Nonlinear, Non-Gaussian, and Non-stationary State Space Models and Applications to Economic and Financial Time Series

Financial markets and the economy are changing rapidly. On financial markets, many <br />financial time series exhibit changes of volatility (variance) over time. Moreover, many <br />financial time series are well known to have non-Gaussian heavy-tailed distributions. <br />These facts indicate that a nonlinear non-Gaussian time series analysis is needed. <br />Regarding the economy, as one example, the Japanese economy has the experience of <br />the "bubble economy" in the late 1980s. After bursting of the "bubble economy", the <br />economy entered a decade o,f economic stagnation, which is often called "the lost <br />decade". These facts indicate that conventional linear regression based on ordinary <br />least squares might be ineffective to analyze a non-stationary economy because the <br />coefficients of linear regression are fixed. This paper shows several statistical <br />approaches based on nonlinear non-Gaussian state space modeling and time-varying <br />coefficient autoregressive modeling. These approaches are novel studies of financial <br />markets and the economy. <br />　In chapter 1, the Monte Carlo filter is introduced. It is a minimal introduction to <br />nonlinear non-Gaussian state-space modeling. <br />　In chapter 2, we propose a method to seek initial distributions of parameters for a <br />self-organizing state space model proposed by Kitagawa]. Our method is based on the <br />simplex Nelder-Mead algorithm for solving nonlinear and discontinuous optimization <br />problems. We show the effectiveness of our method by applying it to a linear Gaussian <br />model, a linear non-Gaussian Model, a nonlinear Gaussian model, and a stochastic <br />volatility model. <br />　In chapter 3, we propose a smoothing algorithm based on the Monte Carlo filter and <br />the inverse function of a system equation (an inverse system function). Our method is <br />applicable to any nonlinear non-Gaussian state space model if an inverse system <br />equation is given analytically. Moreover, we propose a filter initialization algorithm <br />based on a smoothing distribution obtained by our smoothing algorithm and an <br />inverse system equation. <br />　In chapter 4, we illustrate the effectiveness of our approach by applying it to <br />stochastic volatility models and stochastic volatility models with heavy-tailed <br />distributions for the daily return of the Yen/Dollar exchange rate. <br />　In chapter 5, we propose a method that estimates a time-varying linear system <br />equation based on time-varying coefficients' vector autoregressive modeling <br />(time-varying VAR), and which controls the system. In our framework, an optimal <br />feedback is determined using linear quadratic dynamic programming in each period.<br />The coeffients of time-varying VAR are assumed to change gradually (this <br />assumption is widely known as smoothness priors of the Bayesian procedure). The <br />coefficients are estimated using the Kalman filter. In our empirical analyses, we show <br />the effectiveness of our approach by applying it to monetary policy, in particular, the <br />inflation targeting of the United Kingdom and the nominal growth rate targeting of <br />Japan. Furthermore, we emphasize that monetary policy must be forecast-based <br />because transmission lags pertain from monetary policy to the economy. Our approach <br />is convenient and effective for central bank practitioners when they are unaware of <br />the true model of the economy. Additionally, we find that the coefficients of <br />time-varying VAR change in response to changes of monetary policy. <br />　In chapter 6, we estimate the β of a single factor model that is ofben used by <br />financial practitioners. In this chapter, we assume that β changes "gradually" over <br />time; this assumption is identical to that in chapter 5. Using our approach, we can <br />estimate β, even if it is time varying. We apply our approach to the Japanese Stock <br />Markets and show its effectiveness. Although we adopt a very restrictive method (we <br />assume smoothness priors and use the Kalman fiker, which is based on linear state <br />space modeling and the Gaussian distribution), we can obtain good estimates of β.

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