi, Fagiolo and Roventini2010: 1752; Reference Dosi, Napoletano, Roventini and Treibich2017: 68) model consumption-goods production as production of final goods: as production of a homogeneous good by firm j whose level of production (Q _j,t ) depends on its desired level of output ( $$N_{j,t}^{d}$$ ), inventories inherited from the previous period (N _j,t-1), and on “adaptive” demand expectations ( $$D_{j,t}^{e}$$ ), which in turn depend on the demand that the firm faced in the previous two periods (D _j,t-1 and D _j,t-2), on its demand expectations of the previous period ( $$D_{j,t{\hbox{-}}{\rm{1}}}^{e}$$ ), and on the gross domestic product of the previous period (GDP _t-1) (cf. Dosi et al. Reference Dosi, Napoletano, Roventini, Stieglitz and Treibich2020: 1494). The graph in Figure 2 can be used to model consumption-goods production. Note that the curved downward arrow stands for a process of downward causation operating from a macroeconomic variable (GDP _t ) to a microeconomic variable ( $$D_{j,t+1}^{e}$$ ).

Figure 2. Consumption-goods production in the K+S model.

Dosi et al. (Reference Dosi, Napoletano, Roventini and Treibich2017: 68–69) model price formation as the formation of prices of final (i.e. consumption) goods. Prices of final goods (p _j,t ) equal the unit cost of consumption-goods firm j (c _j,t ) plus a variable mark-up (μ _j,t ), which depends on the market share of the firm in t−1 (f _j,t-1), which in turn depends on its competitiveness in t−1 (E _j,t-1), which relates directly to p _j,t-1 and (inversely) to the possible amount of unfilled demand inherited in t−1 from the preceding period (l _j,t-1). In both sectors, the difference between firm revenues and costs accounts for firms’ profits, which change the stock of liquid assets of firms. A firm goes bankrupt if that stock is smaller than 0, or if its market share falls to zero. In both cases, the firm exits the market and is replaced by a new (usually smaller) entrant.

The graph in Figure 3 can be used to illustrate price formation. Here, the fat arrow doesn’t denote a relation of causal dependence, but a relation of constitution or asymmetric supervenience. Y asymmetrically supervenes on X = {X ₁ … X _n } iff changes to Y necessitate changes to X but not vice versa.Footnote ⁴ Price (p _j,t ) clearly supervenes on unit cost (c _j,t ) and mark-up (μ _j,t ) in this sense: changes to p _j,t necessitate changes to c _j,t or μ _j,t but changes to c _j,t or μ _j,t do not necessitate changes to p _j,t (changes to c _j,t and μ _j,t can even out exactly). Note that competitiveness (E _j,t-1), for instance, does not supervene on price (p _j,t-1) or level of unfilled demand (l _j,t-1) in this sense: changes to E _j,t-1 do not necessitate changes to p _j,t-1 or l _j,t-1 (they can be due to all kinds of changes, including a change in productivity). Note further that p _j,t doesn’t causally depend on c _j,t or μ _j,t . In order for p _j,t to causally depend on c _j,t or μ _j,t , changes to p _j,t must not necessitate changes to c _j,t or μ _j,t .Footnote ⁵ As pointed out above, however, changes to p _j,t do necessitate changes to c _j,t or μ _j,t .

Figure 3. Price formation in the K+S model.

The most important relations of supervenience or constitution obtain, of course, between macroeconomic aggregates and the microeconomic quantities that constitute these aggregates. Dosi et al. (Reference Dosi, Napoletano, Roventini and Treibich2017: 70) express GDP, for instance, as the sum of value added of F ₁ capital-goods-firms and F ₂ consumption-goods-firms and assume that national account identities are satisfied, i.e. that GDP equals aggregate consumption (C _t ), investment (I _t ) and change in inventories (N _t ) (Figure 4). Dosi et al. (Reference Dosi, Fagiolo, Napoletano and Roventini2015: 171n) define inflation (π _t ) as variation of the consumer price index (CPI), where the CPI is computed as the sum of prices of F ₂ consumption-good firms weighted for their market shares (Figure 5).

Figure 4. The constitution of GDP in the K+S model.

Figure 5. The constitution of inflation in the K+S model.

Finally, according to the monetary policy rule, interest rates (r _t ) causally depend on the rate of inflation, the unemployment rate (U _t ), and the target rates of interest (r ^T ), inflation (π ^T ), and unemployment (U ^T ) (Figure 6): The policy rule implies that the monetary authority selects the interest rate in response to changes in inflation and unemployment relative to their target levels (cf. Dosi et al. Reference Dosi, Napoletano, Roventini and Treibich2017: 71).

