Alessandra Luati’s contribution to the Discussion of ‘New tools for network time series with an application to COVID-19 hospitalisations’ by Nason et al.

I would like to congratulate the authors for a contribution rich in original ideas, that may pave the way for further advances in the analysis of network time series.In this short comment, I would like to focus on the novel concept of network autocorrelation function, that the authors introduce as a generalised autocorrelation function depending on two parameters, the time-lag h and the r-stage neighbour.The aim is to include information on r-stage dependence while assessing the temporal dependence among random variables.Accordingly, the network autocorrelation function is defined as the ratio of a weighted distance between lagged vectors of variables, where the weighting matrix is a function of each r-stage adjacency matrix and weights assigned to past values of neighbouring nodes, and its absolute maximum.My own research has centred on the development of generalised measures of variance, autocovariance, and autocorrelation, each depending on two parameters.In a univariate setting, the generalised autocovariance function introduced in Proietti and Luati (2015) and its zero-lag related case, the variance profile by Luati et al. (2012), is defined as inverse Fourier transforms of a Box-Cox transformation of the spectral density function of a stationary process.This class of functions depends on the lag h and on a real-valued power parameter p.The underlying rationale, which has a well established tradition in statistics and time series analysis (Tukey, 1957;Box and Cox, 1964), is that applying power transformations to the spectral density function can highlight distinct features of the process.In the paper, the authors develop a generalised network autocorrelation function, and a corresponding partial network autocorrelation function, that is specific to GNAR models, and use it for order selection among different GNAR specifications.In the empirical 1