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2013 | OriginalPaper | Buchkapitel

Consistency of Linear Forecasts in a Nonlinear Stochastic Economy

verfasst von : Cars Hommes, Gerhard Sorger, Florian Wagener

Erschienen in: Global Analysis of Dynamic Models in Economics and Finance

Verlag: Springer Berlin Heidelberg

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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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Vorheriges Kapitel One-Dimensional Discontinuous Piecewise-Linear Maps and the Dynamics of Financial Markets

Nächstes Kapitel A Homoclinic Route to Volatility: Dynamics of Asset Prices Under Autoregressive Forecasting

The model in Grandmont (1985) is deterministic such that REE are actually perfect foresight equilibria.

See, e.g., Box, Jenkins, and Reinsel (1994) for a discussion of these definitions.

Dynamical systems which are uniformly expanding – such as (4) – typically have ‘nice’ invariant probability measures; see, e.g., Lasota and Mackey (1985).

Note that stationarity of the first two moments does not necessarily imply stationarity of the process itself.

See also Hommes and Sorger (1998, footnote 2).

By ‘non-degenerate’ we mean that it has positive variance.

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 \(\bar{{p}}_{t-1}^{+} = [1/(t - 1)]\sum\limits_{i=1}^{t-1}{p}_{i}\).

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.

The case β = 0 must excluded since for β = 0 there is no dynamics in (11). Sorger (1998) presents an example of an OG-model of a more general form than (8) for which a chaotic CEE with β = 0 exists.

Because the model is deterministic, REE are equivalent to perfect foresight equilibria.

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.

See also Medio and Raines (2007) and Gardini, Hommes, Tramontana, and de Vilder (2009) for an extensive discussion and characterization of the forward perfect foresight dynamics in the case of a non-monotonic offer curve.

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.

Note that a steady state SCEE is also an REE.

The reader should have a look at the time series of the learning parameters α_t and β_t in Figs. 6–8 again. In these simulations, in the initial phase of the learning process \(1.8 \leq {\alpha }_{t} \leq 3\), whereas \(-0.6 \leq {\beta }_{t} \leq -0.2\).

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.

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 β^j. 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.

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.

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Titel: Consistency of Linear Forecasts in a Nonlinear Stochastic Economy
verfasst von: Cars Hommes
Gerhard Sorger
Florian Wagener
Verlag: Springer Berlin Heidelberg
Buch: Global Analysis of Dynamic Models in Economics and Finance
Print ISBN: 978-3-642-29502-7

Electronic ISBN: 978-3-642-29503-4

Copyright-Jahr: 2013
DOI: https://doi.org/10.1007/978-3-642-29503-4_10

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