Figure 6. Monetary policy in the K+S model.

The graph in Figure 7 illustrates the core of the K+S model plus the monetary policy rule and minus consumption. I should say that the graph doesn’t capture every detail of even consumption- and capital-goods production: not every relation of causal dependence or supervenience is covered, and I abstracted from functional form at the micro- and macroeconomic level. But I shall be concerned with problems of causal inference: with problems of providing empirical evidence in support of the relations of causal dependence denoted by the thin arrows. And for the purpose of discussing these problems, the graph is sufficiently detailed.

Figure 7. The core of the K+S model plus the monetary policy rule and minus consumption.

Before turning to these problems, however, I will have to present the method that Dosi et al. (Reference Dosi, Napoletano, Roventini and Treibich2017: sections 4 and 5) employ to “validate” their model. They validate their model by (1) selecting initial values for variables, by (2) selecting benchmark parameters, i.e. by calibrating model parameters (by selecting 50 for the number of capital-good firms, 200 for the number of consumption-good firms, 0.30 for ζ_1,2, 0.4 for ϕ, and so on)Footnote ⁶ , by (3) coding the model equations in a structured or object-oriented programming language, by (4) running the model on a computer an arbitrary number of times (e.g. 5000 times), and by (5) reproducing a number of micro- and macroeconomic empirical regularities (so-called “stylized facts”).

The micro- and macroeconomic stylized facts that Dosi et al. (Reference Dosi, Fagiolo and Roventini2008: section 2) say the core of the K+S model is able to reproduce include the heterogeneity of firms in terms of productivity; the fat tails of the GDP growth rate distribution; short-lived recessions; and cross-correlations of macro-variables (more specifically, the pro-cyclicality of consumption, productivity, nominal wages, and inflation, and the counter-cyclicality of unemployment, prices and mark-ups).

Dosi et al. (Reference Dosi, Napoletano, Roventini and Treibich2017: 65) claim that the validated model is “well-equipped to explore … the short and long-run effects of economic policies”. Among the policies whose effects they explore are innovation policies, Keynesian fiscal policies and policies of income redistribution. When exploring the effects of innovation policies, for instance, they (1) take the validated model as a point of departure, (2) manipulate the parameters measuring the firms’ search capabilities (ζ₁ and ζ₂), (3) run the model 100 times, (4) observe that after 100 runs, the GDP growth rate is higher than that of the validated model if ζ_1,2 > 0.3, and lower if ζ_1,2 < 0.3, and (5) conclude that “policies favoring innovation promote faster growth” (Dosi et al. Reference Dosi, Napoletano, Roventini and Treibich2017: 76).

3. Causal Inference in Macroeconomic ACE Modelling

The claim that the validated model is “well-equipped” to explore policy effects is precisely the claim that Guerini and Moneta deny. They argue that the method of stylized fact reproduction is not sufficiently rigorous:

The method of stylized fact reproduction is not “rigorous enough” because for any stylized fact, there is an indefinite number of relations of causal dependence that might have generated the fact. An example of a stylized fact that the K+S model is able to reproduce is the correlation (and pro-cyclicality) of consumption and inflation (cf. section 2). That correlation is compatible with three relations of causal dependence: consumption causally depends on inflation; inflation causally depends on consumption; or both inflation and consumption depend causally on a third variable (or set of variables). The third relation (with the set of variables acting as a confounder) divides into an indefinite number of relations, depending on the relations of causal dependence that obtain between the variables in the set.

The inference method that Guerini and Moneta (Reference Guerini and Moneta2017) employ to provide empirical evidence in support of the relations of causal dependence between the micro- and macro-variables of the K+S model is the five-step procedure of (1) transforming artificial and real-world data in a way that allows them to be directly comparable, of (2) testing for the assumptions of statistical equilibrium and ergodicity, of (3) estimating the parameters of a reduced-form vector autoregression (VAR) model, of (4) running a specific search algorithm to identify the relations of causal dependence generating the artificial data and those generating the real-world data, and by (5) comparing both sets of relations.

The primary purpose of step (1) is to select k variables of interest and to adjust the time series that provide values for these variables in the artificial and real-world data sets. The assumptions to be tested in step (2) need to be satisfied in order for any model to be able to approximate the data generating process. They require that the observed time series have distributional properties that are time-independent (“statistical equilibrium”), and that they are a random sample of a multivariate stochastic process (“ergodicity”). Note that these requirements don’t coincide with the requirement of insensitivity to initial conditions to be discussed in section 5 below.

