Disequilibrium macroeconometrics

Abstract

1. Introduction

We need to think beyond the standard paradigm, to identify and explain the large, deep, and often persistent downturns, accompanied by high levels of unemployment, that episodically afflict capitalist economies, with devastating effects on the lives of so many (Guzman and Stiglitz, 2020: 2).

The standard macroeconomic paradigm, for example, represented by the dynamic stochastic general equilibrium (DSGE) models, has been criticized by empirical macroeconometricians, like myself, for not describing sufficiently well dominant features of our evolving economies, such as strong interrelatedness, dynamism, and complexity, not to mention pervasive disequilibria and recurring crises. That standard mainstream models were not able foresee the financial crisis in 2008 and the ensuing great recession, the worst in a century, should suffice as an example. It made Colander et al. (2009) criticize an economics profession where

the majority failed to warn about the threatening system crisis and ignored the work of those who did and, as the crisis unfolded, had no choice but to abandon their standard models and to produce hand-waving common-sense remedies.

A decade later a pandemic-induced health crisis has rapidly ignited an economic crisis with exploding unemployment rates, growing inequality, and with yet unknown consequences for financial stability. To make things worse, all this is playing out against the backdrop of a climate crisis that cannot be addressed by “business as usual.” The question is whether mainstream models this time are better equipped to solve all these interrelated problems. If they are not—and all evidence points in that direction¹—then there is a significant risk that they will simply be solving problems in one place while creating new ones elsewhere (Stiglitz, 2018).

Regardless of the explosion of economic research since the financial crisis, DSGE models do not seem to have changed fundamentally, except for adding a few exogenous financial frictions to the model. Stiglitz (2018) argues that “the criticism of the DSGE approach is not that it is a too simple model of the economy, but that it is simple in the wrong way.” Guzman and Stiglitz (2020) elucidates:

Such downturns and crises are not consistent with the standard paradigm of a well-functioning competitive economy, and macroeconomic equilibrium models based on that paradigm, including dynamic stochastic general equilibrium (DSGE) models incorporating wage rigidities and other frictions, are widely recognized (except by the practitioners of those models) to have failed to predict the possibility of those downturns, to explain them, or even to design appropriate policy responses. We are forced to think beyond the standard paradigm, to identify which of the many unrealistic assumptions of those models, are the most crucial.

The list of potential candidates responsible for the failure is extensive. Which of them are most crucial should optimally be argued for in empirical models that are able to capture defining features of economic data. In particular, it seems crucial that the question of empirical evidence should be addressed with a methodology that avoids the pitfall of imposing theoretically motivated restrictions on the data prior to estimation. Otherwise, it is not possible to know which results are true empirical findings and which are due to the assumptions made. See Hoover (2006) and Juselius (2011a) for detailed discussions.

Many economists would argue that the quality and the informational content of macroeconomic data are too low for the results to make sense unless the empirical model is constrained by theory from the outset. No doubt, macroeconomic data are often contaminated with measurement errors, but unless they are systematic and accumulate to a nonstationary process it may not be of great concern for the more important long-run analysis. Whatever the case, theoretically correct measurements do not exist and, hence, cannot be used by politicians and decision makers to react on. The forecasts, plans, and expectations that agents make are based on the observed data. To understand the persistent movements away from equilibrium we better understand these data, however imperfect they are. Guzman and Stiglitz (2020) argues that such an understanding has to be in the context of a dynamic disequilibrium theory:

In explaining deep downturns, we thus need to understand (a) how the market economy generates such large fluctuations in aggregate demand, disproportionate to any exogenous shocks in the “real variables,” large enough that they are associated with high unemployment; and (b) the dynamics of adjustment: why they are such that high levels of unemployment can persist. A dynamic disequilibrium theory with randomness provides insights into the underlying economic processes in ways that the DSGE models cannot.

