Suppose that my main purpose is to model (using GLM e.g.) an annual count data by using two predictors one of which is mean annual water level measurement which, in itself, is auto-correlated (i.e. one year's measurement is affected by previous year's measurement). What should I do then? I know what to do when my response variable violates the assumption of independence but what about predictor variables? Should I do something?
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Nothing special . It might suggest that an ARMAX model aka Transfer Function Model might include a lag of the X. Note that when you pre-whiten the two observed series ala https://web.archive.org/web/20160216193539/https://onlinecourses.science.psu.edu/stat510/node/75 your pw filter for X and Y would have an ARMA structure of at least (1,0,0)(0,0,0) .
https://autobox.com/cms/index.php/afs-university/intro-to-forecasting/doc_download/18-regression-vs-box-jenkins/ might also provide some oversight.
Distributional assumptions are all about the residuals from a model NOT the distribution of any of the X's .
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$\begingroup$ Thank you IrishStat, Can I detect this with autocorrelation function (acf) in R? $\endgroup$– KO 88Commented Nov 21, 2018 at 19:40
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1$\begingroup$ if the data is free of pulses and level shifts auto.arima can be sometimes useful. $\endgroup$ Commented Nov 21, 2018 at 21:00
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$\begingroup$ Thank you so much for taking time to write a reply, IrishStat. Should I still use auto.arima function to determine the right ARIMA model even when the ACF plot of the model (built by using the "suspected" autoregressive predictor) shows no significant autocorrelation? So my question is a general one: if ACF plot is all ok, then does this mean I have no problem of autocorrelation/autoregression with my response and predictors? $\endgroup$– KO 88Commented Nov 21, 2018 at 21:15
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1$\begingroup$ If and only if 1) the acf and pacf has 0 significant correlations AND 2) there are no pulses, seasonal pulses,level shifts, local time trends AND 3) there is a constant error process free THEN thw pw model is (0,0,0)(0,0,0). $\endgroup$ Commented Nov 22, 2018 at 0:07
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$\begingroup$ Thank you so much for your answer, Sir. I have one last question. A critical one. So should I check all these ACF, PACF and seasonality of all my predictors + response variables before I build my model or should I check the autocorrelation of the residuals of the model as it does in Zuur et al. 2009? $\endgroup$– KO 88Commented Nov 22, 2018 at 7:57