The VAR model of the artificial and real-world data represents the value of each of the k variables at t as a linear combination of contemporaneous and p lagged values of all k variables:

$${\bf{{{Y}}_t} = {\bf{B}}{{\bf{Y}}_t} + {\bf \Gamma _{\bf{1}}}{{\bf{Y}}_{t - 1}} + \ldots + {\bf\Gamma _{\bf{p}}}{{\bf{Y}}_{{\bf{t - p}}}} + {\bf\varepsilon _t}},$$

where Y _t , Y _t-1 … are vectors of the values of variables Y ₁ … Y _k at t, t−1 … t−p, where B and Γ ₁ … Γ _p are matrices collecting the model parameters, and where ε _t is a vector of error terms. The diagonal elements of B are equal to zero (no value depends on itself), and the error terms in ε _t are assumed to be mutually independent. Subtracting BY _t from both sides of the equation yields the so-called structural vector autoregression (SVAR) model:

$${\bf\Gamma _{\bf 0}}{{\bf{Y}}_{\bf{t}}} = {\bf\Gamma _1}{{\bf{Y}}_{{\bf{t - 1}}}} + \ldots + {\bf\Gamma _{\bf{p}}}{{\bf{Y}}_{{\bf{t - p}}}} + {\bf\varepsilon _{\bf{t}}},$$

where Γ ₀ = I − B. Dividing both sides of the SVAR model by Γ ₀ gives the reduced-form VAR model:

$${{\bf{Y}}_{\bf{t}}} = {\bf\Gamma_{0}^{\bf - 1}}{\bf\Gamma_1}{{\bf{Y}}_{{\bf{t - 1}}}} + \ldots + {\bf\Gamma_{0}^{\bf - 1}}\bf\Gamma_{\bf{p}}{{\bf{Y}}_{{\bf{t - p}}}} + {\Gamma_{0}^{ - 1}}{\bf\varepsilon _{\bf{t}}}$$

$$ = {{\bf{A}}_{\bf{1}}}{{\bf{Y}}_{{\bf{t - 1}}}} + \ldots + {{\bf{A}}_{\bf{p}}}{{\bf{Y}}_{{\bf{t - p}}}} + {{\bf{u}}_{\bf{t}}}.$$

The endogeneity of the contemporaneous variables biases estimations of the parameters of the VAR and SVAR models. It is therefore the parameters contained in A ₁ … A _p that are estimated in step (3).

The parameters contained in A ₁ … A _p , however, are not the ones that figure in the causal model (i.e. the model representing relations of causal dependence between the k variables). In order to identify the parameters of the causal model, the parameters contained in Γ ₀ , Γ ₁ … Γ _p need to be recovered. If the parameters in Γ ₀ are recovered, they can be used (together with the parameters in A ₁ … A _p ) to identify the parameters in Γ ₁ … Γ _p . But the problem with the parameters in Γ ₀ is that any inversion of Γ ₀ is compatible with the parameters contained in A ₁ … A _p (as long as the inverse is unit-diagonal).

Step (4) is meant to solve this problem by running a causal search algorithm: the LiNGAM. The LiNGAM is an algorithm relying on the assumption that the variables in V can be represented as a Linear Non-Gaussian Acyclic Model. Like any other causal search algorithm, the LiNGAM operates by forming a complete undirected graph, and by testing for conditional independence relations to eliminate unnecessary edges and to direct the remaining ones.Footnote ⁷ But other search algorithms result in a (so-called Markov equivalent) class of observationally equivalent directed graphs: they narrow down the class of observationally equivalent equation systems but stop short of fully identifying the parameters of the causal model. The LiNGAM, by contrast, stands out as being capable of identifying the parameters of the causal model completely. Guerini and Moneta run the LiNGAM to discover the causal dependencies among the residuals u _t = ${{\bf\Gamma_{0}^{\bf - 1}}{\bf\varepsilon _t}}$ of the reduced-form VAR model. These causal dependencies give the parameters contained in Γ ₀ , which are then used to identify the parameters in Γ ₁ … Γ _p .

In step (5), they use a “similarity measure”, which indicates, by what percentage the relations generating the artificial and real-world data coincide with one another.

