Shrinkage estimation in ARMA-GARCH regression models with an application in Bitcoin returns

Abstract

We focus on the extensions of autoregressive conditional heteroscedastic (ARCH) models and the generalised autoregressive conditional heteroscedastic (GARCH) models applied to financial data. Volatility is observed in financial time series as a response to information or news, which in most cases is unknown beforehand. Although, in certain situations, the timing of information provided may not be a surprise (e.g. announcements of mergers or initial public offerings (IPOs), etc.), giving rise to some aspects of volatility being predictable. Even though volatility is a latent measure in that it is not directly observable but given ample information, it can be estimated. With the uncertainty of risk on financial assets, it would be an inadequate assumption that a constant variance exists over a given time period which is assumed when using ordinary least squares estimation. In the past, linear regression models were used to predict relationships between macro-economic variables but when heteroscedasticity is present, one might still obtain unbiased regression parameter estimates with too low standard errors, which will influence the true sense of precision. The ARMA-GARCH regression model is one of many extensions of the GARCH process with respect to the conditional mean. This dynamic model allows for both the conditional mean and conditional variance to be modelled by the ARMA process and the GARCH process respectively. More specifically, in this mini-dissertation, we develop shrinkage estimation techniques for the parameter vector of the linear regression model with ARMA-GARCH errors. For the purpose of shrinkage estimation, we will be assuming that some linear restriction hold on the regression parameter space. From a practical point of view, specifying a set of logical restrictions plays an important role in economic and financial modelling. We conducted an extensive Monte Carlo simulation study to assess the relative performance of the proposed estimation techniques compared to the existing likelihood-based estimators. The application of our research is considered in the estimation and modelling of Bitcoin returns and testing the significance of the interest in the topic of cryptocurrencies as well as the impact of which traditional financial markets may have on Bitcoin and the cryptocurrency market. Keywords: ARMA-GARCH regression, Bitcoin return, maximum likelihood estimation, Preliminary test estimator, Shrinkage estimator.