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Previous articleNext article FreeCommentRoxana Mihet and Laura VeldkampRoxana MihetNew York University Search for more articles by this author and Laura VeldkampNew York University, NBER, and CEPR Search for more articles by this author New York UniversityNew York University, NBER, and CEPRPDFPDF PLUSFull Text Add to favoritesDownload CitationTrack CitationsPermissionsReprints Share onFacebookTwitterLinked InRedditEmailQR Code SectionsMoreWhat Are the Origins of Economic Fluctuations?Most macroeconomic models exhibit local stability. Their fluctuations come from exogenous shocks that disturb an otherwise stable system. The effects of small shocks dissipate over time, as the model returns to its steady state or to a stable growth path. The reason why this assumption seems plausible is that macroeconomic data shows a stable-growth trajectory in many developed economies. However, another possibility that fits the data is a system that is locally unstable, but globally stable, and which exhibits endogenous oscillations. Limit cycles are one possible outcome in such a system.A (stable) limit cycle is a trajectory that attracts all neighboring trajectories, as shown in Figure 1. A system with a stable limit cycle exhibits self-sustained oscillations, as all the neighboring trajectories spiral toward the limit cycle. Thus, limit cycles can only occur in nonlinear systems.1 Perturbations of the system away from the limit cycle do not impact its global stability. The system returns to the limit cycle, which is an attractor for the system.Fig. 1. Visual representation of a limit cycleSource: Authors’ representation.Note: Independent of the starting point, the system always converges to the stable limit cycle.View Large ImageDownload PowerPointA handful of economic theories can explain why economic interactions give rise to locally unstable feedbacks and oscillations. One example is the effect of trend-following expectations among market speculators, which tends to amplify price asymmetries away from fundamentals in asset markets. In Grandmont (1985), an OLG model without bequests exhibits limit cycles when there is a conflict between the wealth effect and the intertemporal elasticity of substitution effect associated with interest rate movements and sufficiently high concavity. Beaudry, Galizia, and Portier (2015) demonstrate that a DSGE model with demand complementarities related to an accumulable good can generate nonmonotonic optimal policy functions and limit cycles. Externalities in production or money can create convexities and limit cycles as in Farmer (1986) or Benhabib and Farmer (1994). Models with imperfect financial markets and heterogeneously infinitely-lived agents as in Woodford (1986, 1989), and models with three sectors and Cobb-Douglas technology as in Benhabib and Rustichini (1990) can also generate deterministic limit cycles.While economic systems may exhibit local instability, we rarely see examples of global economic instability. The supply of physical goods is bounded by physical matter on earth. With a fixed money supply, prices are also bounded. Labor is bounded by the time in a day. Such limiting forces rule out explosive economic systems. Modeling an economic system as locally unstable, but globally stable, is an appealing way to reconcile these forces.The theoretical possibility of limit cycles is interesting, but lacks much empirical support. Beaudry, Galizia, and Portier set out to fill that gap. The fact that there is a cycle does not reveal whether it is a limit cycle or not. The authors instead estimate time-series models on macroeconomic data to determine whether the estimated stochastic processes have the properties needed for limit cycles to emerge. By examining the properties of several macroeconomic variables, they find some evidence of the existence of limit cycles.The Economy as a Water WheelLimit cycles arise frequently in science. A classic example of this is a machine called a Lorenz water wheel. Imagine a bicycle wheel with cups tied to its rim. Each cup has a small hole in the bottom. Water is poured into a fixed position just to one side of the top of the wheel. When a cup catches some of the water, the wheel begins to spin around its axle. The system is locally unstable because it does not stay still as long as the water continues to flow. The state of the system (position of the wheel) fluctuates, even though the forces acting on it are constant. Fluctuations arise without shocks. Yet, the system is globally stable because the machine rotates around a fixed axle and never spins infinitely fast. If the wheel were pushed to spin faster, friction would eventually slow it back down to its original rate of rotation. The Lorenz wheel is a metaphor for business cycles in this model. The water that sets the wheel in motion is like the model’s data generating process.Two crucial ingredients create cycles in both the wheel and the business-cycle model. These two features are what this paper is testing for. First, there are nonlinear forces. The fact that the cups are tied to the rim of the wheel means that the wheel itself exerts force on the cups, changing their direction as they rotate around the axis. Without this nonlinear force, the cups might move, but they would not cycle. The second key ingredient is history dependence. On the wheel, the cups drain slowly. Thus, the level of water in any cup is an indicator of its past (as is the momentum of the wheel). If that cup has passed under the water source frequently, the cup will be fuller. If not, it will be less full. If the cups had no bottoms, they would have no history dependence. With bottomless cups and no momentum, the cycles would cease.But nonlinearity and history dependence are not sufficient to create limit cycles. If the water is poured directly at the top of the wheel, the wheel might spin either way and switch direction from time to time in an aperiodic fashion. This is an example of chaotic dynamics, a possibility we return to at the end of this discussion. Furthermore, if we were to tie a heavy weight to one point on the wheel, the weight would fall to the bottom and stay there. Rotation would cease. Even though the nonlinear force and history dependence were still present in the system, cycles would not arise. Therefore, finding nonlinearity and history dependence in a dynamic system is a necessary but not a sufficient condition as it only creates the possibility of limit cycles. After finding evidence of these two features in macroeconomic systems, the authors then simulate the estimated stochastic processes to see whether limit cycles do indeed arise.Is the Macroeconomy Locally Unstable?