When applying their five-step procedure to the K+S model, Guerini and Moneta (Reference Guerini and Moneta2017: section 4) select only the 6 macroeconomic variables (GDP _t , C _t , I _t , U _t , π _t and r _t ) as variables of interest. They use US time series data from the first quarter of 1959 to the second quarter of 2014 to provide values for these variables, and they adjust the length of the artificial data set accordingly. They test for the assumptions of statistical equilibrium and ergodicity and conclude that both assumptions are “reasonable”. They then use the two data sets to specify two VAR models and to estimate the parameters that figure in these models. Next, they employ the LiNGAM to discover the causal dependencies among the residuals u _t = ${{\bf\Gamma_{0}^{\bf - 1}}{\bf\varepsilon _t}}$ of the reduced-form VAR models. Finally, the similarity measure tells them that the percentage by which the relations of causal dependence generating the artificial and real-world data coincide is in the interval between 65% and 80%, depending on the technique that is used to estimate the parameters of the reduced-form VAR models. Guerini and Moneta (Reference Guerini and Moneta2017: 21) say that they “guess that this is a positive result” for the K+S model.

Dosi et al. (Reference Dosi, Napoletano, Roventini, Stieglitz and Treibich2020: 1500) agree that this is a positive result: they say that Guerini and Moneta (Reference Guerini and Moneta2017) have “externally validated” the relations of causal dependence “between macroeconomic variables within the K+S model”. I am not entirely sure, however, whether this is a positive result. It implies that at least 20% of the relations of causal dependence generating the artificial data have no counterpart in reality, or that at least 20% of the relations of causal dependence generating the real-world data have no counterpart in the model.

I also want to emphasize that the causal inference method that Guerini and Moneta (Reference Guerini and Moneta2017) propose is meant to be an inference method for the K+S model as a whole, and not only for the macro-part of the model. Their method is meant to provide empirical evidence in support of all relations of causal dependence that obtain in the target system of the K+S model, and not only in support of the relations of causal dependence that obtain at the macro-level of that system.

I finally want to point out that Guerini and Moneta need to provide empirical evidence in support of the relations of causal dependence that obtain at the micro- and macro-level of the system if that evidence is supposed to be evidence in support of the efficacy of the policies that Dosi et al. explore. These (innovation, Keynesian fiscal, and income redistribution) policies all involve, after all, manipulations of benchmark parameters at the micro-level.

Guerini and Moneta (Reference Guerini and Moneta2017: 5) suggest that there are problems for the application of their method to the K+S model as a whole, and that these problems are practical in nature and not of the in-principle kind: they say that “representing into a unique model every single micro-mechanism at work in a complex economy is a very difficult task”. I agree that there are practical problems for the application of their method to the K+S model as a whole. But I also think that in addition to these practical problems, there are in-principle problems for the application of the inference method to the K+S model as a whole: three problems on which the following three sections will elaborate.

4. The Non-measurability of Expectations

The first problem has to do with our inability to measure individual or aggregate expectations. The claim that expectations cannot be measured may come as a surprise to theorists who conduct surveys to measure expectations on a regular basis. But Kevin Hoover (Reference Hoover2001: 137) explains why we cannot measure expectations by conducting surveys:

Hoover suggests that expectations fall into the same category as preferences. In revealed-preference theory, a consumer’s preference is reconstructed from her behaviour (from her “revealed” preference). Statements about what she thinks she prefers are to be dismissed as neither verifiable nor trustworthy. Similarly, expectation variables cannot be measured because a subject’s statement about what she expects can neither be verified nor trusted.

Assume for the sake of argument that a subject’s statement about what she expects can be verified or trusted. Manski (Reference Manski2018: section VII) lists a number of problems that would obtain even if that assumption were true: the expectations of firms are difficult to measure; the practice of rounding complicates the interpretation of responses; probabilistic expectations may be ambiguous (to the extent that they express, for instance, ignorance or uncertainty); respondents sometimes confound beliefs and preferences etc. Manski (Reference Manski2018: 423) believes that these problems can be solved. But even if they could be solved, there would be the further problem that survey studies cannot keep track of the way, in which agents form (i.e. revise or update) individual or aggregate expectations. Agents seem to revise or update their expectations more often than they are surveyed. We accordingly need to understand the way, in which they revise or update their expectations. And survey studies cannot keep track of the way, in which they form (i.e. revise or update) individual or aggregate expectations.

One could argue that the way, in which agents form individual or aggregate expectations, can be investigated in laboratory experiments. But laboratory experiments are unlikely to teach us anything about the formation of individual or aggregate expectations. They are unlikely to teach us anything about the formation of individual expectations because in real life, the formation of individual expectations requires that expectations be revised and updated in light of information that (like government announcements, media reports, personal observations etc.) is often generated or collected in obscure ways, and because these obscure ways are difficult (or impossible) to mimic in laboratory experiments. These experiments are also unlikely to teach us anything about the formation of aggregate expectations because the dynamics of the formation of aggregate expectations in small groups and economic systems have little in common.

