Cointegration models with non Gaussian GARCH innovations
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Cointegration models with non Gaussian GARCH innovations Nimitha John1
· Balakrishna Narayana1
Received: 10 January 2017 / Accepted: 11 November 2017 © Sapienza Università di Roma 2017
Abstract This paper presents the estimation procedures for a bivariate cointegration model when the errors are generated by a constant conditional correlation model. In particular, the method of maximum likelihood is discussed when the errors follow Generalised Autoregressive Conditional Hetroskedastic (GARCH) models with Gaussian and some non Gaussian innovations. The method of estimation is illustrated using simulated observations. Data analysis is provided to highlight the applications of the proposed models. Keywords Cointegration · Fisher scoring algorithm · Generalised autoregressive conditional heterosedasticity · Volatility Models
1 Introduction Despite the extensive literature on cointegration and heteroskedastic models, little attention is given to the issue of testing cointegration in the presence of hetroskedastic errors. The available works are mostly based on simulation techniques and hence no formal theories have been developed, see for instance [8] and [9]. Perhaps, the work by [10] was the first one to perceive the issues of cointegration and heteroskedasticity together. To our knowledge, there is no systematic studies on cointegration under hetroskedasticity when the innovations are generated from non Gaussian distributions. So, the main objective of the present study is to explore the possibility of modelling cointegrating time series when the errors are generated by non Gaussian Generalised Autoregressive Conditional Heterosedasticity (GARCH) models. We propose the modelling of two cointegrating time series with GARCH errors, based on the heteroskedastic model proposed by [3]. Since most of the empirical distributions of
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Nimitha John [email protected] Balakrishna Narayana [email protected]
1
Department of Statisics, Cochin University of Science and Technology, Cochin 682022, India
123
N. John, B. Narayana
financial time series show deviations from normality, a joint modelling of bivariate cointegration in the presence of heteroskedasticity using non Gaussian innovations is also called for. The model applicability is then discussed with the modelling of several price series. The rest of the paper is organised as follows. In Sect. 2, we introduce the cointegrating GARCH model. Sect. 3 deals with the inference procedures of the cointegrating GARCH model under Gaussian and non Gaussian set up. A simulation study is presented in Sect. 4. Sect. 5 illustrate the application of the proposed model by analysing certain commodity price series. In Sect. 6, we summarise the conclusions of the study.
2 Cointegrating Models with GARCH innovations Following [7], we use the Phillip’s triangular representation for the purpose of defining cointegration model with GARCH innovations. Suppose that x t = (x1t x2t ) is a vector of non stationary series with order of integration one, and let the cointegrating vecto
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