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I'm new to time-series analysis and am currently having some issues understanding which model would be appropriate for what I am looking at. I tried looking at other questions about time-series and regression equivalents or hierarchical forecasting. I also looked at questions that addressed multiple predictor variables here and here. The question that is most closely related to mine is this one but I'm still unsure if it's the appropriate one.

To start, I'm not interested in doing any future forecasting (hence the hesitation I had in looking at other questions). Rather, I'm interested in looking at how multiple predictor/independent variables relate to the outcome/dependent variable.

For instance: I have the outcome variable: VAR_O and I have three predictor variables where VAR_C is the control variable and VAR_P1 and VAR_P2 are the predictor variables of interest.

In a cross-sectional analysis, I would use a hierarchical regression where block 1 consists of VAR_C and VAR_P1 to 'predict' VAR_O. The second block would consist of VAR_C, VAR_P1, and VAR_P2 and I would examine the change in R sq that resulted from the addition of VAR_P2.

However, suppose that my data is longitudinal (with the unit of analysis being years or months); since the observations/cases are not independent, my understanding is that I can't use regression and that I should use time-series analysis.

So my questions are:

1) What would be the appropriate time-series analysis to examine whether changes in VAR_P1 and VAR_P2 across time predict changes in VAR_O, while controlling for VAR_C? (in other words, just doing the same as a cross-sectional hierarchical regression but with longitudinal data?)

2) Also, what if there are seasonal trends (in the case of months) 3) or there aren't (in the case of years)?

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