Motivated by the above, the idea of this paper is, therefore, to propose an alternative empirical methodology, the cointegrated VAR (CVAR) approach, based on scientifically sound principles for how to identify and analyze empirical regularities in nonstationary economic data. Since the CVAR can offer a detailed and accurate description of typical features of economic data, such as strong interdependencies, important feedback dynamics, structural breaks, and pervasive disequilibria, it provides a powerful methodology for checking the empirical adequacy of theoretical assumptions. But the CVAR, as such, is not a substitute for a more structural approach to macroeconomic modeling. Its strength lies in its ability to uncover patterns in the data, some of which are consistent, others untenable, with the underlying assumptions of structural economic models. Hence, the CVAR can often suggest how to improve such models.

At the first sight, the CVAR may seem quite different from the DSGE approach, but a second look shows that it builds on similar principles: the economy is basically a dynamic, stochastic system driven by pushing and pulling forces. Exogenous stochastic trends push the system out of equilibrium and endogenous adjustment dynamics pull the system back to equilibrium. Nevertheless, the two approaches are fundamentally different in other dimensions. For example, a DSGE model is typically taken to the data assuming that the chosen theoretical model is a true representation of the empirical reality, whereas the CVAR model is a general-to-specific approach, starting from a statistical representation of all information in the data and then testing down until no more restrictions can be imposed on the “final” parsimonious model. This implies that underlying theoretical assumptions—such as long-run price neutrality, exogeneity—are imposed only after having been tested, whereas imposed from the outset in the DSGE model. If the underlying theoretical assumptions are empirically correct, the CVAR model is likely to yield similar answers as the DSGE model, if they are not, the CVAR results would show this. The CVAR analysis is, therefore, able to provide a reality check of the theoretical model.

In practice, theoretically motivated restrictions imposed on DSGE models seldom survive a CVAR reality check. For example, Juselius and Franchi (2007) performed a reality check of an RBC theory model in Ireland (2004) which was taken to the data using the DSGE methodology. The CVAR check showed that essentially all RBC assumptions lacked empirical support, that all major conclusions were reversed when the theoretically motivated restrictions were loosened in the data, and that the empirical reality was characterized by pervasive disequilibria rather than stationary equilibria. Ireland’s empirical findings reflected the assumptions made rather than the empirical reality.

The rest of the article is organized around four thematic parts. The first one defines the CVAR model and discusses its potential as an empirical methodology for macroeconomic disequilibrium models. The second part considers the long spells of unemployment—a typical problem of our time—and introduces the Phillips curve with a Phelpsian natural rate as a potentially relevant economic model for describing this phenomenon. This part also discusses how to take the Phillips curve model to the data based on a so-called “theory consistent CVAR scenario.” Furthermore, it demonstrates how to translate all basic assumptions of the economic model into a set of testable hypotheses in the CVAR. The third part deals with the empirical analysis, starting with a presentation of the data, the sample period, the most extreme events in that period and continues with an analysis of the statistical properties of the VAR system. This is followed by an analysis of the exogenous forces that have pushed unemployment out of equilibrium for long periods of time and of the adjustment forces that sluggishly have pulled it back again. The forth part examines a set of a priori assumptions underlying the Phillips curve model that are empirically untenable and suggests how the assumptions can be replaced by new ones. Finally, this part ends with a more general discussion of which theoretical assumptions in mainstream economic models are the most difficult to reconcile with today’s nonstationary economic data.

Throughout the paper, Guzman and Stiglitz (2020) are extensively cited, thereby emphasizing the close correspondence between disequilibrium modeling and the CVAR.