Expectations, therefore, cannot be measured. There is currently no way to tell whether the expectations that agents form with respect to the behaviour of macroeconomic variables (or indeed any variable) should be modelled as “rational”, “adaptive”, or in some alternative way. Expectations are rational if whatever information is available is completely exploited (Muth Reference Muth1961; Lucas and Prescott Reference Lucas and Prescott1971). Expectations are adaptive if agents correct past forecasting mistakes when forming expectations.Footnote ⁸

Critics of rational expectations argue that agents rarely exploit the available information completely, or that they tend to make systematic mistakes when exploiting that information. But proponents of rational expectations respond that systematic mistakes cancel out each other at the aggregate level, and that expectations are correct on average. The critics point out that the response remains unsupported by experimental studies (Anufriev and Hommes Reference Anufriev and Hommes2012) or observational evidence. But the point does not imply that agents never form rational expectations. One may assume, for instance, that in the face of competition, firms have a strong incentive to exploit the available information completely.

In addition to the “adaptive” expectations of consumption-goods producing firms, Dosi et al. (Reference Dosi, Napoletano, Roventini, Stieglitz and Treibich2020, section III.C) consider three alternative heuristics that likewise qualify as adaptive. Dosi et al. (Reference Dosi, Napoletano, Roventini, Stieglitz and Treibich2020, section V) conduct a number of simulated experiments to show that the stability of macroeconomic system dynamics increases with the degree to which the heuristics of forming demand expectations is “naïve”. They also state, however, that their model is able to reproduce the same number of stylized facts under all four heuristics (cf. Dosi et al. Reference Dosi, Napoletano, Roventini, Stieglitz and Treibich2020: 1499). They do not show, that is, how individual agents form expectations as a matter of fact.

Why does the non-measurability of expectations pose a problem for causal inference? It poses a problem because in order to provide conclusive evidence in support of a relation of causal dependence between X and Y, one needs to rule out that there is a confounder: a third variable (or set of variables) Z, on which both X and Y causally depend. In order to rule out that there is such a variable (or set of variables) Z, one needs to control for Z, i.e. include Z in the set of preselected variables and observe that Z remains (largely) unchanged, while X and Y change.

Expectations are often believed to act as confounders in just this sense. Of expectations, however, we cannot know whether they can be controlled for because they cannot be measured. Consider, for instance, the claim that inflation causally depends on aggregate demand. In order to provide conclusive evidence in support of that claim, one would need to rule out that inflation and aggregate demand causally depend on inflation expectations. That they causally depend on inflation expectations is, however, exactly what one would expect when understanding the new Keynesian IS curve and the new Phillips curve of the canonical DSGE model as expressing causal hypotheses. Thus, in order to rule out that inflation and aggregate demand causally depend on inflation expectations, one would need to control for inflation expectations. The problem is that inflation expectations cannot be measured, and that we cannot tell whether we are able to control for them. In other words: the empirical evidence that we can provide in support of the claim that inflation causally depends on aggregate demand remains inconclusive.

One might think that the case of expectations is an exotic one, and that the non-measurability of expectations does not get in the way of causal inference in macroeconomics in general. But (considered generally and in its most uncontroversial form) the famous Lucas critique says that each of the equations of models representing relations between macroeconomic aggregates needs to be “derived from decision rules … of agents in the economy”, that “some view of the behavior of the future values of variables of concern to them …, in conjunction with other factors, determines their optimum decision rules”, and that the assumption that this view remains invariant under alternative policy rules is an “extreme assumption” (Lucas Reference Lucas1976: 25).

The Lucas critique says, in other words, that variables Z change whenever policy interventions target X to influence Y, where Z represents an expectational aggregate like inflation or aggregate demand expectations. In the community of agent-based macroeconomists, you sometimes hear voices saying that the empirical relevance of the Lucas critique is questionable in light of the results of empirical procedures of causal inference.Footnote ⁹ But the point is that if Z cannot be measured, the Lucas critique denies that there are reliable empirical procedures that can be employed to provide conclusive empirical evidence in support of causal relations between X and Y, when X and Y stand for macroeconomic aggregates. Thus, the non-measurability of expectations seems to get in the way of causal inference in macroeconomics in general.

5. Sensitive Dependence on Initial Conditions

The second problem arises if the macroeconomy is a chaotic system. I’m aware that a variety of definitions have been proposed for “chaos”, and that many of these definitions account for the appearance of randomness in terms of predictability or information theory (cf. e.g. Frigg Reference Frigg2006; Werndl Reference Werndl2009). But fortunately, I don’t need to adopt any precise definition of “chaos”. I will instead rely on a necessary condition of any notion of chaos: on the condition of sensitive dependence on initial conditions.

