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Let $X_t=1+3t+0.5X_{t-1}+ \epsilon_t$ be a trend-stationary model, where $\epsilon$ is a white noise, which has zero expected value and standard deviation.

Which line is the time series fluctuating around? And why?

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    $\begingroup$ Is this a homework question? If so, please add the self-study tag and describe what you’ve tried so far. $\endgroup$ – The Laconic Dec 31 '18 at 13:09
  • $\begingroup$ I added the tag. I don't know how to start or approach the problem. $\endgroup$ – nyaki Dec 31 '18 at 13:19
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    $\begingroup$ But you didn’t get this question out of the blue; you must have some existing knowledge about time series etc. So where are you stuck specifically? Make sure you understand what trend-stationary means. Then determine what “fluctuate around” means specifically, in mathematical terms. $\endgroup$ – The Laconic Dec 31 '18 at 13:31
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    $\begingroup$ What would your answer be if all the $\epsilon_t$ were zero? That might provide a good guess. BTW, that's exactly the case with "zero ... standard deviation" as stated in your question, but I suspect you might have omitted a crucial word, such as "constant standard deviation." $\endgroup$ – whuber Dec 31 '18 at 16:41
  • $\begingroup$ I gave x(0) a value and calculated the first 20. From this, I got -4+6t. Does this might be a good approach? $\endgroup$ – nyaki Dec 31 '18 at 18:25

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