2. The CVAR: an empirical methodology for macroeconomic modeling

The CVAR is inherently consistent with a world where unanticipated shocks accumulate over time to generate stochastic trends that move economic equilibria - the pushing forces - and where deviations from these equilibria are corrected by the dynamics of the adjustment mechanisms - the pulling forces. Thus, the CVAR model has a good chance of nesting a multivariate, path-dependent data-generating process and relevant dynamic macroeconomic theories (Hoover et al., 2008)

While the CVAR is consistent with basic features of economic data, the question of how to link the statistically defined CVAR to a theoretically defined economic model is still an intriguing one. Is it at all possible to confront stylized economic models with the complex macroeconomic reality without compromising high scientific standards? Historically, this question was already extensively debated in the Cowles Commission in particular by the econometric pioneer and Nobel Prize winner, Trygve Haavelmo, in his famous monograph, “The Probability Approach to Economics” (Haavelmo, 1944). One of his most inspiring ideas was to introduce a design of experiment for the case of non-experimental data obtained by “passive” observations—data for which there is no control of how they have been generated. Haavelmo argued that the validity of inference on structural economic parameters is in this case only granted to the extent that the probabilistic assumptions underlying the model are satisfied vis-a-vis the data in question. See also Spanos (2009).

Unfortunately, macroeconomic time-series data are difficult from a probability point of view as each observation corresponds to just one realization from an underlying stochastic distribution. We know the outcome of unemployment in a certain period, but not what it would have been had the economy experienced different shocks. Also, to make things worse, economic time-series data are strongly path dependent. Where we are today depends on where we were yesterday. Hence, the analysis of macroeconomic data is typically associated with problems, such as multicollinearity, spurious correlation and regression, dynamics, simultaneity, autonomy, and identification. While these problems were recognized by Trygve Haavelmo and his mentor Ragnar Frish, they were not able to satisfactorily solve all of them. Juselius (2015) argues that a successful solution would have required access to the theory of nonstationary processes, which was developed many decades later (Phillips,1987; Johansen, 1988, 1996). The problem of valid inference in economic models needed the concept of cointegration to be adequately solved.

2.1 The vector autoregressive model

If P is a normal distribution and the lag structure can be truncated at lag k without loss of information, then the mean of the conditional probabilities corresponds to the mean of the VAR model. In this sense, the VAR model is just a convenient reformulation of the covariances of the original data. If the normality assumption holds, the covariances are time invariant, and if the lag truncation is adequate, then the realization of the $p×1$ vector x_t should not deviate systematically from its expected value given the information at time t. This is similar to the premise behind the rational expectations hypothesis (REH) that “all that could have been known about the structure of the economy was actually known at time t when plans were made.” But contrary to most REH-based models, the VAR model does not assume a priori a theoretical structure to be imposed on the data.

Furthermore, if $Π≠0$ but of reduced rank, then $xt∼I(1)$ and (3) defines a first-order cointegrated process. If additionally $Γ≠0$ and of reduced rank, then $xt∼I(2)$ and (4) defines a second-order multicointegrated process.

Thus, the VAR model represents a set of possible economic models. Its usefulness comes from the possibility to test relevant hypotheses based on scientifically valid principles. Hypotheses not rejected by the tests are sequentially imposed on the model, thereby narrowing down the search for the most empirically relevant model among the possible set. If a theoretical model is empirically relevant, its major features would become increasingly visible during the testing-down process. The “final” model satisfies the probabilistic assumptions and is, therefore, consistent with Haavelmo’s vision of a probability approach to economics. But, since the VAR is a broader description of the economic reality than postulated by the simplified theory, the analysis often produce new results not formulated before. Such hypotheses have to be tested on new data. In this sense, the CVAR approach resembles a progressive research program.

2.2 The CVAR model for I(1) data

The impulse response functions, $C∗(L)(Φ1Dt+εt),$ describe how shocks to the variables permeate through the system until they settle down at the level given by the long-run impact matrix $C=β⊥(α⊥′Γβ⊥)−1α⊥′$⁠.

2.3 The CVAR model for I(2) data

Economic data frequently exhibit too much persistence to be tenable with the I(1) assumption. In contrast to the I(1) model, where deviations from equilibrium are only moderately persistent and, hence, stationary, the I(2) model can account for disequilibria which are indistinguishable from a nonstationary process. Many economists argue that economic data cannot be I(2) as economic variables cannot drift away forever from their equilibrium values in contrast to a nonstationary process. While this is obviously correct, it does not exclude the possibility that over finite samples variables may exhibit a persistency profile that is empirically indistinguishable from a unit root/double unit root process. In this sense, an empirical unit root is not considered a structural parameter but rather a convenient statistical approximation that helps to classify economic variables/relations into more homogeneous groups. For a detailed discussion, see Juselius (2013).

