# Running an exponential random graph model (ERGM) with lagged covariates

at the moment I am trying to fit an exponential random graph model (ERGM) in R with the function 'ergm' that comes with the 'statnet' package. Here I have two questions. One is of technical nature and one of pracital nature.

First of all, I am interested whether it is wise to include lagged covariates in an ERGM model. To formalize this question: Assume that I observe da collection of random networks and covariates on node and edge level in time: $\{(Y_t,X_t):t=1,...,T\}$ where $X_t$ represents the covariates and $Y_t$ is a random variable that represents a directed network at time $t$, defined on fixed nodes in form of an adjacency matrix with elements $y_{t,ij}$ that are one if an directed edge from $i$ to $j$ is there, and zero if not. Then define a probabilisitic model by

$\mathbb{P}(Y_t=y_t)=\exp\{\theta'u(y_t,x_t)-\psi(\theta)\}$,

where $y_t$ represents a realization of $Y_t$, $u(y_t,x_t)$ represents sufficient statistics, $\theta$ gives the parameters of interest and $\psi(\theta)$ is just the normalization constant defined on the set of all possible networks that are permutations of $Y$ with fixed node number. Now is my question whether it is a good idea to include lagged statistics $u_i(y_{t-1})$ or $u_i(x_{t-1})$ in $u(y_t,x_t)$? I suspect that lagged endogenous network statistics are a bad idea, but what about lagged covariates on edge or node level? Does such an inclusion lead to a higher probability of degeneracy? Is there a similar problem as multicolinearity in linear models if $x_t\approx x_{t-1}$?

Second , I want to ask whether it is possible to fit a Temporal Exponential Random Graph Models (TERGM) model like in the 'btergm' package with the 'ergm' function implemented in 'statnet'. The reason of my desire to us the function 'ergm' instead of 'btergm' is because the 'statnet' package comes with superior fitting algorithms (Hummels Stepping algorithm).