Abstract
Linear regression with distributed-lags is a consolidated methodology in time series analysis to assess the impact of several explanatory variables on an outcome that may persist over several periods.Finite polynomial distributed-lags have a long tradition due to a good flexibility accompanied by the advantage of a linear representation, which allows parameter estimation through Ordinary Least Squares (OLS).However, they require to specify polynomial degree and lag length, and entail the loss of some initial observations.Gamma distributed-lags overcome these problems and represents a good compromise between flexibility and number of parameters, however they have not a linear representation in the parameters and currently available estimation methods, like OLS-based grid search and non-linear least squares, are unsatisfactory in the case of multiple explanatory variables.For these reasons, the Gamma lag distribution has not been able to replace finite polynomial lags in applied time series analysis, and it has been mostly employed in the case of a single explanatory variable.In this paper, we propose a hill climbing algorithm for maximum likelihood estimation of multiple linear regression with Gamma distributed-lags.The proposed algorithm is applied to assess the dynamic relationship between Bitcoin's price and three composite indices of the US stock market.
Highlights
Linear regression with distributed-lags is a consolidated methodology in time series analysis to assess the impact of several explanatory variables on an outcome that may persist over several periods
Polynomial distributed-lags (Almon 1965; Andrews and Fair 1992) have a long tradition in applied time series analysis due to a good flexibility accompanied by the advantage of a linear representation, which allows the regression model to be estimated through Ordinary Least Squares (OLS)
Despite its desirable statistical properties, the geometric lag distribution has been criticized to be too restrictive for several applications, mainly because the weights do not increase towards a peak before decreasing to zero
Summary
Linear regression with distributed-lags is a consolidated methodology in time series analysis to assess the impact of several explanatory variables on an outcome that may persist over several periods. From the analysis of the literature it is apparent that, since its first proposal, the Gamma lag distribution has found relatively little application compared to the finite polynomial one It has been employed mostly in the case of a single explanatory variable, with a predominant use of OLS-GS compared to NLS estimation and Markov Chain Monte Carlo (MCMC) simulation. It is true that even MLE of linear regression with finite polynomial distributed-lags requires to compare several different models with an exponential time complexity in order to select degree and lag length, several shortcuts are in use to effectively reduce the number of comparisons, like assuming that degree and lag length are the same for all the explanatory variables.
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