The polynomially cointegrated relations, $β′xt−1+d′Δxt−1,$ capture a setting where the disequilibrium error, $β′xt−1,$ and the differenced relation, $d′Δxt−1,$ both are very persistent (near) I(1) processes that cointegrate to I(0). Such polynomially cointegrated relations can frequently be interpreted in terms of dynamic equilibrium relations. The coefficients α and d represent two levels of equilibrium correction: the α adjustment describing how the acceleration rates, $Δ2xt,$ adjust to the dynamic equilibrium relation and the d adjustment how the growth rates, $Δxt,$ adjust to the long-run disequilibrium, $β′xt.$ The relationships, $ζτ′Δxt−1,$ describe adjustment to stationary medium-run relations, $τ′Δxt−1,$ among differenced variables. Johansen (2006) derives likelihood based tests for hypotheses on β and $τ.$

Self-reinforcing feedback behavior tend to generate persistent disequilibria as a result of equilibrium error increasing (positive feedback) in the medium run and error correcting behavior (negative feedback) in the very long run. The signs and significance of the coefficients, $β,d,α,ζ,$ determine where in the system there is positive versus negative feed-back. The I(2) model is, therefore, informative of how each variable responds to imbalances in the system both in the medium and the long run. See Juselius and Assenmacher (2017) for further details.

Such long-lasting swings in the data, due to various positive feedbacks, can often be interpreted as shifts between multiple, stochastically evolving, equilibria. Examples of variables that have exhibited this kind of behavior is the real exchange rate, the real interest rate, the real stock price as well as the unemployment rate, all of them variables that standard models would assume to be I(0) or at most $I(1).$

In a disequilibrium economics world such persistent behavior is far from puzzling. For example Guzman and Stiglitz (2020) argue that costly bankruptcies, in a world of interdependent firms, financial institutions, and households, entail multiple equilibria as a consequence of large macroeconomic externalities. Another example is financial behavior that frequently drives asset prices so far away from equilibrium that any explanation—except that market participants’ plans/forecasts must have been based on imperfect/incomplete knowledge/beliefs—seems tenable with empirical evidence.

2.4 Economic theory, the empirical reality, and the CVAR

In a theoretical model, it is customary to distinguish from the outset between endogenous and exogenous variables. In an empirical model, the distinction between endogenous and exogenous variables may not be as clear-cut because also exogenous variables tend to exhibit feedback effects from changes in the endogenous variables. But, even though the probability approach entails that the full system is being modeled, a division into endogenous and exogenous variables can nonetheless be established, provided weak—or preferably strong—exogeneity is supported by the data. If exogeneity is rejected, then only the endogenous variables can a priori be expected to be satisfactorily modeled. This is because the choice of information set was motivated by the choice of economic model. The empirical results would in this case often be a mixture of fully and partially specified relationships. See Juselius (2017) for a theoretically motivated example.

In a theoretical model, it is also customary to use the ceteris paribus assumption, keeping certain variables fixed, to focus on those variables of specific interest. But, most ceteris paribus variables are stochastic, rather than constant, and they may also be subject to important feed-back effects. In a probability based approach, such as the CVAR, highly relevant ceteris paribus variables must, therefore, be brought into the empirical analysis by conditioning. And if the ceteris paribus variables are nonstationary, then they might be very influential for the empirical conclusions of the study as will be illustrated in the subsequent sections.

