A hill climbing algorithm for maximum likelihood estimation of the Gamma distributed-lag model with multiple explanatory variables

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.