I have a panel data structure. I want to compare the forecast of simple panel regression with individual fixed effects with the same individual fixed effects Spatial Autoregressive Model (SAR) with only spatial autoregressive of the dependent Y variable. I use plm package for panel regression and splm package for spatial ones. I am new to spatial econometrics so I don't fully understand the math beyond spatial regression. The biggest problem I face so far is that predict method in either stats, plm and spmod is not designed for spatial kinds of models. The only spatial model I can't use for forecasting is the Spatial Lag Model (SLX) since it is estimated by the simple plm method with some spatial lag of the independent variable\s.

As an example, I use US States Production from plm package.

library(tidyverse)
library(plm)
library(splm)

data(Produc, package = "plm")
data(usaww)

formula <- log(gsp) ~ log(pcap) + log(pc) + log(emp) + unemp

# panel regression
plm_reg <- plm::plm(formula = formula, data = filter(Produc, year != 1986), model = "within", effect = "individual", index = c("state", "year"))
summary(plm_reg)

# SAR
splm_reg <- splm::spml(formula = formula, data = filter(Produc, year != 1986), listw = spdep::mat2listw(usaww), model = "within", effect = "individual", spatial.error = "none", lag = TRUE, index = c("state", "year"))
summary(splm_reg)

# panel forecast
plm_pred = plm:::predict.plm(object = plm_reg, newdata = pdata.frame(Produc, index = c("state", "year")))
View(plm_pred)

# SAR forecast
splm_pred = predict(object = splm_reg, pdata.frame(Produc, index = c("state", "year"))) # error 

The second issue I faced is that the right-hand side of the equation contains a dependent variable, so if I want to predict for t+1 period I need to have Y variable, which is pointless since I want to predict the value of Y variable in the period t+1. The answer to the question Spatial econometrics -- computing residuals suggests, that I can reduce ρWy by (I - ρW)^-1 after some algebraic manipulations: $$ y=ρWy+xb $$ to $$ y=(I−ρW)^{−1} (xb+e) $$

Does it mean, that I can use plm with (I - ρW)^-1 where ρ is a Moran's I (for instance) to estimate the SAR model?

Anyway, I am looking for suggestions on performing a forecast with spatial panel data in R or Python. I would appreciate for any help.

I have a panel data structure. I want to compare the forecast of simple panel regression with individual fixed effects with the same individual fixed effects Spatial Autoregressive Model (SAR) with only spatial autoregressive of the dependent Y variable. I use plm package for panel regression and splm package for spatial ones. I am new to spatial econometrics so I don't fully understand the math beyond spatial regression. The biggest problem I face so far is that predict method in either stats, plm and spmod is not designed for spatial kinds of models. The only spatial model I can't use for forecasting is the Spatial Lag Model (SLX) since it is estimated by the simple plm method with some spatial lag of the independent variable\s.

As an example, I use US States Production from plm package.

library(tidyverse)
library(plm)
library(splm)

data(Produc, package = "plm")
data(usaww)

formula <- log(gsp) ~ log(pcap) + log(pc) + log(emp) + unemp

# panel regression
plm_reg <- plm::plm(formula = formula, data = filter(Produc, year != 1986), model = "within", effect = "individual", index = c("state", "year"))
summary(plm_reg)

# SAR
splm_reg <- splm::spml(formula = formula, data = filter(Produc, year != 1986), listw = spdep::mat2listw(usaww), model = "within", effect = "individual", spatial.error = "none", lag = TRUE, index = c("state", "year"))
summary(splm_reg)

# panel forecast
plm_pred = plm:::predict.plm(object = plm_reg, newdata = pdata.frame(Produc, index = c("state", "year")))
View(plm_pred)

# SAR forecast
splm_pred = predict(object = splm_reg, pdata.frame(Produc, index = c("state", "year"))) # error 

The second issue I faced is that the right-hand side of the equation contains a dependent variable, so if I want to predict for t+1 period I need to have Y variable, which is pointless since I want to predict the value of Y variable in the period t+1. The answer to the question Spatial econometrics -- computing residuals suggests, that I can reduce ρWy to (I - ρW)^-1 after some algebraic manipulations: $$ y=ρWy+xb $$ to $$ y=(I−ρW)^{−1} (xb+e) $$

Does it mean, that I can use plm with (I - ρW)^-1 where ρ is a Moran's I (for instance) to estimate the SAR model?

Anyway, I am looking for suggestions on performing a forecast with spatial panel data in R or Python. I would appreciate for any help.