Mathematical analysis of chaotic behavior in monetary policy rules

Chaotic behavior which is based on nonlinearity and deterministic process has captured the attention of many mathematical economists in recent years. This behavior can put limits of predictability on the future behavior of a series. Thus, detecting the chaos in economics can help the policy designer to have a better understanding about the impact of monetary policy on real economy. This thesis tries to find the chaotic behavior in an economic dynamical system by performing both theoretical and empirical investigation. For the theoretical part Hopf bifurcation theorem is used. The Existence of bifurcation ensure us the availability of being limited cycles. The results of this part show that there exists the Hopf bifurcation between the parameters of the model. Empirically this thesis used the Brock, Dechert and Scheinkman (BDS) test to detect the chaotic behavior in the system. The BDS is a portmanteau test which can be used against a variety of possible independence including nonlinearity and chaos. Generally, this test is based on the correlation dimension, and the null hypothesis is tested that a time series data came from a data generation process that is independent and identical distribution. Asymptotically, this test statistic has a standard normal distribution under the pure whiteness hypothesis. If we remove the linear dependency and conditional heteroscedasticity from the time series data this test can show whether that data come from a chaotic data generation process. We used the quarterly data of the United State of America and the period is from 1980:1-2010:4. We employed the Kwiatkowski, Phillips, Schmidt and Shin (KPSS) test for checking the unit root and Generalized Method of Moments (GMM) estimator for estimating the coefficients of the system of equations. Autoregressive Moving Average (ARMA) is used for removing the linear relations. The main finding of the BDS test confirms the existence of chaotic behavior in the simulated outcomes (output gap) obtained from the estimated system of equations. The general conclusion drawn from this thesis indicates that the rules with feedback may create the chaotic behavior in the real world, and seems the rules without feedback can be a better choice for conducting monetary policy.

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