3. The Phillips curve and the long spells of unemployment

Unemployment and inflation are two important policy variables typically considered to be tied together by the Phillip’s curve. It was first established by A. W. Phillips as an empirical regularity describing wage inflation to be negatively associated with unemployment. Samuelson and Solow (1960) respecified the Phillips curve as a relationship between price inflation and unemployment and Friedman (1968) and Phelps (1967) introduced the expectations augmented Phillips curve. That inflation expectations had become increasingly important became obvious during the so-called stagflation period of the seventies, even to the extent that the previously negative relationship between inflation and unemployment became positive. Starting from the mid-1980s, inflation rate kept steadily declining while at the same time unemployment exhibited long persistent swings and the standard Phillip’s curve seemed again to have broken down. Europe, in particular, experienced this kind of inflation-unemployment pattern, labeled unemployment hysteresis.

The structural slumps theory, developed by Edmund Phelps in the early 1990s, was an impressive attempt to explain the persistent fluctuations in the unemployment rates. The idea was to explain how open economies connected by the world real interest rate—set in a global capital market—and the real exchange rate—set in a global customers market for tradables—can be hit by long spells of unemployment. The theoretical implication for the Phillips curve was that the natural rate of unemployment became a function of the real interest rate and/or the real exchange rate.

In Phelps’ work, real interest rates and real exchange rates were assumed stationary whereas empirical evidence shows them to be indistinguishable from a unit root process. Frydman and Goldberg (2007) argues that financial behavior under imperfect knowledge can drive asset prices, such as nominal exchange rates and long-term bond rates, persistently away from their long-run benchmark values. Juselius (2013) therefore argues that the structural slumps theory based on imperfect knowledge expectations would be a more adequate way to explain the long persistent movements in the unemployment data.

Numerous CVAR analyses suggest, however, that inflation expectations by the market have had a rather insignificant effect on the determination of price inflation, in particular after financial deregulation gained momentum in the mid-1980s. Instead, central banks’ inflation expectations may have taken on a more prominent role, notably when the money targeting regime in the first part of the eighties in practice—albeit never formally so—was replaced by inflation targeting. While expected inflation by nature is unobservable, the spread between the federal funds rate and the long bond rate is likely to be a good proxy for the Fed’s inflation expectations.

Finally, Phelps’ structural slumps theory assumes that the unemployment rate and the real interest rate are comoving. The long swings typical of the real interest rate would imply that the unemployment gap, $u−un,$ is less persistent than the unemployment rate so that $Δp$ and $(u−un)$ could be cointegrated even though $Δp$ and u may seem unrelated. This may explain the lack of empirical support for the Phillips curve over the last several decades.

4. Linking theory to evidence: a bridging principle

The question of how to take a theoretical model to the data is a difficult one. This is because a statistically well-specified empirical model and a theoretically well-specified economic model often represent two different entities without any clear link between the two (Johansen and Juselius, 2006; Juselius, 2011b). A so-called theory-consistent CVAR scenario (Juselius, 2006; Juselius and Franchi, 2007) offers a bridging principle by translating basic assumptions about the shock structure and steady-state behavior into testable hypotheses on the pulling and pushing forces of a CVAR model (Johansen and Juselius, 1992). Such a CVAR scenario describes a set of testable empirical regularities one should find in the data provided the basic assumptions of the theoretical model are empirically valid.

A CVAR scenario takes as a starting point the number of explicitly or implicitly assumed exogenous trends, a testable hypothesis in the CVAR. Once the division into exogenous and endogenous forces is determined, the full set of irreducible cointegration relations can be derived (Davidson, 1998). An irreducible cointegration relation consists of exactly the right number of variables needed to make the relation stationary: if cointegration is lost when a variable is omitted from the relation, then the relation was indeed irreducible, otherwise it was not. One may say that irreducible cointegration relations are stationary building blocks that can be linearly combined to produce a long-run structure of interpretable equilibrium relationships. While in a stationary world individual variables are the building blocks, in a nonstationary world irreducible cointegration relations take over this role. In some cases, an irreducible cointegration relation may directly correspond to a plausible economic relation, but as will be shown below, most economic relations are a weighted sum of two—or several—irreducible cointegration relations.

Thus, the first two irreducible cointegration relations, tied together by the α coefficients, reproduce the Phillips curve with a Phelpsian natural rate.

