A one sided formula of the form ~ t, or ~ t | g, specifying a time covariate t and, optionally, a grouping factor g. A covariate for this correlation structure must be integer valued. When a grouping factor is present in formula, the correlation structure is assumed to apply only to observations within the same grouping level; observations with different grouping levels are assumed to be uncorrelated. Defaults to ~ 1, which corresponds to using the order of the observations in the data as a covariate, and no groups.

A non-negative integer specifying the autoregressive (AR) order of the ARMA structure. Default is 0.

A non-negative integer specifying the moving average (MA) order of the ARMA structure. Default is 0.

A non-negative integer specifying the autoregressive response (ARR) order. See 'Details' for differences of AR and ARR effects. Default is 0.

A flag indicating whether ARMA effects should be estimated by means of residual covariance matrices (currently only possible for stationary ARMA effects of order 1). If FALSE (the default) a regression formulation is used that is considerably faster and allows for ARMA effects of order higher than 1 but cannot handle user defined standard errors.