# the difference between using an AR(1) term (as in GAMM) versus using PM lag variable (in GAM)

I conducted an experiment to predict particulate matter (PM) level using a GAM. To do so I included the lag1 PM (PM value of day before) as well as few meteorological terms. In my second experiment using GAMM model, I know I need to incorporate the auto-correlation into the model. My question is what is the difference between using an AR(1) term (as in GAMM) versus using PM lag variable (in GAM)? I appreciate your respond.

Think of the error term $$\epsilon_t$$ in a GAMM model with a continuous response (i.e., $$PM$$) as being an umbrella term which captures the effect of all other factors NOT included in your model on the $$PM$$ value recorded at time $$t$$. Usually, we might not even know what these factors are, though there are cases where we know what some of these factors are but we decided not to include them in the model even though we measured them.
If you assume this error term to follow an AR(1) process, you are simply stating that you believe the error term at time t depends on the error term at time t - 1. In other words, you believe that knowing how these other factors affect PM at time $$t-1$$ will allow you to predict how they influence $$PM$$ at time $$t$$.
When you include $$PM_{t-1}$$ in the model, you are assuming that the level of $$PM$$ at time $$t-1$$ influences the level of PM at time $$t$$, which is a different thing altogether.