Taken together, the scenario describes the hypothetical results one would expect to find in a CVAR analysis, provided the Phillips curve with a Phelpsian natural rate, the expectations hypothesis of the interest rates and the Fisher parity represent valid empirical regularities in the data. If the model passes this first check, then the theory model would potentially have strong empirical support in the data.

5. The choice of data, sample period, and extreme events

The complexity of the empirical reality often surpasses the theory supposed to explain it and the theory-consistent scenario can be rejected even though the theoretical model is basically sound. Therefore, one should always check (i) whether the chosen sample period is representative for the theory, (ii) whether the theoretically motivated information set is sufficiently broad, and (iii) whether the most important institutional changes have been controlled for. By sidestepping any of these checks, the empirical analysis may—and often does—produce meaningless results.

A first CVAR analysis of the four scenario variables based on the sample 1971:1-2020:1 showed essentially no support for the theory consistent CVAR scenario in (10). Rather than one trend and three cointegration relations, the test suggested three common trends and one cointegration relation. The latter resembled the policy relation (15), rather than the Phillips curve (12). To check the sensitivity of the results to the choice of sample period was, therefore, well-motivated. We considered two likely candidates for the empirical failure: the stagflation period in the seventies and the money targeting regime in the first half of the eighties, both of them may not represent a standard Phillips curve model. The choice of 1984:1-2020:1 as a sample period avoids the econometric problem of mixing these different regimes.

But, while the Phillips curve now became more visible, the empirical evidence was still rather weak. This raised the next question whether any ceteris paribus variables should be added to the empirical model. Inspired by a similar analysis in Juselius and Dimelis (2018), a variable measuring consumer confidence was added to the information set. This significantly improved the evidence for (13), thus illustrating that a relevant, but omitted, ceteris paribus variable can change the conclusions when added to the model.²

Inflation is measured by CPI inflation, rather than by producer price inflation. The motivation for this choice is that other studies of the Phillips curve also use this measure, for example Juselius (2006, Chapter 20) for Danish data, Juselius and Ordóñez (2009) for Spanish data, Juselius and Juselius (2013) for Finnish data, and Juselius and Dimelis (2018) for Greek data. It demonstrates that a CVAR analysis can mimic a designed experiment where similarities and dissimilarities in the results can be related to institutional differences among the economies.

The five variables in the CVAR vector x_t are defined as follows:

• Infl_t = the annual change in log(CPI) multiplied by 100 to make it comparable with annual interest rates in percentage,

• Unr_t = the number of unemployed divided by the labor force in percentage rates,

• $Tb3mt$ = the three months treasury bill rate in annual percentage rates,

• $FfB10t$ = the spread between the federal funds rate, $Fft,$ and the 10 year bond rate, $B10t,$ both in annual percentage rates,

• Conf_t = Consumer confidence index.

All variables have been selected from the FRED database of Federal Reserve Bank St Louis. Graphs of the five variables in levels are given in Figure 1 and of their differences in Figure 2, all of them for the period 1984:1-2020:1. The variables in levels exhibit long persistent swings whereas the variables in differences look stationary in the mean. Some of them—in particular the treasury bill rate—seem to have a nonconstant variance. The graphs in Figure 2 show several spikes (shocks) that potentially might indicate extreme events.

Figure 1.

The data in levels, 1984:1-2020:1.

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Figure 2.

The data in differences, 1984:1-2020:1.

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To satisfy the normality assumption of the CVAR, dummy variables are often needed to control for unanticipated large effects associated with extreme events. Some of these are exogenous and the shocks would be truly unanticipated, others are endogenously induced, such as sudden bursts of expectational bubbles. Guzman and Stiglitz (2020: 21–22) elucidates:

But, occasionally there may be large changes in expectations that in turn trigger changes in the perceptions about the sustainability of the credit relations. Such changes in perceptions often then trigger a reassessment of understandings of the economic and social system, in ways that can lead to or deepen or prolong a crisis. This, for instance, is a typical characteristic in financial crises - sudden changes in the sets of beliefs of market participants lead to the revelation of large individual and aggregate wealth misperceptions.

