Making predictions with a distributed lag model
So the short answer is yes, you can generate some predictions based on your model. Whether it will generate good predictions is another question. Basically you are only using the spatial variation to estimate your model and no time variation. So your predictions would most likely be very poor unless the relationship is very stable over time. There's also no way of knowing which lagged values of the explanatory variable would be useful in prediction, unless there is some spatial structure that you can use (i.e. some regions are leading indicators for other regions)