The initial conditions of a system can be characterized as a vector of values that the variables of the system attain “initially”, i.e. before the interactions between the system components denoted by the variables are iterated. Let S ₀ be this vector of initial values, and let S _t be the vector of values that the variables of the system attain after t iterations. Then for initial conditions S ₀, an arbitrary (positive) number δ > 0 and slightly modified initial conditions T ₀, | S ₀ − T ₀| < δ will evolve such that | S _t − T _t | ≈ | S ₀ − T ₀|e ^λt , where λ is the (largest) Lyapunov exponent of the system, which measures the average rate of divergences between neighbouring trajectories that depart from values close to those in S ₀. We say that a system is (not) sensitive to initial conditions if λ > (<) 0, i.e. if neighbouring trajectories (don’t) diverge exponentially. We also say, of course, that the greater (smaller) λ, the more (less) the system is sensitive to initial conditions.

Dosi et al. (Reference Dosi, Fagiolo and Napoletano2013, Reference Dosi, Fagiolo, Napoletano and Roventini2015, Reference Dosi, Napoletano, Roventini and Treibich2017) point out on several occasions that they think of the target system of the K+S model as a complex system. There is no universally agreed upon definition of ‘complex system’, and there is unlikely to be any (single) such definition. But chaos is among the conditions that are regularly proposed as conditions for the complexity of a system. Ask a physicist or engineer what a complex system is, and they are likely to respond that a complex system is chaotic. If chaos is a necessary condition for the complexity of a system, and if the target system of the K+S model is a complex system, then the target system of the K+S model will be chaotic and sensitive to initial conditions, for that matter.

Things are not that easy, however. Ladyman and Wiesner (Reference Ladyman and Wiesner2020: 78) argue that chaos is not a necessary condition for complexity: that there are complex systems that are not chaotic. Whether a system is chaotic (or sensitive to initial conditions) is, more importantly, an empirical matter. We can use time series for the variables of the system to estimate the Lyapunov exponent: if the estimator is positive, the system will be sensitive to initial conditions; if it is negative, the system will be insensitive to initial conditions. The definition of the Lyapunov exponent involves the infinite time limit (such that, strictly speaking, does not characterize exponential growth for any finite time limit). But an asymptotic distribution of a nonparametric estimator of the Lyapunov exponent can be derived, and that estimator can be used to measure sensitivity to initial conditions (cf. Shintani and Linton Reference Shintani and Linton2003). The estimator has been used to reject the null of chaos (or sensitivity to initial conditions) for low-dimensional macroeconomic systems (more specifically, for time series of output in industrialized nations). But the target system of the K+S model (or indeed any macroeconomic ACE model) qualifies as high-dimensional (as including many variables), and the problem with testing for high-dimensional chaos is that one would need extremely long time series to distinguish chaos from randomness. Up to now, these time series are not available for any of the variables included in macroeconomic agent-based models. We accordingly don’t know (yet) whether the target system of the K+S model is sensitive to initial conditions.

Why is it important whether the target system of the K+S model is sensitive to initial conditions? Because causal inference will be a lot harder if it is. If the target system of the K+S model is sensitive to initial conditions, then all potential confounding (sets of) variables Z that have been omitted from the model will have to retain exactly the same value z over the course of history that the iterated model is supposed to target. If any of these values deviates from z, the sensitive dependence of the target system on initial conditions will imply that the relation of causal dependence that obtains between X and Y according to the model might fail to have a counterpart in reality.

Consider the relation of causal dependence that obtains between the labour demanded by capital-goods producing firm i at t ( $$L_{i,t}^{D}$$ ) and the quantity (machines) produced by that firm at t (Q _i,t ) according to the K+S model. Critics of the K+S model might argue that the model should include a variable Z denoting the demand for the machines that firm i produced at t−1 because Z acts as a potential confounder: because both the labour demanded by firm i ( $$L_{i,t}^{D}$$ ) and the quantities produced by that firm (Q _i,t ) causally depend on the demand for the machines that firm i produced at t−1 (Z) (Figure 8).

Figure 8. Does the demand for machines act as a confounder?

As long as the target system of the K+S model is insensitive to initial conditions, defenders of the K+S model can point out that the demand for the machines produced at t−1 (Z) is negligible: that it can be neglected because the values of Z remain roughly constant over all iterations, while the values demanded labour and produced quantity can be observed to change. But if the target system of the K+S model is sensitive to initial condition, then the defenders of the K+S model can no longer claim that Z is negligible: then any slight deviations of the values of Z from z can mean that Z acts as a confounder, and that the relation of causal dependence that obtains between demanded labour and produced quantity according to the K+S model fails to have a counterpart in reality.

