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I want to perform a regression between 3 variables [x1,x2,x3] that have no trend and no seasonality across their time observations and a variable [Y] that has trend and seasonality. For [Y] I've deseasonalized, removed the residuals, then averaged the Trend component values and grouped it by unique combination of x1,x2,x3 observations. Then I ran the regression. The philosophy behind is that changes in x1, x2 x3 affect the trend (slope+absolute) of the variable [Y]

Is it a correct approach or am I mistreating the time series?

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I would say you are (potentially -most likely) mistreating time series data. Regression methods and the underlying assumptions of no auto-correlation within the series are well developed here http://users.stat.umn.edu/~helwig/notes/mvlr-Notes.pdf . BUT since you have time series data play close attention to this (well-written If I must say so myself) web-gem https://autobox.com/pdfs/regvsbox-old.pdf . Don't try and go back and criticize your teachers or your text , they told you the assumptions but perhaps you didn't hear them.

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  • $\begingroup$ Thank you @IrishStat I'll read the materials. Could it be the case that the Trend component of the time serie is not autocorrelated, hence a usable variable? $\endgroup$
    – nba2020
    Aug 17 '19 at 19:23
  • $\begingroup$ the trend component (1,2,3,...) is definitely autocorrelated. I should have written "no autocorrelation with the user-specified stochastic series" . A final model reflecting three trends might be x1=1,2,3,4,...t x2=0,0,0,0,1,2,3,...t-4 and x3=0,0,0,0,0,0,0,1,2,3..t-6 $\endgroup$
    – IrishStat
    Aug 17 '19 at 21:29
  • $\begingroup$ sorry .. let me rephrase ... I should have written "no autocorrelation with the suitable filtered user-specified stochastic series" . This means that model identification is done by pre-whitening stats.stackexchange.com/questions/221072/… $\endgroup$
    – IrishStat
    Aug 18 '19 at 4:51
  • $\begingroup$ Thanks for your comments, appreciate the effort :) What I haven't understood still is how can I resolve/take into account the fact that Y is autocorrelated. Should I include an independent Y_dummy that takes 1,2,3,4... per sequential time observation next to Y_trend variable? The x variables aren't autocorrelated. $\endgroup$
    – nba2020
    Aug 18 '19 at 11:10
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    $\begingroup$ The probable answer is to incorporate some lag of Y which might aid the "discovery of the effect of the candidate X's AND their lags " AND possible latent deterministic structure which untreated might obfuscate model identification. Only your data knows for sure ! If you post it I will try and help further in my infinite spare time. $\endgroup$
    – IrishStat
    Aug 18 '19 at 12:49

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