After the burst, the shocks are no longer unanticipated and should, therefore, be explained by the dynamics of the model. Exceptionally, an extraordinary event—such as the financial crisis—is so fundamental that also its lagged effects are large and unanticipated.

The present sample period has witnessed several periods of long-lasting disequilibria, for example the Savings and Loans Crisis in the beginning of the nineties; the information technology (IT) bubble in the second half of the nineties; the so-called Great Moderation period that ended with the house price bubble in 2007; the financial crisis in 2008. All of them were in varying degree the result of persistent misconceptions in the market.

The econometric challenge is to model the dynamics of long-lasting imbalances combined with occasional unanticipated large shocks. Such large shocks can be identified ex post and controlled for by adequately specified dummies, but are generally not predictable. Since a disequilibrium in one sector is typically off-set by a similar disequilibrium in another sector, such disequilibria can persist for surprisingly long periods. Depending on the degree of their persistence they can either be modeled by the I(1) or by the I(2) cointegration model. However, to spot the growing imbalances well in advance to prevent them from growing further, seems much more important. For this purpose the CVAR can provide useful information about the system’s feed-back dynamics describing where in the system the offsetting adjustment takes place and about the exogenous shocks that have triggered the movements away from equilibrium.

The economic system can also be hit by transitory shocks. By definition they have no permanent effect on the process and are often just “mistakes.” Financial markets frequently drive interest rates up and down due to expectations that ex post turned out to be wrong. With high frequency interest rate data, such “mistakes” are frequent and tend to produce long tails in the error distribution. But, because they are symmetrical (+/-) and because excess kurtosis is less problematic than skewness, they may be left unaccounted for in the model, provided they are not huge (Nielsen, 2004).

Based on the above considerations, the dummy variables entering D_t in (3) are defined as follows:

• $Dp1984.5$ is an impulse dummy (1 at 1984:5, 0 otherwise) accounting for a large drop in the spread between the federal funds rate and the bond rate.

• $Dp2006.9$ is an impulse dummy (1 at 2006:9, 0 otherwise) accounting for a large drop in inflation rates as a result of the bursting house price bubble.

• $Ds2008.10$ is a step dummy (0 until 2008:10, 1 otherwise) restricted to be the cointegration relations. It allows us to test whether the equilibrium mean has changed as a consequence of the financial crisis.

• $Dp2008.10$ is a permanent impulse dummy (1 at 2008:10, 0 otherwise) needed to control for a large drop in inflation rate, a strong increase in unemployment rate, a large drop in the short-long interest rate spread, and a large drop in consumer confidence, triggered by the outbreak of the financial crisis.

• $Dp2008.11$ is an impulse dummy (1 at 2008:11, 0 otherwise) accounting for a large drop in the inflation rate.

• $Dp2009.10$ and $Dp2009.11$ are impulse dummies accounting for two consecutive increases in the inflation rate.

A CVAR(3) with x_t and D_t defined as above was able to explain the variation in $Δxt$ with a trace correlation of 0.39 for the full system. The R² for the individual equations varied from 0.24 (unemployment) to 0.78 (confidence). All residual distributions were symmetrical but the interest rate residuals were moderately long-tailed.

6. General features of the CVAR system

The first important step of the analysis is to determine the division into the pulling and pushing forces by the cointegration rank, r. A priori, we expect to find two cointegration relations and three common trends. This is because the CVAR analysis of the first four variables suggested one cointegrating relation and, hence, three common stochastic trends. Adding the confidence variable to the system will produce one new cointegration relation (unless consumer confidence is long-run excludable) without increasing the number of common stochastic trends (Juselius, 2006, Chapter 19).

Table 1 reports the eigenvalues, $λi,$$i=1,…,5,$ and the trace tests with p-values in brackets.³ The trace test rejects $p−r=5$ (no cointegration, five trends) and $p−r=4$ (one cointegration relation, four trends), but cannot reject our prior $p−r=3$ (two cointegration relations, three trends).