Similar considerations apply in the case of the relation of causal dependence that obtains between the rate of inflation (π _t ) and the federal funds rate t (r _t ) according to the K+S model if the confounding variable Z stands for inflation expectations. Critics of the K+S model might argue that the model should include a variable Z denoting inflation expectations because Z acts as a potential confounder to π _t and r _t (because both π _t and r _t causally depend on Z) (Figure 9).

Figure 9. Do inflation expectations act as a confounder?

That Z acts as a confounder to π _t and r _t follows, again, from the canonical DSGE model if the new Philipps curve and the monetary policy rule included in that model are interpreted as expressing causal hypotheses. As long as the target system of the K+S model is insensitive to initial conditions, defenders of the K+S model can point out that inflation expectations are negligible: that Z can be neglected because the values of Z remain the same over all iterations, while the values of π _t and r _t can be observed to change. But if the target system of the K+S model is sensitive to initial condition, then the defenders of the K+S model can no longer claim that Z is negligible: then any slight divergences of the values of Z from z can mean that Z acts as a confounder to π _t and r _t , and that the relation of causal dependence that obtains between π _t and r _t according to the K+S model fails to have a counterpart in reality.

Note that I’m not saying that the target system of the K+S model is sensitive to initial conditions, or that causal inference is impossible if the target system is sensitive to initial conditions. I’m saying that causal inference is harder if the target system of the K+S model is sensitive to initial conditions: that it is the harder, the greater the Lyapunov exponent (λ). The greater the Lyapunov exponent (λ), and the greater the number of iterations (t), the less the values of Z may diverge from a constant value z, and the more potential confounders need to be included in the set of preselected variables if causal inference is to be successful: if causal inference is supposed to demonstrate that the relation of causal dependence between X and Y does not only obtain in the model.

I would accordingly describe the second problem of causal inference in agent-based macroeconomics as follows: We first need to decide whether the target system of the K+S model is sensitive to initial conditions. If it is, we need to specify the conditions, under which we can say that the Lyapunov exponent and number of iterations are sufficiently small to allow for successful causal inference. The problem is not unsolvable, but it currently seems impossible to decide whether the target system of the K+S model is sensitive to initial conditions (remember that we need long time series to decide that question, and that these time series are currently unavailable). It also seems that there is no general solution to the problem of specifying the conditions, under which we can say that the Lyapunov exponent and number of iterations are sufficiently small.

6. Top-down Causation and Causal Exclusion

The third problem is well-known to philosophers of mind. It arises when philosophers advance the causal exclusion argument in favour of epiphenomenalism, which is the position that mental events supervene on (or are constituted by) physical events and do not cause any physical events (cf. Kim Reference Kim1989). The causal graph that corresponds to the argument is shown in Figure 10.

Figure 10. Constitution and causation in the causal exclusion argument.

Here, P ₁ and P ₂ are physical events, while M is a mental event, which supervenes on (or is constituted by) P ₁. Thus, the fat arrow denotes that P ₁ constitutes M (or that M supervenes on P ₁). In the original version of the argument, the thin arrows stand for relations of causal sufficiency. And the conclusion of the original version states that M cannot be a cause of P ₂ (that there cannot be any top-down causation from M to P ₂) if P ₁ is a sufficient cause of P ₂ because there would be nothing left for M to contribute to the occurrence of P ₂.

In a more recent variant of the causal exclusion argument, the thin arrows stand for relations of causal dependence in the sense of Woodward’s interventionist definition of causal dependence (cf. section 3 above). In its interventionist version and as expressed by Baumgartner (Reference Baumgartner2010: section III), the causal exclusion argument runs as follows: If P ₂ causally depends on P ₁, and if M supervenes on P ₁, then P ₂ cannot causally depend on M because in order for P ₂ to causally depend on M, there would have to be a possible intervention on M that changes P ₂ while P ₁ is held fixed, and because such an intervention is impossible: it is impossible to change M without changing P ₁, and vice versa (cf. Baumgartner Reference Baumgartner2010: section III).

Woodward (Reference Woodward2015: sections 6–9) responds that in order for P ₂ to causally depend on M, there does not need to be any possible intervention on M that changes P ₂ while P ₁ is held fixed. He in fact agrees that there cannot be such an intervention. He argues that in order to test whether P ₂ causally depends on M, one will have to manipulate an intervention variable that type-level causes both M and P ₁, and he expresses the causal dependence of both M and P ₁ on the intervention variable by a curly bracket (cf. Woodward Reference Woodward2015: 331) (Figure 11).

