Application of cointegrated state-space models to financial and economic data

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Al-qurashi, Miaad Hamad
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University College Cork
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In the dynamic stochastic modeling of financial and economic time series, the concept of cointegration plays an important role. It refers to the existence of long-term equilibrium relations between variables in a dynamic environment. The cointegration theory posits that, in a non-stationary environment, the long-term equilibrium relations will show up as stationary relations between certain variables. In the context of linear models, this translates into the existence of so-called cointegrating linear relations and corresponding cointegrating vectors. The error correction model was introduced by Engle and Granger(1987) for estimating models exhibiting cointegration. However, it was not until Johansen(1988, 1991) that a rigorous methodology was proposed for estimation and analysis of cointegrated models in a multivariate dynamic setting. The iconic research by Johansen sets out the Vector Error Correction Model (VECM) approach to cointegration, by applying maximum likelihood estimation. The VECM is based on the Vector Autoregressive (VAR) model. The VAR model has several shortcomings for cointegration studies. Several authors have explored estimation of cointegration using VARMA and state-space models, e.g. Ribarits and Hanzon (2014b,a), Yap and Reinsel (1995), Lütkepohl and Claessen (1997), Poskitt (1994, 2006), Bauer and Wagner (2002), Kascha and Trenkler (2011), among others. This research study focuses on the discrete-time state-space model as well as on a continuous-discrete time state-space model. This research study proposes a parameterization and an estimation method that translates the cointegration property into a low-rank constraint on a resulting likelihood optimization. First, the study partially optimizes the likelihood function. The resulting criterion is a function of eigenvalues of a matrix due to the low-rank constraint. This raises the challenge of calculating the derivatives of the parameterized matrix eigenvalues. The research study addresses this problem by applying the envelope theorem. The study applies a gradient method to optimize the remaining model parameters using a new parametrization. This parameterization has bounded and numerically stable parameters. The method is illustrated by some simulated examples and examples involving stock price index data, oil price data, and more. Comparisons are made against a classical VECM, and in many cases, the state-space model yields better results.
Cointegrated state-space models
Al-qurashi, M. H. 2021. Application of cointegrated state-space models to financial and economic data. PhD Thesis, University College Cork.