“Physicists like to think that all you have to do is say, these are the conditions, now what happens next?”—Richard Feynman (1965)There are two necessary conditions for the existence of limit cycles: first, limit cycles require a nonlinear system (higher-order terms); second, they require history dependence. Nonlinear systems can give rise to a remarkable set of spatial-temporal phenomena, including periodic or chaotic system dynamics. Contrasting with the much simpler linear systems often used in macroeconomics because of the easiness of their computation, nonlinear systems may appear chaotic, unpredictable, and often even counterintuitive. Beaudry, Galizia, and Portier argue that the local stability of the macroeconomic system should not be evaluated using linear time-series methods, even if the nonlinearities are minor, or away from the steady state. Doing so could result in wrongly classifying the system as locally stable, when in fact the system is endogenously unstable.In their paper, Beaudry, Galizia and Portier carry three different exercises. They first test for nonlinearities, and second for history dependence. Finally, they perform various simulations to assess whether the estimated coefficients induce limit cycles.Testing for Nonlinearities and History DependenceFirst, Beaudry, Galizia and Portier test for nonlinearities and history dependence by comparing the performance of a standard linear autoregressive model to various extended AR models that include nonlinear terms. The hypothesis is that including nonlinear terms causes the system to switch from local stability to local instability. The aim is to look for values on the coefficients of the nonlinear terms that can support limit cycles. Practically, the authors estimate the univariate process below with ordinary least squares and then test whether the coefficients on the nonlinear terms are different from zero. They also compute the largest eigenvalue of each univariate process when linearized around its steady state and check whether it is larger than one, which suggests local instability.The authors allow for different nonlinear forces of the following general form:(1)xt=a0+a1xt−1+a2xt−2+a3Xt+F(xt−1,xt−2,Xt−1)+εtwhere Xt–1 is an accumulation term of lags of discounted infinite lags of xt and F(…) is a multivariate polynomial containing second- and third-order terms in all its arguments, including the accumulation term. The authors posit that this formulation allows more distant lags to play a role without having to estimate too many parameters. More specifically, they estimate a minimal model that includes only a third-order term xt−13 an intermediate model that includes all the third-order terms without any second-order terms, and a full model including all third-and second-order terms.The estimation uses detrended quarterly US data from 1950 to 2014 on labor market variables (log hours worked per capita, nonfarm business hours, the job-finding rate, and the unemployment rate) and goods market variables (output, consumption, durable goods, fixed investment, and capacity utilization). All data is detrended using a high-pass filter that removes fluctuations with periods longer than 20 years.The results are mixed; sometimes they work, other times they do not. Nonlinearity and history dependence appear for most of the labor market variables. There is evidence from three particular time-series specifications, which seem thoughtfully chosen. For other macroeconomic variables, there is limited evidence for limit cycles. Now, whether other models would deliver the same results, or which other models the authors have tried, is unknown. Overall, these estimation results are suggestive of limit cycles.However, testing for nonzero coefficients is a weak test. The paper claims success for limit cycles if the t-statistics on nonlinear or history-dependent terms are sufficiently high. But these statistics are for tests whose null hypothesis is that the coefficients are zero. The objective of the paper is to test for local instability. Even with nonlinear and history-dependent terms, a system can still be locally stable (recall the water wheel with the heavy weight). The null hypothesis that should be tested instead is local stability. This would amount to checking whether the coefficients of a group of nonlinear parameters are jointly not too large and positive.These results do speak more clearly about nonlinearity. Most of the models used in macroeconomics are linear or near-linear, typically locally stable, and converge to a point. Beaudry, Galizia, and Portier reject such linear models for most of the series they examine. Standard DSGE models are usually linearized around the steady state. Standard growth models with utility over consumption and concave technology also do not exhibit limit cycles. Limit cycles are inconsistent with a representative infinitely lived agent, maximizing a concave, time-separable utility function in consumption only. But they need not be inconsistent with optimal growth models where the utility function has a more complicated form, for example, if it includes the capital stock as an additional argument (see Grandmont, 1985).Simulating to Check for Limit CyclesFinally, to quantify how strongly the data supports the presence of limit cycles, the authors use a bootstrap method. By estimating a model on subsets of the data, they produce a set of parameters whose frequency approximates the parameter distribution conditional on the data. Then for each combination of estimated parameters in this bootstrap set, they simulate the stochastic