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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.

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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.

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    $\begingroup$ Thank you Isabella for explanation. One more question: If my time series of PM is not stationary do I need to make it stationary before using GAMM with AR(1)? The acf and pacf plot of the residuals show non-stationary if I don't use differenced PM. $\endgroup$ – Saraz Aug 7 at 2:27
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    $\begingroup$ If you look at a plot of PM versus time, as well as plots of your meteorological predictors versus time, you will likely notice that all of these plots exhibit temporal patterns (e.g., temporal cycles). In that case, you should include a smooth term of time in your model, along with (linear or smooth) terms for the meteorological variables. You can then plot the residuals from this model against lag-1 PM to see if there's evidence of a systematic trend in this plot. If there is, include lag-1 PM in your model and then examine the new residuals for any evidence of autocorrelation. $\endgroup$ – Isabella Ghement Aug 7 at 3:07
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    $\begingroup$ thank you Isabella for great help. $\endgroup$ – Saraz Aug 7 at 10:36

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