In spatial statistics and spatial econometrics, very commonly models like the following are employed. Let $Y$ represent the response, $X$ some predictors, $\beta$ the vector of coefficients, $\rho$ the spatial autocorrelaton and $W$ some spatial weight matrix, put together to a linear model with a spatial lag:
$$Y=X\beta + \rho WY+\eta, \text{ }(1)$$ with $\eta \sim N(0,\sigma^2_{\eta}I_{N})$ being the error term. I am searching for a model that allows the spatial correlation (maybe also the fixed effects coefficents) to vary with time, i.e. something like $$Y_t=X_t\beta(t) + \rho(t) W_tY_t+\eta_t,$$ being a possible specification of the model (1) with $Y=(Y_1,...,Y_T)$, $X=(X_1,...,X_T)$, $\eta=(\eta_1,...,\eta_T)$, $W=\text{blockdiag}(W_1,...,W_T)$ and $\rho(t)$ and $\beta(t)$ being some (smooth) functions of time.
I know that this works for models without spatial lags, termed Varying-Coefficients model (Trevor Hastie and Robert Tibshirani (1993)) but I have never seen something alike employed for spatial models and the spatial autocorrelation. Does anyone know a model or a paper that is able to capture time-varying spatial dependence-structures?