As a safeguard against an incorrect choice of rank and, in particular, against undetected I(2) persistence in the model, Table 1 also reports the five largest characteristic roots of the model for r = 5, 3, and 2. The unrestricted VAR model (column $p−r=0)$ contains three characteristic roots very close to the unit circle (0.98) and two large roots (0.91). Imposing two unit roots leaves a near unit root (0.97) in the model, while three unit roots (column $p−r=3)$ replicates almost exactly the unrestricted VAR results. However the two largest unrestricted roots (0.91) are quite close to the unit circle, suggesting that the empirical model might be borderline I(2).⁴ The I(2) hypothesis (7) was, however, rejected and the subsequent analyses are based on the I(1) model.⁵

Conditional on the choice of $r=2,p−r=3,$Table 2 report tests of long-run exclusion, stationarity, weak exogeneity, and endogeneity. See Juselius (2006) for an exposition of the tests.

The long-run exclusion test asks whether a variables is excludable from the long-run structure without significant loss of information. If the test rejects, then the variable in question is causally related to at least one of the other variables. While none of the five variables are excludable, the step dummy, $Ds2008.10,$ could possibly be left out based on a 6% P-value. It was, however, found to add useful information about the effect of the financial crisis and was, therefore, allowed to remain in the model.

The stationarity test checks whether a variable can be considered stationary around a mean with a mean shift in 2008:10. Formally, the test asks whether a specific variable is a unit vector in the space spanned by the cointegration vectors. The tests show that stationarity was rejected for all the five variables.

The tests of weak exogeneity and endogeneity—formally a test of a zero row in α and a unit vector in $α,$ respectively—show that neither $Tb3m$ nor Conf can be rejected individually as weakly exogenous, and, similarly, neither FfB10 nor Unr as purely endogenous variables. However, the joint exogeneity of $Tb3m$ and Conf is rejected, as is the joint endogeneity of FfB10 and Unrt. Hence, a division into endogenous and exogenous variables, typically assumed in theoretical models, was not empirically acceptable in this system. Such a division is empirically rare due to the empirical complexity typical of the data generating process.

Applying the above tests may imply the difference between empirical success and failure. For example, if one of the assumed endogenous variables is found to be weakly exogenous or long-run excludable, then it is a strong message that the economic model needs to be modified in one way or the other. In the present application, the tests tell us that the main drivers of this system are shocks to the three months treasury bill rate and consumer confidence, that unemployment and the interest rate spread are the main adjusting variables, and that the status of inflation rate is somewhat undetermined. Based on this information, a more detailed study of the pushing and pulling forces will take place.

7. The pushing force

…in moments of high distress in which pre-established perceptions are questioned, and in which everyone is learning not just about how the economy works but also about how the others think that the economy works and about how the inconsistencies that get triggered or revealed after large changes in beliefs will be resolved, uncertainty will grow and discrepancies of beliefs may become more acute, as it becomes less clear for market participants which model best represents the workings of the economy (Guzman and Stiglitz, 2020: 12).

Standard economic (DSGE) models typically assume that the exogenous driving trends are determined outside the economic system. If they are nonstationary, then the endogenous variables will inherit the nonstationarity, if they are stationary, then the endogenous variables will be stationary. But several applications have shown that the nonstationarity of the variables also can be endogenously induced and often associated with self-reinforcing feedback dynamics inside the economic system (Juselius and Assenmacher, 2017; Juselius, 2017; Juselius and Stillwagon, 2018).

Table 3 reports the estimates of the identified stochastic trends and their loadings, $β˜⊥,$ onto the variables.⁶ Inspired by the weak exogeneity test results in Table 2, the first stochastic trend is normalized on inflation, the second on the treasury bill rate, and the third on consumer confidence. All three stochastic trends exhibit small additional effects from shocks to the other variables as the rejection of the joint exogeneity hypotheses would predict.