Abstract
The notion of consistent expectations equilibrium is extended to economies that are described by a nonlinear stochastic system. Agents in the model do not know the nonlinear law of motion and use a simple linear forecasting rule to form their expectations. Along a stochastic consistent expectations equilibrium (SCEE), these expectations are correct in a linear statistical sense, i.e., the unconditional mean and autocovariances of the actual (but unknown) nonlinear stochastic process coincide with those of the linear stochastic process on which the agents base their beliefs. In general, the linear forecasts do not coincide with the true conditional expectation, but an SCEE is an ‘approximate rational expectations equilibrium’ in the sense that forecasting errors are unbiased and uncorrelated. Adaptive learning of SCEE is studied in an overlapping generations framework.
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Notes
- 1.
The model in Grandmont (1985) is deterministic such that REE are actually perfect foresight equilibria.
- 2.
See, e.g., Box, Jenkins, and Reinsel (1994) for a discussion of these definitions.
- 3.
- 4.
Note that stationarity of the first two moments does not necessarily imply stationarity of the process itself.
- 5.
See also Hommes and Sorger (1998, footnote 2).
- 6.
By ‘non-degenerate’ we mean that it has positive variance.
- 7.
The OLS-estimate for α is identical to (14). The OLS-estimate for β is slightly different from (15), namely \({\beta }_{t-1} = [\sum\limits_{i=0}^{t-2}({p}_{i} -\bar{ {p}}_{t-1}^{-})({p}_{i+1} -\bar{ {p}}_{t-1}^{+})]/[\sum\limits_{i=0}^{t-2}{({p}_{i} -\bar{ {p}}_{t-1}^{-})}^{2}]\) for t ≥ 3, where \(\bar{{p}}_{t-1}^{-} = [1/(t - 1)]\sum\limits_{i=0}^{t-2}{p}_{i}\) and \(\bar{{p}}_{t-1}^{+} = [1/(t - 1)]\sum\limits_{i=1}^{t-1}{p}_{i}\).
- 8.
Here we mean orbital convergence, that is, the existence of a 2-cycle \(\{{p}_{1}^{{_\ast}},{p}_{2}^{{_\ast}}\}\) such that \(\lim_{t\rightarrow +\infty }{p}_{2t} = {p}_{1}^{{_\ast}}\) and \(\lim_{t\rightarrow +\infty }{p}_{2t+1} = {p}_{2}^{{_\ast}}\), or vice versa. Since \({p}_{1}^{{_\ast}}\neq {p}_{2}^{{_\ast}}\) the sequence \({({p}_{t})}_{t=0}^{+\infty }\) is not convergent in the usual sense.
- 9.
- 10.
Because the model is deterministic, REE are equivalent to perfect foresight equilibria.
- 11.
We have chosen F to be symmetric around 0 but, without loss of generality, we could have chosen F to be symmetric around any fixed point α.
- 12.
- 13.
The name ‘noisy 2-cycle’ SCEE captures the feature that price fluctuations look like a noisy 2-cycle. In Sect. 5.3 we will see that the underlying (deterministic) limiting law of motion of this economy has indeed a stable 2-cycle.
- 14.
Note that a steady state SCEE is also an REE.
- 15.
- 16.
For example, for the initial state \(({p}_{0},{\alpha }_{0},{\beta }_{0}) = (2.25, 2.25,-1)\) as in the simulation of the ‘noisy 2-cycle’ SCEE in Fig. 8, for different realizations of the random shocks η t we have indeed also observed converge to the steady state SCEE of Fig. 6. OLS-learning in this model exhibits the same type of path dependence.
- 17.
See also Jungeilges (2007) for a similar example in the cobweb framework, where under SAC-learning the first order sample autocorrelation coefficient may converge to its correct value β while higher order sample autocorrelation coefficients need not converge to the correct values βj. In a related paper Tuinstra (2003) shows that, under OLS-learning in an OG-model with money growth and inflation, ‘beliefs equilibria’ may arise where the first order autocorrelation coefficient converges while prices fluctuate on a quasi-periodic or chaotic attractor. A beliefs equilibrium is in fact a first order CEE, where agents have fitted the correct regression line to a quasi-periodic or chaotic attractor.
- 18.
The first order autocorrelation coefficient is significant, but recall that first order autocorrelation can not be exploited, since agents have to make a 2-period ahead forecast.
- 19.
The scatter plot technique works very well for 1-dimensional systems but becomes less informative for higher-dimensional systems. Brock, Hsieh, and LeBaron (1991), Brock and Dechert (1991) and Barnett et al. (1998) contain extensive discussions of the sensitivity to increasing dimension and the sensitivity to noise of nonlinear time series embedding methods.
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Acknowledgements
Earlier versions of this paper have been presented at the IFAC symposium Computational Economics, Finance and Engineering, Barcelona, July 6–8, 2000, the CeNDEF workshops on Economic Dynamics, Amsterdam, January 4–6, 2001 and Leiden, June 17–28, 2002, the 8th Viennese workshop on Optimal Control, Dynamic Games and Nonlinear Dynamics, Vienna, May 14–16, 2003 as well as various department seminars. Stimulating discussions with participants are gratefully acknowledged. In particular, we would like to thank Buz Brock, Jim Bullard, George Evans, Jean-Michel Grandmont and Seppo Honkapohja for stimulating discussions. This research has been supported by the Netherlands Organization for Scientific Research (NWO).
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Hommes, C., Sorger, G., Wagener, F. (2013). Consistency of Linear Forecasts in a Nonlinear Stochastic Economy. In: Bischi, G., Chiarella, C., Sushko, I. (eds) Global Analysis of Dynamic Models in Economics and Finance. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29503-4_10
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