process described by those parameters and check to see if it exhibits cycles. They do this exercise for the three different model specifications. For the labor market variables, between 50% and 78% of the parameter estimates in the bootstrap set generate limit cycles. The interpretation of this finding is that limit cycles are a robust feature of the data.To explore whether their detecting procedure or their detrending might generate spurious limit cycles when the data-generating process is actually locally stable, the authors use Monte Carlo analysis. They construct an artificial series that surely cannot have limit cycles, such as a linear AR(2). They detrend the series, estimate the minimal, intermediate, and full models, and then check for limit cycles. They spuriously detect limit cycles less than 5% of the time. Based on this finding, the authors argue that they are unlikely to detect limit cycles that are not present. However, demonstrating that the procedure does not in correctly detect limit cycles on an AR(2) series, which cannot exhibit a limit cycle by construction, does not imply that the procedure never produces false positives for other types of processes. It does suggest that macroeconomic data is unlikely to be an AR(2).Results for a broader class of macroeconomic variables are mixed. After estimating processes for detrended output, total hours, total consumption, durable goods, investment, and the capacity-utilization rate, and simulating those processes, limit cycles emerge in a few cases. For most variables, limit cycles appear for one or two of the three econometric specifications. For structures, all three models generate cycles. No limit cycles are detected for equipment investment.Cycles or Chaos?“The basic idea of Western science is that you don’t have to take into account the falling of a leaf on some planet in another galaxy when you’re trying to account for the motion of a billiard ball on a pool table on earth. Very small influences can be neglected. There’s a convergence in the way things work, and arbitrarily small influences don’t blow up to have arbitrarily large effects.”—Gleick (2001)An important future test for this research agenda is to determine whether macroeconomic data can rule out chaotic behavior, or distinguish between chaos and limit cycles. Chaos theory generalizes the properties of local instability and global stability of limit cycles. Chaotic systems exhibit oscillations in a neighborhood of an unstable equilibrium (or shift between neighborhoods of different unstable equilibria). Similar to limit cycles, chaotic systems also have an attracting set of orbits. However, in contrast to limit cycles, this attracting set of orbits is aperiodic. In other words, the variable never returns to exactly the same state.This distinction is important because forecasting in a chaotic system is nearly impossible. Forecasting is premised on the idea of local stability: Small differences in states will diminish over time. If differences diminish, then if we see a state today that is similar, but not exactly identical, to an economic state observed in the past, we can infer that the future path of the economy will be similar to the path observed in the past. Thus, the past is a reasonable forecast of the future, even though no state is exactly the same.In contrast, in a chaotic system, the uncertainty in a forecast increases exponentially with elapsed time. Tiny differences in initial conditions can lead to hugely different forecast results. For example, starting from nearly the same initial point, Edward Lorenz, an MIT mathematician and meteorologist, tried to use a computer to simulate the basic dynamics of weather patters. Then he resimulated the same system from what, up to machine accuracy, was the same starting point. He was surprised to find that his forecasts grew farther and farther apart until all resemblance disappeared (figure 2). If every run of a simulation can produce an entirely different dynamic, stemming only from the most minute differences in starting conditions or rounding error, then none are reliable forecasts.Fig. 2. Visual representation of chaotic behavior. Starting from a very similar initial point, the chaotic systems diverge.Source: Gleick (2001).View Large ImageDownload PowerPointFurthermore, inferring the nature of a chaotic system from data is virtually impossible. There is no existing procedure for taking chaotic data patterns and working backward to the underlying mathematical relationship. So, if chaotic patterns appear in macroeconomic data, finding the underlying equation that predicts future prices may not be possible for now.These inference and forecasting problems pose a severe challenge for macroeconomics. If forecasting is hopeless and deducing models or mechanisms is impossible, is macroeconomics useful? The authors do not argue that the economy is chaotic. But much of their evidence of nonlinearity and history dependence supports chaos, as well as limit cycles. Distinguishing the two can be difficult. But the challenges raised by chaotic nonlinear systems are much greater than those raised by their periodic cousins, limit cycles. Because of these important differences, more evidence should eventually be brought to bear on this question, perhaps disciplined by a quantitative structural model.In conclusion Beaudry, Galizia, and Portier study the time-series behavior of some macroeconomic aggregates, primarily labor series, and find some evidence consistent with limit cycles. This is an important topic that taps into one of the most fundamental questions in macroeconomics: Why does the macroeconomy exhibit recurrent fluctuations? The authors bring new evidence to bear on the possibility of limit cycles. Although the evidence for many macroeconomic series is mixed, the paper offers a new approach, and hopefully opens up a new research area that tackles a timeless question.EndnotesFor acknowledgments, sources of research support, and disclosure of the authors’ material financial relationships, if any, please see http://www.nber.org/chapters/c13773.ack.1. 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