Figure 11. An intervention variable as common type-level cause.

An ensuing change in P ₂ would not be attributable to the change in M unambiguously. But an ensuing change in P ₂ would at least allow for the possibility of there being a process of top-down causation operating from M to P ₂.

All sides agree that a process of top-down causation operating from M to P ₂ will be impossible if M causally depends on P ₁. If M causally depends on P ₁, P ₁ will act as a confounder and stand in need of control in an experiment carried out to check whether P ₂ causally depends on M. Woodward (Reference Woodward2015: 316–317) argues that M cannot be regarded as causally depending on P ₁ because P ₁ and M fail to satisfy the “independent fixability” assumption: because P ₁ cannot be manipulated without manipulating M, and vice versa. But Gebharter (Reference Gebharter2017: section 3) points out that M causally depends on P ₁ in the sense that in a causal Bayes net, there will be an arrow departing from P ₁ and directed into M (that M will be probabilistically independent of its non-descendants given P ₁).

Stern and Eva (2021: section 3) respond that the arrows in a causal Bayes net should be interpreted as representing “e-parenthood”: X is an e-parent of Y and Y an e-child of X if either (i) Y causally depends on X, or (ii) Y asymmetrically supervenes on X. If the arrows are interpreted as representing “e-parenthood”, Woodward’s conclusion, according to which a process of top-down causation operating from M to P ₂ is at least possible, is re-established. Stern and Eva (2021: section 6) also point out, however, that in the special sciences, there are cases in which sets of variables violate a condition they refer to as “difference maker (DM) sufficiency”. The condition requires essentially that an upper-level variable X screen off a lower-level variable Y from a lower-level variable L that constitutes X: P(Y∣X ∧ L) = P(Y∣X).Footnote ¹⁰

As a well-known example of a set violating DM sufficiency, Stern and Eva quote {TC, D}, where TC and D stand for total cholesterol and heart disease, respectively, and where TC is the sum of high-density cholesterol (HDC) and low-density cholesterol (LDC). {TC, D} violates DM sufficiency because TC does not screen off HDC and LDC from D: it matters a great deal to D whether or not LDC is large because LDC influences D negatively, while HDC influences D positively.

In macroeconomics, there are, regrettably, many cases, in which DM sufficiency is violated. Consider, for instance, the set of variables standing for GDP at t and the demand expectations that firm j forms at t+1: {GDP _t , $$D_{j,t+1}^{e}$$ }. This set violates DM sufficiency because, arguably, GDP _t does not screen off the quantity produced by firm j at t (Q _j,t ) from the demand expectations formed at t+1 ( $$D_{j,t+1}^{e}$$ ): it matters a great deal whether or not Q _j,t is large because Q _j,t is likely to influence demand expectations formed at t+1 more strongly than the quantities that any of the other capital- or consumption-goods producing firms produce in t to constitute GDP _t (Figure 12).

Figure 12. A violation of difference-maker sufficiency in the K+S model.

More macroeconomic examples of sets violating DM sufficiency are easy to find. Consider e.g. the set {π _t , $$P_{m,\,t + 1}^e$$ }, where π _t is inflation in t and $$P_{m,\,t + 1}^e$$ the price that agents expect to pay for commodity m (e.g. real estate) in t+1. This set violates DM sufficiency because π _t does not screen off $$P_{m,\,t + 1}^e$$ from P _m,t , where P _m,t is the price that agents pay for the same commodity in t: it matters a great deal to $$P_{m,\,t + 1}^e$$ whether or not P _m,t is large because P _m,t is likely to influence $$P_{m,\,t + 1}^e$$ more strongly than most of the other prices that form the supervenience base of inflation in t.

When DM sufficiency is violated, manipulations of X will be too “fat-handed” to allow for the conclusion that Y causally depends on X. Manipulations of TC will be too fat-handed to allow for the conclusion that D causally depends on TC. Similarly, manipulations of GDP _t (or π _t ) will be too fat-handed to allow for the conclusion that there is a process of top-down causation operating from GDP _t (or π _t ) to $$D_{j,t+1}^{e}$$ (or $$P_{m,\,t + 1}^e$$ ). For causal search algorithms this means that they will result in directed graphs that fail to contain arrows departing from upper-level variables and directed into lower-level ones. For the LiNGAM that Guerini and Moneta (Reference Guerini and Moneta2017) suggest can be employed to discover the relations of causal dependence obtaining in the target system of the K+S model, this means, in particular, that it will result in a directed graph that fails to contain an arrow departing from GDP _t and directed into $$D_{j,t+1}^{e}$$ .